Difference between revisions of "Multiplication of matrices"

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In chapter 3 of the robotics script some examples of matrices multiplied with vectors appear. On page 3-28 a transformation equation is presented for a rotation  
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In chapter 3 of the robotics script some examples of matrices multiplied with vectors appear. On page 3-28 a two-dimensional transformation equation is presented for a rotation about <math>\varphi</math> about the origin (see figure) in combination with a translation. The resulting vector is the following:<br/>
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:<math>
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\left[\begin{array}{c}
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    x_1 \\
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    y_1 \\
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    1
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\end{array}\right]=
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\left[\begin{array}{ccc}
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    \cos\varphi & -\sin\varphi & m\\
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    \sin\varphi & \cos\varphi & n\\
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    0 & 0 & 1
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\end{array}\right]
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\cdot
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\left[\begin{array}{c}
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    x_0 \\
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    y_0 \\
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    1
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\end{array}\right] =
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\left[\begin{array}{c}
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    \cos\varphi\cdot x_0-\sin\varphi\cdot y_0+m\\
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    \sin\varphi\cdot x_0+\cos\varphi\cdot y_0+n\\
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    0\cdot x_0+0\cdot y_0+1\cdot 1
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\end{array}\right]=
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\left[\begin{array}{c}
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    \cos\varphi\cdot x_0-\sin\varphi\cdot y_0+m\\
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    \sin\varphi\cdot x_0+\cos\varphi\cdot y_0+n\\
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    1
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\end{array}\right]
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</math>
 
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Revision as of 16:25, 16 May 2014

← Back: Addition of matrices Overview: Matrices Next: Matrix inversion →

Two matrices can be multiplied if the number of colums of the left matrix equals the number of rows of the right matrix. The result of the multiplication of an l-by-m matrix \mathbf{A}=(a_{ij})_{i=1...l,j=1...m} with an m-by-n matrix \mathbf{B}=(b_{ij})_{i=1...m,j=1...n} is an l-by-n matrix \mathbf{C}=(c_{ij})_{i=1...l,j=1...n}. The components of the resulting matrix are comuputed as follows:

c_{ij}=\sum^{m}_{k=1}a_{ik}\cdot b_{kj}

For example the multiplication of a 2-by-3 matrix with a 3-by-2 matrix results in a 2-by-2 matrix and is computed as follows:

\mathbf{A}\cdot\mathbf{B}= \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{array}\right]\cdot \left[\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22}\\ b_{31} & b_{32} \end{array}\right]= \left[\begin{array}{cc} a_{11}\cdot b_{11}+a_{12}\cdot b_{21}+a_{13}\cdot b_{31} & a_{11}\cdot b_{12}+a_{12}\cdot b_{22}+a_{13}\cdot b_{12} \\ a_{21}\cdot b_{11}+a_{22}\cdot b_{21}+a_{23}\cdot b_{31} & a_{21}\cdot b_{12}+a_{22}\cdot b_{22}+a_{23}\cdot b_{12} \end{array}\right]

Some further rules for matrix multiplications are:

\begin{align} \mathbf{A}\cdot\mathbf{B}&\ne\mathbf{B}\cdot\mathbf{A} \\ (\mathbf{A}\cdot\mathbf{B})\cdot\mathbf{C}&=\mathbf{A}\cdot(\mathbf{B}\cdot\mathbf{C}) \\ (\mathbf{A}+\mathbf{B})\cdot\mathbf{C}&=\mathbf{A}\cdot\mathbf{C}+\mathbf{B}\cdot\mathbf{C} &\text{for all l-by-m matrices } \mathbf{A},\mathbf{B} \text{ and m-by-n matrices } \mathbf{C}\\ \mathbf{A}\cdot(\mathbf{B})+\mathbf{C})&=\mathbf{A}\cdot\mathbf{B}+\mathbf{A}\cdot\mathbf{C} &\text{for all l-by-m matrices } \mathbf{A} \text{ and m-by-n matrices }\mathbf{B},\mathbf{C} \end{align}

Example: Multiplication of matrices

A good example for the multiplication of several matrices in the context of robotics and transformations is presented in the robotics script. Please have a look in chapter 3 on page 3-35

Multiplication of matrices with vectors

A vector is a just special form of a matrix with either only one row or one column. Because an l-by-m matrix can only be multiplied by an m-by-n matrix, there are two possibilities of multiplying matrices and vectors. The first possibility is a 1-by-m row vector multiplied with an m-by-n matrix which results in a 1-by-n row vector:

\left[\begin{array}{ccc} v_{11} & \dots & v_{1m} \end{array}\right] \cdot \left[\begin{array}{ccc} a_{11} & \dots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{m1} & \dots & a_{mn} \end{array}\right] = \left[\begin{array}{ccc} v_{11}a_{11}+\dots+v_{1m}a_{m1} & \dots & v_{11}a_{1n}+\dots+v_{1m}a_{mn} \end{array}\right]

The second possibility is a l-by-m matrix multiplied with an m-by-1 column vector which results in a l-by-1 column vector:

\left[\begin{array}{ccc} a_{11} & \dots & a_{1m}\\ \vdots & \ddots & \vdots\\ a_{l1} & \dots & a_{lm} \end{array}\right] \cdot \left[\begin{array}{c} v_{11}\\ \vdots \\ v_{m1} \end{array}\right] = \left[\begin{array}{c} a_{11}v_{11}+\dots+a_{1m}v_{m1}\\ \vdots \\ a_{l1}v_{11}+\dots+a_{lm}v_{m1} \end{array}\right]

Example: Multiplication of matrices and vectors
Matrix vector mult.png

In chapter 3 of the robotics script some examples of matrices multiplied with vectors appear. On page 3-28 a two-dimensional transformation equation is presented for a rotation about \varphi about the origin (see figure) in combination with a translation. The resulting vector is the following:

\left[\begin{array}{c} x_1 \\ y_1 \\ 1 \end{array}\right]= \left[\begin{array}{ccc} \cos\varphi & -\sin\varphi & m\\ \sin\varphi & \cos\varphi & n\\ 0 & 0 & 1 \end{array}\right] \cdot \left[\begin{array}{c} x_0 \\ y_0 \\ 1 \end{array}\right] = \left[\begin{array}{c} \cos\varphi\cdot x_0-\sin\varphi\cdot y_0+m\\ \sin\varphi\cdot x_0+\cos\varphi\cdot y_0+n\\ 0\cdot x_0+0\cdot y_0+1\cdot 1 \end{array}\right]= \left[\begin{array}{c} \cos\varphi\cdot x_0-\sin\varphi\cdot y_0+m\\ \sin\varphi\cdot x_0+\cos\varphi\cdot y_0+n\\ 1 \end{array}\right]
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