[translate] matrices
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\subsubsection{Operations with vectors}
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\subsubsection{Operations with vectors}
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Mit Vektoren kann sehr einfach gerechnet werden. Listing
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Similarly to the scalar variables discussed above we can work with
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\ref{vectorscalarlisting} zeigt die Verrechnung von Vektoren mit Skalaren
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vectors and do calculations. Listing~\ref{vectorscalarlisting} shows
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mit den Operatoren \code[Operator!arithmetischer!1add@+]{+},
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how vectors and scalars can be combined with the operators \code[Operator!arithmetic!1add@+]{+},
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\code[Operator!arithmetischer!2sub@-]{-},
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\code[Operator!arithmetic!2sub@-]{-},
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\code[Operator!arithmetischer!3mul@*]{*},
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\code[Operator!arithmetic!3mul@*]{*},
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\code[Operator!arithmetischer!4div@/]{/}
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\code[Operator!arithmetic!4div@/]{/}
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\code[Operator!arithmetischer!5powe@.\^{}]{.\^}.
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\code[Operator!arithmetic!5powe@.\^{}]{.\^}.
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\begin{lstlisting}[caption={Rechnen mit Vektoren und Skalaren.},label=vectorscalarlisting]
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\begin{lstlisting}[caption={Cancluating with vectors and scalars.},label=vectorscalarlisting]
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>> a = (0:2:8)
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>> a = (0:2:8)
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a =
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a =
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0 2 4 6 8
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0 2 4 6 8
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>> a + 5 % Addition von einem Skalar
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>> a + 5 % adding a scalar
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ans =
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ans =
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5 7 9 11 13
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5 7 9 11 13
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>> a - 5 % Subtraktion von einem Skalar
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>> a - 5 % subtracting a scalar
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ans =
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ans =
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-5 -3 -1 1 3
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-5 -3 -1 1 3
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>> a * 2 % Multiplikation mit einem Skalar
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>> a * 2 % multiplication
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ans =
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ans =
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0 4 8 12 16
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0 4 8 12 16
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>> a / 2 % Division mit einem Skalar
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>> a / 2 % division
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ans =
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ans =
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0 1 2 3 4
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0 1 2 3 4
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>> a .^ 2 % Potenzierung mit einem Skalar
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>> a .^ 2 % exponentiation
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ans =
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ans =
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0 4 16 36 64
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0 4 16 36 64
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\end{lstlisting}
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\end{lstlisting}
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Bei der elementweisen Verrechnung von zwei Vektoren muss
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When calculating with scalars and vectors the same mathematical
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sichergestellt werden, dass sie die gleiche L\"ange und das gleiche
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operation is done to each element of the vector. In case of, e.g. an
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Layout (Spalten- oder Zeilenvektor) haben. Addition und Subtraktion
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addition this is called an element-wise addition.
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erfolgt immer elementweise (Listing~\ref{vectoradditionlisting}).
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Care has to be taken when you do calculations with two vectors. For
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element-wise operations of two vectors, e.g. each element of vector
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\varcode{a} should be added to the respective element of vector
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\varcode{b} the two vectors must have the same length and the same
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layout (row- or column vectors). Addition and subtraction are always
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element-wise (listing~\ref{vectoradditionlisting}).
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\begin{lstlisting}[caption={Elementweise Addition und Subtraktion von
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\begin{lstlisting}[caption={Element-wise addition and subtraction of two vectors.},label=vectoradditionlisting]
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Vektoren.},label=vectoradditionlisting]
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>> a = [4 9 12];
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>> a = [4 9 12];
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>> b = [4 3 2];
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>> b = [4 3 2];
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>> a + b % Addition von 2 Vektoren
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>> a + b % addition
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ans =
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ans =
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8 12 14
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8 12 14
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>> a - b % Subtraktion von 2 Vektoren
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>> a - b % subtraction
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ans =
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ans =
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0 6 10
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0 6 10
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>> c = [8 4];
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>> c = [8 4];
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>> a + c % Beide Vektoren muessen gleich gross sein!
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>> a + c % both vectors must have the same length!
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Error using +
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Error using +
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Matrix dimensions must agree.
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Matrix dimensions must agree.
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>> d = [8; 4; 2];
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>> d = [8; 4; 2];
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>> a + d % Beide Vektoren muessen das gleiche Layout haben!
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>> a + d % both vectors must have the same layout!
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Error using +
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Error using +
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Matrix dimensions must agree.
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Matrix dimensions must agree.
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\end{lstlisting}
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\end{lstlisting}
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Bei der Multiplikation, der Division und der Potenzierung mu{\ss} mit
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Element-wise multiplication and division and exponentiation requires a
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vorangestellem '.' angezeigt werden, dass es sich um eine
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different operator with preceding '.'. \matlab{} defines the
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\emph{elementweise} Verarbeitung handeln soll. F\"ur diese
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following operators for element-wise operations on vectors
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elementweisen Operationen kennt \matlab{} die Operatoren
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\code[Operator!arithmetic!3mule@.*]{.*},
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\code[Operator!arithmetischer!3mule@.*]{.*},
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\code[Operator!arithmetic!4dive@./]{./} and
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\code[Operator!arithmetischer!4dive@./]{./} und
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\code[Operator!arithmetic!5powe@.\^{}]{.\^{}}
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\code[Operator!arithmetischer!5powe@.\^{}]{.\^{}}
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(listing~\ref{vectorelemmultiplicationlisting}).
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(Listing~\ref{vectorelemmultiplicationlisting}).
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\begin{lstlisting}[caption={Element-wise multiplication, division and
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\begin{lstlisting}[caption={Elementweise Multiplikation, Division und
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exponentiation of two vectors.},label=vectorelemmultiplicationlisting]
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Potenzierung von Vektoren.},label=vectorelemmultiplicationlisting]
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>> a .* b % element-wise multiplication
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>> a .* b % Elementweise Multiplikation
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ans =
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ans =
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16 27 24
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16 27 24
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>> a ./ b % Elementweise Division
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>> a ./ b % element-wise division
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ans =
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ans =
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1 3 6
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1 3 6
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>> a ./ b % Elementweise Potenzierung
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>> a ./ b % element-wise exponentiation
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ans =
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ans =
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256 729 144
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256 729 144
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>> a .* c % Beide Vektoren muessen gleich gross sein!
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>> a .* c % both vectors must have the same size!
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Error using .*
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Error using .*
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Matrix dimensions must agree.
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Matrix dimensions must agree.
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>> a .* d % Beide Vektoren muessen das gleiche Layout haben!
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>> a .* d % Both vectors must have the same layout!
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Error using .*
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Error using .*
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Matrix dimensions must agree.
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Matrix dimensions must agree.
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\end{lstlisting}
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\end{lstlisting}
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Die einfachen Operatoren \code[Operator!arithmetischer!3mul@*]{*},
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The simple operators \code[Operator!arithmetic!3mul@*]{*},
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\code[Operator!arithmetischer!4div@/]{/} und
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\code[Operator!arithmetic!4div@/]{/} and
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\code[Operator!arithmetischer!5pow@\^{}]{\^{}} sind mit den
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\code[Operator!arithmetic!5pow@\^{}]{\^{}} execute the respective
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entsprechenden Matrixoperationen aus der linearen Algebrar belegt (Box
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matrix-operations known from linear algebra (Box~
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\ref{matrixmultiplication}). Insbesondere ist die Multiplikation eines
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\ref{matrixmultiplication}). As a special case is the multiplication
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Zeilenvektors $\vec a$ mit einem Spaltenvektor $\vec b$ das Skalarprodukt
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of a row-vectors $\vec a$ with a column-vector $\vec b$ the
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$\sum_i = a_i b_i$.
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scalar-poduct (or dot-product) $\sum_i = a_i b_i$.
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\begin{lstlisting}[caption={Multiplikation von Vektoren.},label=vectormultiplicationlisting]
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\begin{lstlisting}[caption={Multiplication of vectors.},label=vectormultiplicationlisting]
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>> a * b % Multiplikation zweier Zeilenvektoren
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>> a * b % multiplication of two vectors
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Error using *
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Error using *
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Inner matrix dimensions must agree.
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Inner matrix dimensions must agree.
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>> a' * b' % Multiplikation zweier Spaltenvektoren
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>> a' * b' % multiplication of column-vectors
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Error using *
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Error using *
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Inner matrix dimensions must agree.
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Inner matrix dimensions must agree.
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>> a * b' % Multiplikation Zeilenvektor mit Spaltenvektor
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>> a * b' % multiplication of a row- and column-vector
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ans =
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ans =
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67
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67
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>> a' * b % Multiplikation Spaltenvektor mit Zeilenvektor
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>> a' * b % multiplication of a column- and a row-vector
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ans =
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ans =
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16 12 8
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16 12 8
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36 27 18
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36 27 18
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48 36 24
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48 36 24
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\end{lstlisting}
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\end{lstlisting}
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\pagebreak[4]
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\pagebreak[4]
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Zum Entfernen von Elementen aus einem Vektor, wird den
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entsprechenden Zellen ein leeren Wert (\code[Operator!Matrix!{[]}]{[]}) zugewiesen:
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To remove elements from a vector an empty value
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\begin{lstlisting}[label=vectoreraselisting, caption={L\"oschen von Elementen aus einem Vektor.}]
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(\code[Operator!Matrix!{[]}]{[]}) is assigned to the respective
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elements:
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\begin{lstlisting}[label=vectoreraselisting, caption={Deleting elements of a vector.}]
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>> a = (0:2:8);
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>> a = (0:2:8);
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>> length(a)
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>> length(a)
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ans = 5
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ans = 5
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>> a(1) = [] % loesche das erste Element
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>> a(1) = [] % delete the 1st element
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a = 2 4 6 8
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a = 2 4 6 8
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>> a([1 3]) = [] % loesche das erste und dritte Element
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>> a([1 3]) = [] % delete the 1st and 3rd element
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a = 4 8
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a = 4 8
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>> length(a)
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>> length(a)
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ans = 2
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ans = 2
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\end{lstlisting}
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\end{lstlisting}
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Neben dem L\"oschen von Vektorinhalten k\"onnen Vektoren auch
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In addition to deleting of vector elements one also add new elements
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erweitert oder zusammengesetzt werden. Auch hier muss das Layout der
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or concatenate two vectors. When performing a concatenation the two
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Vektoren \"ubereinstimmen (Listing \ref{vectorinsertlisting}, Zeile
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concatenated vectors must match in their layout
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10). Zum Erweitern eines Vektors kann \"uber das Ende hinaus
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(listing~\ref{vectorinsertlisting}, Line 11). To extend a vector we
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zugewiesen werden (Zeile 20). \matlab{} erweitert dann die Variable
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can simply assign values beyond the end of the vector (line 21 in
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entsprechend. Dieser Vorgang ist rechenintensiv da der ganze Vektor
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listing~ \ref{vectorinsertlisting}). \matlab{} will automatically
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an eine neue Stelle im Arbeitsspeicher kopiert wird und sollte, soweit
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adjust the variable. This way of extending a vector on-the-fly is
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m\"oglich, vermieden werden.
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however expensive. In the background \matlab{} has to reserve new
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memory of the appropriate size and then copies the contents into
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\begin{lstlisting}[caption={Zusammenf\"ugen und Erweitern von Vektoren.}, label=vectorinsertlisting]
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it. If possible this should be avoided (the \matlab{} editor will warn
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you).
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\begin{lstlisting}[caption={Concatenation and extension of vectors.}, label=vectorinsertlisting]
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>> a = [4 3 2 1];
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>> a = [4 3 2 1];
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>> b = [10 12 14 16];
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>> b = [10 12 14 16];
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>> c = [a b] % erstelle einen Vektor aus einer Liste von Vektoren
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>> c = [a b] % create a new vector by concatenation
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c =
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c =
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4 3 2 1 10 12 14 16
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4 3 2 1 10 12 14 16
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>> length(c)
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>> length(c)
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@ -532,45 +541,44 @@ ans = 8
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>> length(a) + length(b)
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>> length(a) + length(b)
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ans = 8
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ans = 8
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>> c = [a b']; % Vektorlayout muss uebereinstimmen
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>> c = [a b']; % vector layouts must match
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Error using horzcat
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Error using horzcat
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Dimensions of matrices being concatenated are not consistent.
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Dimensions of matrices being concatenated are not consistent.
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>> a(1:3) = [5 6 7] % Weise den ersten drei Elementen neue Werte zu
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>> a(1:3) = [5 6 7] % assign new values to elements of the vector
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a =
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a =
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5 6 7 1
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5 6 7 1
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>> a(1:3) = [1 2 3 4]; % Laenge der Vektoren muss uebereinstimmen
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>> a(1:3) = [1 2 3 4]; % range of elements and number of new values must match
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In an assignment A(I) = B, the number of elements in B and I must be the same.
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In an assignment A(I) = B, the number of elements in B and I must be the same.
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>> a(3:6) = [1 2 3 4] % Zuweisung ueber die Laenge des Vektors hinweg
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>> a(3:6) = [1 2 3 4] % extending a vector by assigning beyond its bounds
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a =
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a =
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5 6 1 2 3 4
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5 6 1 2 3 4
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\end{lstlisting}
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\end{lstlisting}
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\subsection{Matrizen}
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\subsection{Matrices}
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Vektoren sind 1-dimensionale Spezialf\"alle von $n$-dimensionalen
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Vectors are a special case of the more general data structure,
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Matrizen. Matrizen k\"onnen in \matlab{} beliebig viele Dimensionen
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i.e. the matrix. Vectors are matrices in which one dimension is a
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haben. Von praktischer Bedeutung sind allerdings nur Matrizen mit bis
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singleton dimension (length of 1). While matrices can have an almost
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zu vier Dimensionen. Meist beschr\"ankt es sich jedoch auf 2- bis 3-d
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arbitrary number of dimensions the most common matrices are 2-3
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Matrizen (Abbildung \ref{matrixfig} A,B).
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dimensional (figure~\ref{matrixfig} A, B).
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\begin{figure}
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\begin{figure}
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\includegraphics[width=0.5\columnwidth]{matrices}
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\includegraphics[width=0.5\columnwidth]{matrices}
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\titlecaption{Matrizen.}{\textbf{A)} Eine Variable (``test'') die eine
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\titlecaption{Matrices.}{\textbf{A)} 2-dimensional matrix with the
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2-dimensionale Matrize ist. \textbf{B)} Illustration einer
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name ``test''. \textbf{B)} Illustration of a 3-dimensional
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3-dimensionalen Matrize. Die Pfeile zeigen den Rang der
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matrix. Arrows indicate the rank across the dimensions.}\label{matrixfig}
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Dimensionen an.}\label{matrixfig}
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\end{figure}
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\end{figure}
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Erzeugt werden Matrizen sehr \"ahnlich zu den Vektoren (Listing
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Matrices can be created similarly to vectors
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\ref{matrixListing}). Die Definition einer Matrize wird, wie beim
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(listing~\ref{matrixlisting}). The definition of a matrix is enclosed
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Vektor, durch \code[Operator!Matrix!{[]}]{[]} eingeschlossen. Das
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into the square braces \code[Operator!Matrix!{[]}]{[]} the semicolon
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Semikolon \code[Operator!Matrix!;]{;} trennt die einzelnen Zeilen der
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operator \code[Operator!Matrix!;]{;} separates the individual rows of
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Matrize.
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a matrix.
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\begin{lstlisting}[label=matrixListing, caption={Erzeugen von Matrizen.}]
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\begin{lstlisting}[label=matrixlisting, caption={Creating matrices.}]
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>> a = [1 2 3; 4 5 6; 7 8 9]
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>> a = [1 2 3; 4 5 6; 7 8 9]
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>> a =
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>> a =
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1 2 3
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1 2 3
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@ -588,39 +596,40 @@ b(:,:,2) =
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1 1 1 1
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1 1 1 1
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\end{lstlisting}
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\end{lstlisting}
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Zur Defintion von mehr-dimensionalen Matrizen ist die Notation in
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The notation shown in line 1 is not suited to create matrices of
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Zeile 1 nicht geeignet. Es gibt allerdings eine Reihe von
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higher dimensions. For these, \matlab{} provides a number of
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Helferfunktionen, die $n$-dimensionale Matrizen erstellen k\"onnen
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creator-functions that help creating n-dimensional matrices
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(z.B. \code{ones()}, Zeile 7). Die \code{cat()}-Funktion kann
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(e.g. \code{ones()}, line 7 called with 3 arguments creates a 3-D
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mehrdimensionale Matrizen zusammenzuf\"ugen.
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matrix). The function \code{cat()} allows to concatenate n-dimensional
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matrices.
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Um Informationen \"uber die Gr\"o{\ss}e einer Matrize zu bekommen ist
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To request the length of a vector we used the function
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die Funktion \code{length()} nicht geeignet. Wie oben erw\"ahnt gibt sie
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\code{length()}. This function is no longer suited to request
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die Gr\"o{\ss}e der l\"angsten Dimension aus. Die \code{size()}-Funktion
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information about the size of a matrix. As mentioned above,
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gibt dagegen die L\"ange jeder Dimension als Vektor zur\"uck.
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\code{length()} would return the length of the largest dimension. The
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function \code{size()} however, returns the length in each dimension
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and should be always preferred over \code{length()}.
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\begin{figure}
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\begin{figure}
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\includegraphics[width=0.9\columnwidth]{matrixIndexing}
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\includegraphics[width=0.9\columnwidth]{matrixIndexing}
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\titlecaption{Indices von Matrizen.}{Jedes Feld einer Matrize
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\titlecaption{Indices in matrices.}{Each element of a matrix is
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wird durch einen Index individuell angesprochen. Der Index setzt
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identified by its index. The index is a tuple of as many numbers
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sich aus so vielen Zahlen zusammen wie es Dimensionen gibt (links
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as the matrix has dimensions. The first coordinate in this tuple
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2, rechts 3). Dabei steht die 1. Stelle immer f\"ur die Zeile, die
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counts the row, the second the column and the third the page,
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2. f\"ur die Spalte und die dritte f\"ur das Blatt,
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etc. }\label{matrixindexingfig}
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etc.. }\label{matrixindexingfig}
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\end{figure}
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\end{figure}
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Der Zugriff auf Inhalte von Matrizen erfolgt \"uber den Index
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Analogous to the element access in vectors we can address individual
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(Abbildung \ref{matrixindexingfig}, Listing
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elements of a matrix by it's index. Similar to a coordinate system
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\ref{matrixIndexing}). \"Ahnlich zu den Positionen in einem
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each element is addressed using a n-tuple whit n the number of
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Koordinatensystem wird jede Zelle einer Matrize mit einem Index
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dimensions (figure~\ref{matrixindexingfig},
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angesprochen, der aus $n$ Zahlen besteht wobei $n$ die
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listing~\ref{matrixIndexing}). This type of indexing is called
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Dimensionalit\"at der Matrize ist. Diese Art des Zugriffs wird
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\codeterm{subscript indexing}. The first coordinate refers always to
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\codeterm{subscript indexing} genannt. Dabei bestimmt die errste Zahl
|
the row, the second to the column, the third to the page, and so on.
|
||||||
die Zeilennumer, die zweite die Splatennumer.
|
|
||||||
|
|
||||||
\begin{lstlisting}[caption={Zugriff auf Inhalte von Matrizen,
|
\begin{lstlisting}[caption={Indexing in matrices,
|
||||||
Indizierung.}, label=matrixIndexing]
|
Indizierung.}, label=matrixIndexing]
|
||||||
>> x=rand(3,4) % 2-D Matrix mit Zufallszahlen mit 3 Zeilen und 4 Spalten
|
>> x=rand(3, 4) % 2-D matrix filled with random numbers
|
||||||
x =
|
x =
|
||||||
0.8147 0.9134 0.2785 0.9649
|
0.8147 0.9134 0.2785 0.9649
|
||||||
0.9058 0.6324 0.5469 0.1576
|
0.9058 0.6324 0.5469 0.1576
|
||||||
@ -629,137 +638,146 @@ x =
|
|||||||
ans =
|
ans =
|
||||||
3 4
|
3 4
|
||||||
|
|
||||||
>> x(1,1) % obere linke Ecke
|
>> x(1,1) % top left corner
|
||||||
ans =
|
ans =
|
||||||
0.8147
|
0.8147
|
||||||
>> x(2,3) % Element der 2. Zeile, 3. Spalte
|
>> x(2,3) % element in the 2nd row, 3rd column
|
||||||
ans =
|
ans =
|
||||||
0.5469
|
0.5469
|
||||||
|
|
||||||
>> x(1,:) % erste Zeile
|
>> x(1,:) % the first row
|
||||||
ans =
|
ans =
|
||||||
0.8147 0.9134 0.2785 0.9649
|
0.8147 0.9134 0.2785 0.9649
|
||||||
>> x(:,2) % zweite Spalte
|
>> x(:,2) % second column
|
||||||
ans =
|
ans =
|
||||||
0.9134
|
0.9134
|
||||||
0.6324
|
0.6324
|
||||||
0.0975
|
0.0975
|
||||||
\end{lstlisting}
|
\end{lstlisting}
|
||||||
|
|
||||||
Alternativ zum \codeterm{subscript indexing} k\"onnen die Zellen einer
|
Subscript indexing is very intuitive but offers not always the most
|
||||||
Matrize auch \emph{linear} angesprochen werden (Abbildung
|
straight-forward solution to the problem. Consider for example that
|
||||||
\ref{matrixlinearindexingfig}). Diese Art der Adressierung ist nicht
|
you have a 3-D matrix and you want the minimal number in that
|
||||||
so intuitiv verst\"andlich, kann aber sehr hilfreich sein. Der lineare
|
matrix. An alternative way is the so called \emph{linar indexing} in
|
||||||
Index einer Zelle reicht von 1 bis \code{numel()} Elemente. Wobei
|
which each element of the matrix is addressed by a single number. The
|
||||||
dieser erst entlang der 1. Dimension, dann der 2., 3. etc. Dimension
|
linear index thus ranges from 1 to \code{numel(matrix)}. The linear
|
||||||
ansteigt. Listing \ref{matrixLinearIndexing} zeigt ein Beispiel f\"ur
|
index increases first along the 1st, 2nd, 3rd etc. dimension
|
||||||
den Einsatz des linearen Indizierens, z.B. zum Ermitteln des kleinsten
|
(figure~\ref{matrixlinearindexingfig}). It is not as intuitive but can
|
||||||
Wertes in einer Matrize.
|
be really helpful (listing~\ref{matrixLinearIndexing}).
|
||||||
|
|
||||||
|
|
||||||
\begin{figure}
|
\begin{figure}
|
||||||
\includegraphics[width=0.9\columnwidth]{matrixLinearIndexing}
|
\includegraphics[width=0.9\columnwidth]{matrixLinearIndexing}
|
||||||
\titlecaption{Lineares Indizieren von Matrizen.}{Der Index steigt
|
\titlecaption{Linear indexing in matrices.}{The linear index in a
|
||||||
linear von 1 bis zur Anzahl Elemente in der Matrize an. Dabei
|
matrix increases from 1 to the number of elements in the
|
||||||
steigt der Index zuerst entlang der ersten, zweiten, dritten und
|
matrix. It increases first along the first dimension, then the
|
||||||
weiterer Dimensionen an.}\label{matrixlinearindexingfig}
|
rows in each column and so on.}\label{matrixlinearindexingfig}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
|
|
||||||
\begin{lstlisting}[label=matrixLinearIndexing, caption={Lineares Indizieren in Matrizen.}]
|
\begin{lstlisting}[label=matrixLinearIndexing, caption={Lineares indexing in matrices.}]
|
||||||
>> x = randi(100, [3, 4, 5]); % 3-D Matrix mit Zufallszahlen
|
>> x = randi(100, [3, 4, 5]); % 3-D matrix filled with random numbers
|
||||||
>> size(x)
|
>> size(x)
|
||||||
ans =
|
ans =
|
||||||
3 4 5
|
3 4 5
|
||||||
>> numel(x)
|
>> numel(x)
|
||||||
ans =
|
ans =
|
||||||
60
|
60
|
||||||
>> min(min(min(x))) % Minimum ueber die Zeilen, Spalten, Blaetter...
|
>> min(min(min(x))) % minimum across rows, then columns, then pages
|
||||||
ans =
|
ans =
|
||||||
4
|
4
|
||||||
>> min(x(:)) % oder so
|
>> min(x(1:numel(x))) % or like this
|
||||||
ans =
|
ans =
|
||||||
4
|
4
|
||||||
|
>> min(x(:)) % or even simpler
|
||||||
|
ans =
|
||||||
|
4
|
||||||
\end{lstlisting}
|
\end{lstlisting}
|
||||||
|
|
||||||
\begin{ibox}[t]{\label{matrixmultiplication} Matrixmultiplikation.}
|
\begin{ibox}[t]{\label{matrixmultiplication} The matrix-multiplication.}
|
||||||
Die Matrixmuliplikation aus der linearen Algebra ist nicht eine
|
The matrix-multiplication from linear algebra is \textbf{not} an
|
||||||
elementweise Multiplikation. Die Matrixmultiplikation ist nur dann
|
element-wise multiplication of each element in a matrix \varcode{A}
|
||||||
m\"oglich, wenn die Anzahl Spalten der ersten Matrize gleich der
|
and the respective element from matrix \varcode{B}. It is something
|
||||||
Anzahl Zeilen in der zweiten Matrize ist. Formaler: zwei Matrizen
|
completely different. Confusing element-wise and
|
||||||
$\mathbf{A}$ und $\mathbf{B}$ k\"onnen mulipiziert $(\mathbf{A}
|
matrix-multiplication is one of the most common mistakes in
|
||||||
\cdot \mathbf{B})$ werden, wenn $\mathbf{A}$ die Gr\"o{\ss}e $(m \times n)$ und
|
\matlab{}. \linebreak
|
||||||
$\mathbf{B}$ die Gr\"o{\ss}e $(n \times k)$ hat. Die Mulitplikation ist
|
|
||||||
m\"oglich wenn die \determ{inneren Dimensionen} $n$ gleich sind.
|
The matrix-multiplication is only possible if the number of columns
|
||||||
|
in the first matrix agrees with the number of rows in the other. More
|
||||||
Dann sind die Elemente $c_{i,j}$ des Matrixprodukts $\mathbf{C} =
|
formal: $\mathbf{A}$ and $\mathbf{B}$ can be multiplied $(\mathbf{A}
|
||||||
\mathbf{A} \cdot \mathbf{B}$ gegeben durch das Skalarprodukt jeder
|
\cdot \mathbf{B})$, if $\mathbf{A}$ has the size $(m \times n)$ and
|
||||||
Zeile von $\mathbf{A}$ mit jeder Spalte aus $\mathbf{B}$:
|
$\mathbf{B}$ the size $(n \times k)$. The multiplication is possible
|
||||||
\[ c_{i,j} = \sum_{k=1}^n a_{i,k} \; b_{k,j} \; . \]
|
if the \enterm{inner dimensions} $n$ agree.
|
||||||
|
|
||||||
Die Matrixmultiplikation ist im Allgemeinen auch nicht kommutativ:
|
Then, the elements $c_{i,j}$ of the product $\mathbf{C} = \mathbf{A}
|
||||||
|
\cdot \mathbf{B}$ are given as the scalar product (dot-product) of
|
||||||
|
each row in $\mathbf{A}$ with each column in $\mathbf{B}$: \[
|
||||||
|
c_{i,j} = \sum_{k=1}^n a_{i,k} \; b_{k,j} \; . \]
|
||||||
|
|
||||||
|
The matrix-multiplication is not commutative, that is:
|
||||||
\[ \mathbf{A} \cdot \mathbf{B} \ne \mathbf{B} \cdot \mathbf{A} \; . \]
|
\[ \mathbf{A} \cdot \mathbf{B} \ne \mathbf{B} \cdot \mathbf{A} \; . \]
|
||||||
|
|
||||||
Als Beispiel betrachten wir die beiden Matrizen
|
Consider the matrices:
|
||||||
\[\mathbf{A}_{(3 \times 2)} = \begin{pmatrix} 1 & 2 \\ 5 & 4 \\ -2 & 3 \end{pmatrix}
|
\[\mathbf{A}_{(3 \times 2)} = \begin{pmatrix} 1 & 2 \\ 5 & 4 \\ -2 & 3 \end{pmatrix}
|
||||||
\quad \text{und} \quad \mathbf{B}_{(2 \times 2)} = \begin{pmatrix}
|
\quad \text{and} \quad \mathbf{B}_{(2 \times 2)} = \begin{pmatrix}
|
||||||
-1 & 2 \\ -2 & 5 \end{pmatrix} \; . \]
|
-1 & 2 \\ -2 & 5 \end{pmatrix} \; . \] The inner dimensions of
|
||||||
F\"ur das Produkt $\mathbf{A} \cdot \mathbf{B}$ stimmen die inneren
|
these matrices match ($(3 \times 2) \cdot (2 \times 2)$) and the
|
||||||
Dimensionen der Matrizen \"uberein ($(3 \times 2) \cdot (2
|
product of $\mathbf{C} = \mathbf{A} \cdot \mathbf{B}$ can be
|
||||||
\times 2)$), die Matrixmultiplikation ist also m\"oglich. Nachdem
|
calculated. Following from the number of rows in $\mathbf{A}$ (3)
|
||||||
$\mathbf{A}$ drei Zeilen und $\mathbf{B}$ zwei Spalten hat, hat das
|
and the number of columns in $\mathbf{B}$ (2) the resulting matrix
|
||||||
Ergebnis von $\mathbf{A} \cdot \mathbf{B}$ die Gr\"o{\ss}e $(3
|
$\mathbf{C}$ will have the size $(3 \times 2)$:
|
||||||
\times 2)$:
|
|
||||||
\[ \mathbf{A} \cdot \mathbf{B} = \begin{pmatrix} 1 \cdot -1 + 2 \cdot -2 & 1 \cdot 2 + 2\cdot 5 \\
|
\[ \mathbf{A} \cdot \mathbf{B} = \begin{pmatrix} 1 \cdot -1 + 2 \cdot -2 & 1 \cdot 2 + 2\cdot 5 \\
|
||||||
5 \cdot -1 + 4 \cdot -2 & 5 \cdot 2 + 4 \cdot 5\\
|
5 \cdot -1 + 4 \cdot -2 & 5 \cdot 2 + 4 \cdot 5\\
|
||||||
-2 \cdot -1 + 3 \cdot -2 & -2 \cdot 2 + 3 \cdot 5 \end{pmatrix}
|
-2 \cdot -1 + 3 \cdot -2 & -2 \cdot 2 + 3 \cdot 5 \end{pmatrix}
|
||||||
= \begin{pmatrix} -5 & 12 \\ -13 & 30 \\ -4 & 11\end{pmatrix} \; . \]
|
= \begin{pmatrix} -5 & 12 \\ -13 & 30 \\ -4 & 11\end{pmatrix} \; . \]
|
||||||
|
|
||||||
Das Produkt $\mathbf{B} \cdot \mathbf{A}$ ist dagegen nicht
|
The product of $\mathbf{B} \cdot \mathbf{A}$, however, is not
|
||||||
definiert, da die inneren Dimensionen nicht \"ubereinstimmen
|
defined since the inner dimensions do not agree ($(2 \times 2) \cdot
|
||||||
($(2 \times 2) \cdot (3 \times 2)$).
|
(3 \times 2)$).
|
||||||
\end{ibox}
|
\end{ibox}
|
||||||
|
|
||||||
Beim Rechnen mit Matrizen gelten die gleichen Regeln wie bei
|
Calculations on matrices apply the same rules as the calculations with
|
||||||
Vektoren. Matrizen k\"onnen solange elementweise miteinander
|
vectors. Element-wise computations are possible as long as the
|
||||||
verrechnet werden, wie die Dimensionalit\"aten
|
matrices have the same dimensionality. It is again important to
|
||||||
\"ubereinstimmen. Wichtig ist auch hier wieder die Unterscheidung
|
distinguish between the element-wise
|
||||||
zwischen elementweiser Multiplikation
|
(\code[Operator!arithmetic!3mule@.*]{.*} operator, listing
|
||||||
(\code[Operator!arithmetischer!3mule@.*]{.*} Operator, Listing
|
\ref{matrixOperations} line 10) and the operator for
|
||||||
\ref{matrixOperations} Zeile 10) oder Matrixmultiplikation
|
matrix-multiplication (\code[Operator!arithmetic!3mul@*]{*},
|
||||||
(\code[Operator!arithmetischer!3mul@*]{*} Operator, Listing
|
listing~\ref{matrixOperations} lines 14, 17 and 21,
|
||||||
\ref{matrixOperations} Zeile 14, 17 und 21, Box~\ref{matrixmultiplication}).
|
box~\ref{matrixmultiplication}). To do a matrix-multiplication the
|
||||||
Bei der Matrixmultiplikation m\"ussen die inneren Dimensionen der Matrizen \"ubereinstimmen
|
inner dimensions of the matrices have to agree
|
||||||
(Box~\ref{matrixmultiplication}).
|
(box~\ref{matrixmultiplication}).
|
||||||
|
|
||||||
\pagebreak[4]
|
\pagebreak[4]
|
||||||
\begin{lstlisting}[label=matrixOperations, caption={Zwei Arten der Multiplikation von Matrizen.}]
|
\begin{lstlisting}[label=matrixOperations, caption={Two kinds of multiplications of matrices.}]
|
||||||
>> A = randi(5, [2, 3]) % 2-D Matrix
|
>> A = randi(5, [2, 3]) % 2-D matrix
|
||||||
A =
|
A =
|
||||||
1 5 3
|
1 5 3
|
||||||
3 2 2
|
3 2 2
|
||||||
>> B = randi(5, [2, 3]) % dito
|
>> B = randi(5, [2, 3]) % dto.
|
||||||
B =
|
B =
|
||||||
4 3 5
|
4 3 5
|
||||||
2 4 5
|
2 4 5
|
||||||
|
|
||||||
>> A .* B % elementweise Multiplikation
|
>> A .* B % element-wise multiplication
|
||||||
ans =
|
ans =
|
||||||
4 15 15
|
4 15 15
|
||||||
6 8 10
|
6 8 10
|
||||||
>> A * B % Matrixmultiplikation
|
>> A * B % invalid matrix-multiplication
|
||||||
Error using *
|
Error using *
|
||||||
Inner matrix dimensions must agree.
|
Inner matrix dimensions must agree.
|
||||||
>> A * B' % Matrixmultiplikation
|
>> A * B' % valid matrix-multiplication
|
||||||
ans =
|
ans =
|
||||||
34 37
|
34 37
|
||||||
28 24
|
28 24
|
||||||
>> A' * B % Matrixmultiplikation
|
>> A' * B % matrix-multiplication is not commutative
|
||||||
ans =
|
ans =
|
||||||
10 15 20
|
10 15 20
|
||||||
24 23 35
|
24 23 35
|
||||||
16 17 25
|
16 17 25
|
||||||
\end{lstlisting}
|
\end{lstlisting}
|
||||||
|
|
||||||
\section{Boolesche Operationen}
|
\section{Boolean Operations}
|
||||||
|
|
||||||
Boolesche Ausdr\"ucke sind Anweisungen, die zu \codeterm{wahr} oder
|
Boolesche Ausdr\"ucke sind Anweisungen, die zu \codeterm{wahr} oder
|
||||||
\codeterm{falsch} ausgewertet werden. Man kennt sie z.B. aus der
|
\codeterm{falsch} ausgewertet werden. Man kennt sie z.B. aus der
|
||||||
|
|||||||
Reference in New Issue
Block a user