Difference between revisions of "Multiplication of matrices"
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− | {{Navigation|before=[[Addition of matrices]]|overview=[[Matrices]]|next=[[ | + | {{Navigation|before=[[Addition of matrices]]|overview=[[Matrices]]|next=[[Minors and cofactors]]}} |
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 <math>\mathbf{A}=(a_{ij})_{i=1...l,j=1...m}</math> with an m-by-n matrix <math>\mathbf{B}=(b_{ij})_{i=1...m,j=1...n}</math> is an l-by-n matrix <math>\mathbf{C}=(c_{ij})_{i=1...l,j=1...n}</math>. The components of the resulting matrix are comuputed as follows:<br/> | 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 <math>\mathbf{A}=(a_{ij})_{i=1...l,j=1...m}</math> with an m-by-n matrix <math>\mathbf{B}=(b_{ij})_{i=1...m,j=1...n}</math> is an l-by-n matrix <math>\mathbf{C}=(c_{ij})_{i=1...l,j=1...n}</math>. The components of the resulting matrix are comuputed as follows:<br/> |
Revision as of 13:26, 19 May 2014
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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: |