Difference between revisions of "Multiplication of matrices"
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|Contents= | |Contents= | ||
[[File:matrix_vector_mult.png|right|200px]] | [[File:matrix_vector_mult.png|right|200px]] | ||
− | 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 | + | 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/> |
+ | :<math> | ||
+ | \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] | ||
+ | </math> | ||
}} | }} |
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 with an m-by-n matrix is an l-by-n matrix . The components of the resulting matrix are comuputed as follows:
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:
Some further rules for matrix multiplications are:
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:
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:
Example: Multiplication of matrices and vectors
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 about the origin (see figure) in combination with a translation. The resulting vector is the following: |