Difference between revisions of "Minors and cofactors"
From Robotics
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The '''minor <math>M_{ij}(\mathbf{A})</math>''' of an n-by-n square matrix <math>\mathbf{A}</math> is the [[Determinant of a matrix|determinant]] of a smaller square matrix obtained by removing the row <math>i</math> and the column <math>j</math> from <math>\mathbf{A}</math>.<br/><br/> | The '''minor <math>M_{ij}(\mathbf{A})</math>''' of an n-by-n square matrix <math>\mathbf{A}</math> is the [[Determinant of a matrix|determinant]] of a smaller square matrix obtained by removing the row <math>i</math> and the column <math>j</math> from <math>\mathbf{A}</math>.<br/><br/> | ||
− | Multiplying the minor with <math>(-1)^{i+j}</math> results in the '''cofactor <math>C_{ | + | Multiplying the minor with <math>(-1)^{i+j}</math> results in the '''cofactor <math>C_{ij}(\mathbf{A})</math>''':<br/><br/> |
− | <math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})</math | + | <math>C_{i,j}(\mathbf{A})=(-1)^{i+j}M_{ij}(\mathbf{A})</math> |
{{Example | {{Example | ||
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<br/> | <br/> | ||
:<math> | :<math> | ||
− | \mathbf{A} | + | \mathbf{A}_3 = |
\left[\begin{array}{ccc} | \left[\begin{array}{ccc} | ||
1&0&1\\ | 1&0&1\\ | ||
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1&0&2 | 1&0&2 | ||
\end{array}\right]</math><br/><br/> | \end{array}\right]</math><br/><br/> | ||
− | The minors <math>M_{22}(\mathbf{A} | + | The minors <math>M_{22}(\mathbf{A}_3)</math> and <math>M_{31}(\mathbf{A}_3)</math> for example are defined as<br/><br/> |
− | :<math>M_{22}(\mathbf{A} | + | :<math>M_{22}(\mathbf{A}_3)= |
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
1&\Box&1\\ | 1&\Box&1\\ | ||
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</math><br/><br/> | </math><br/><br/> | ||
:<math> | :<math> | ||
− | M_{31}(\mathbf{A} | + | M_{31}(\mathbf{A}_3)= |
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
\Box & 0&1\\ | \Box & 0&1\\ | ||
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</math><br/><br/> | </math><br/><br/> | ||
The corresponding cofactors in that case are<br/><br/> | The corresponding cofactors in that case are<br/><br/> | ||
− | :<math>C_{22}(\mathbf{A} | + | :<math>C_{22}(\mathbf{A}_3)=(-1)^{2+2}M_{22}(\mathbf{A}_e)=(-1)^4\cdot1=1</math><br/><br/> |
− | :<math>C_{31}(\mathbf{A} | + | :<math>C_{31}(\mathbf{A}_3)=(-1)^{3+1}M_{31}(\mathbf{A}_e)=(-1)^4\cdot-1=-1</math> |
}} | }} | ||
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|Title=Minors and cofactors of a 4-by-4 matrix | |Title=Minors and cofactors of a 4-by-4 matrix | ||
|Contents= | |Contents= | ||
− | <br/><math> | + | This example uses the transformation matrix <math>^R\mathbf{T}_N</math> that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages. |
− | \mathbf{ | + | <br/> |
+ | :<math> | ||
+ | ^R\mathbf{T}_N = | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} | ||
− | 1 | + | 0 & 1 & 0 & 2a\\ |
− | + | 0 & 0 & -1 & 0\\ | |
− | 0 | + | -1 & 0 & 0 & 0\\ |
− | 0 & 0 & | + | 0 & 0 & 0 & 1 |
\end{array}\right]</math><br/><br/> | \end{array}\right]</math><br/><br/> | ||
− | The minors <math>M_{ | + | The minors <math>M_{31}(^R\mathbf{T}_N)</math> and <math>M_{42}(^R\mathbf{T}_N)</math> for example are defined as<br/><br/> |
− | <math>M_{ | + | :<math> |
+ | M_{31}(^R\mathbf{T}_N)= | ||
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
+ | \Box & 1 & 0 & 2a\\ | ||
+ | \Box & 0 & -1 & 0\\ | ||
\Box & \Box & \Box & \Box\\ | \Box & \Box & \Box & \Box\\ | ||
− | + | \Box & 0 & 0 & 1 | |
− | |||
− | 0 & | ||
\end{array}\right|= | \end{array}\right|= | ||
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
− | + | 1&0&2a\\ | |
− | 0 & 1 & 0\\ | + | 0&-1&0\\ |
− | 0 & 0 & | + | 0&0&1 |
− | \end{array}\right|= | + | \end{array}\right|=-1-0=-1 |
− | </math><br/> | + | </math><br/><br/> |
− | <math> | + | :<math> |
− | M_{ | + | M_{42}(^R\mathbf{T}_N)= |
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
− | \Box & | + | 0&\Box&0&2a\\ |
− | \Box & 0 & | + | 0&\Box&-1&0\\ |
+ | -1&\Box&0&0\\ | ||
\Box & \Box & \Box & \Box\\ | \Box & \Box & \Box & \Box\\ | ||
− | |||
\end{array}\right|= | \end{array}\right|= | ||
\left|\begin{array}{ccc} | \left|\begin{array}{ccc} | ||
− | + | 0&0&2a\\ | |
− | 0 & 1 & | + | 0&-1&0\\ |
− | 0 & | + | -1&0&0 |
− | \end{array}\right|= | + | \end{array}\right|=0-2a=-2a |
− | </math><br/><br/> | + | </math><br/> |
+ | <br/><br/> | ||
The corresponding cofactors in that case are<br/><br/> | The corresponding cofactors in that case are<br/><br/> | ||
− | <math>C_{ | + | :<math>C_{31}(^R\mathbf{T}_N)=(-1)^{3+1}M_{31}(^R\mathbf{T}_N)=(-1)^4\cdot(-1)=-1</math> |
− | <math>C_{ | + | :<math>C_{42}(^R\mathbf{T}_N)=(-1)^{4+2}M_{42}(^R\mathbf{T}_N)=(-1)^6\cdot(-2a)=-2a</math> |
}} | }} | ||
[[Category:Article]] | [[Category:Article]] | ||
[[Category:Matrices]] | [[Category:Matrices]] |
Latest revision as of 18:10, 13 November 2015
← Back: Determinant of a matrix | Overview: Matrices | Next: Matrix inversion → |
The minor of an n-by-n square matrix is the determinant of a smaller square matrix obtained by removing the row and the column from .
Multiplying the minor with results in the cofactor :
Example: Minors and cofactors of a 3-by-3 matrix
The minors and for example are defined as The corresponding cofactors in that case are |
Example: Minors and cofactors of a 4-by-4 matrix
This example uses the transformation matrix that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.
The minors and for example are defined as
|