Difference between revisions of "Matrix inversion"

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2-2 & -3+4
 
2-2 & -3+4
 
\end{array}\right]\\&=
 
\end{array}\right]\\&=
−
\left[\begin{array}{cccc}
+
\left[\begin{array}{cc}
 
{\color{Green}\mathbf{1}} & 0\\
 
{\color{Green}\mathbf{1}} & 0\\
 
0 & {\color{Green}\mathbf{1}}
 
0 & {\color{Green}\mathbf{1}}
 
\end{array}\right]=
 
\end{array}\right]=
 
\mathbf{I}_2
 
\mathbf{I}_2
 +
\end{align}</math>}}
 +
 +
{{Example
 +
|Title=inverse of a 3-by-3 matrix
 +
|Contents=
 +
This is an example for the inverse of a 3-by-3 matrix:<br/><br/>
 +
:<math>
 +
\mathbf{A}_e =
 +
\left[\begin{array}{ccc}
 +
1&0&1\\
 +
3&1&0\\
 +
1&0&2
 +
\end{array}\right]
 +
,\quad
 +
{\mathbf{A}_e}^{-1}  =
 +
\left[\begin{array}{ccc}
 +
2&0&-1\\
 +
-6&1&3\\
 +
-1&0&1
 +
\end{array}\right]
 +
</math><br/><br/>
 +
:<math>\begin{align}
 +
{\mathbf{A}_e}{\mathbf{A}_e}^{-1}  &=
 +
\left[\begin{array}{ccc}
 +
1&0&1\\
 +
3&1&0\\
 +
1&0&2
 +
\end{array}\right]\cdot
 +
\left[\begin{array}{ccc}
 +
2&0&-1\\
 +
-6&1&3\\
 +
-1&0&1
 +
\end{array}\right]\\&=
 +
\left[\begin{array}{ccc}
 +
1\cdot2+0\cdot(-6)+1\cdot(-1) & 1\cdot0+0\cdot1+1\cdot0 & 1\cdot(-1)+0\cdot3+1\cdot1\\
 +
3\cdot2+1\cdot(-6)+0\cdot(-1) & 3\cdot0+1\cdot1+0\cdot0 & 3\cdot(-1)+1\cdot3+0\cdot1\\
 +
1\cdot2+0\cdot(-6)+2\cdot(-1) & 1\cdot0+0\cdot1+2\cdot0 & 1\cdot(-1)+0\cdot3+2\cdot1
 +
\end{array}\right]\\&=
 +
\left[\begin{array}{ccc}
 +
2+0+1-1 & 0+0+0 & -1+0+1\\
 +
6-6+0 & 0+1+0 & -3+3+0\\
 +
2+0-2 & 0+0+0 & -1+0+2
 +
\end{array}\right]\\&=
 +
\left[\begin{array}{ccc}
 +
{\color{Green}\mathbf{1}} & 0 & 0\\
 +
0 & {\color{Green}\mathbf{1}} & 0\\
 +
0 & 0 & {\color{Green}\mathbf{1}}
 +
\end{array}\right]=
 +
\mathbf{I}_3
 
\end{align}</math>}}
 
\end{align}</math>}}
  

Revision as of 16:42, 22 May 2014

← Back: Minors and cofactors Overview: Matrix inversion Next: Gauß-Jordan-Algorithm →

The inverse of an n-by-n square matrix \mathbf{A} is denoted as \mathbf{A}^{-1} and defined such that

\mathbf{A}\mathbf{A}^{-1}=\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}_n

where \mathbf{I}_n is the n-by-n identity matrix.
Prerequesite for the inversion is, that \mathbf{A} is an n-by-n square matrix and that \mathbf{A} is regular. Regular means that the row and column vectors are linearly independent and so the determinant is nonzero:

det(\mathbf{A})\ne0

Otherwise the matrix is called singular.

Before determining the inverse of a matrix it is always useful to compute the determinant and check whether the matrix is regular or singular. If it is singular it is not possible to determine the inverse because there is no inverse. The following two subarticles describe two of the common procedures to determine the inverse of a matrix.

  1. Gauß-Jordan-Algorithm
  2. Adjugate Formula


Example: inverse of a 2-by-2 matrix

This is a simple example for the inverse of a 2-by-2 matrix:

\mathbf{A}_e = \left[\begin{array}{cc} 2 & 3\\ 1 & 2 \end{array}\right] ,\quad {\mathbf{A}_e}^{-1} = \left[\begin{array}{cc} 2 & -3\\ -1 & 2 \end{array}\right]

\begin{align} {\mathbf{A}_e}{\mathbf{A}_e}^{-1} &= \left[\begin{array}{cc} 2 & 3\\ 1 & 2 \end{array}\right]\cdot \left[\begin{array}{cc} 2 & -3\\ -1 & 2 \end{array}\right]\\&= \left[\begin{array}{cc} 2\cdot2+3\cdot(-1) & 2\cdot(-3)+3\cdot2\\ 1\cdot2+2\cdot(-1) & 1\cdot(-3)+2\cdot2 \end{array}\right]\\&= \left[\begin{array}{cc} 4-3 & -6+6\\ 2-2 & -3+4 \end{array}\right]\\&= \left[\begin{array}{cc} {\color{Green}\mathbf{1}} & 0\\ 0 & {\color{Green}\mathbf{1}} \end{array}\right]= \mathbf{I}_2 \end{align}
Example: inverse of a 3-by-3 matrix

This is an example for the inverse of a 3-by-3 matrix:

\mathbf{A}_e = \left[\begin{array}{ccc} 1&0&1\\ 3&1&0\\ 1&0&2 \end{array}\right] ,\quad {\mathbf{A}_e}^{-1} = \left[\begin{array}{ccc} 2&0&-1\\ -6&1&3\\ -1&0&1 \end{array}\right]

\begin{align} {\mathbf{A}_e}{\mathbf{A}_e}^{-1} &= \left[\begin{array}{ccc} 1&0&1\\ 3&1&0\\ 1&0&2 \end{array}\right]\cdot \left[\begin{array}{ccc} 2&0&-1\\ -6&1&3\\ -1&0&1 \end{array}\right]\\&= \left[\begin{array}{ccc} 1\cdot2+0\cdot(-6)+1\cdot(-1) & 1\cdot0+0\cdot1+1\cdot0 & 1\cdot(-1)+0\cdot3+1\cdot1\\ 3\cdot2+1\cdot(-6)+0\cdot(-1) & 3\cdot0+1\cdot1+0\cdot0 & 3\cdot(-1)+1\cdot3+0\cdot1\\ 1\cdot2+0\cdot(-6)+2\cdot(-1) & 1\cdot0+0\cdot1+2\cdot0 & 1\cdot(-1)+0\cdot3+2\cdot1 \end{array}\right]\\&= \left[\begin{array}{ccc} 2+0+1-1 & 0+0+0 & -1+0+1\\ 6-6+0 & 0+1+0 & -3+3+0\\ 2+0-2 & 0+0+0 & -1+0+2 \end{array}\right]\\&= \left[\begin{array}{ccc} {\color{Green}\mathbf{1}} & 0 & 0\\ 0 & {\color{Green}\mathbf{1}} & 0\\ 0 & 0 & {\color{Green}\mathbf{1}} \end{array}\right]= \mathbf{I}_3 \end{align}
Example: inverse of a 4-by-4 matrix

This example is a proof of equation 3.40 in the robotics script (see page 3-61):

^R\mathbf{T}_N = \left[\begin{array}{cccc} 0 & 1 & 0 & 2a\\ 0 & 0 & -1 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right] ,\quad {^R\mathbf{T}_N}^{-1} = \left[\begin{array}{cccc} 0 & 0 & -1 & 0\\ 1 & 0 & 0 & -2a\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]

\begin{align} {^R\mathbf{T}_N}{^R\mathbf{T}_N}^{-1} &= \left[\begin{array}{cccc} 0 & 1 & 0 & 2a\\ 0 & 0 & -1 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]\cdot \left[\begin{array}{cccc} 0 & 0 & -1 & 0\\ 1 & 0 & 0 & -2a\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]\\&= \left[\begin{array}{cccc} 0\cdot0+1\cdot1+0\cdot0+2a\cdot0 & 0\cdot0+1\cdot0+0\cdot(-1)+2a\cdot0 & 0\cdot(-1)+1\cdot0+0\cdot0+2a\cdot0 & 0\cdot0+1\cdot(-2a)+0\cdot0+2a\cdot1\\ 0\cdot0+0\cdot1+(-1)\cdot0+0\cdot0 & 0\cdot0+0\cdot0+(-1)\cdot(-1)+0\cdot0 & 0\cdot(-1)+0\cdot0+(-1)\cdot0+0\cdot0 & 0\cdot0+0\cdot(-2a)+(-1)\cdot0+0\cdot1\\ (-1)\cdot0+0\cdot1+0\cdot0+0\cdot0 & (-1)\cdot0+0\cdot0+0\cdot(-1)+0\cdot0 & (-1)\cdot(-1)+0\cdot0+0\cdot0+0\cdot0 & (-1)\cdot0+0\cdot(-2a)+0\cdot0+0\cdot1\\ 0\cdot0+0\cdot1+0\cdot0+1\cdot0 & 0\cdot0+0\cdot0+0\cdot(-1)+1\cdot0 & 0\cdot(-1)+0\cdot0+0\cdot0+1\cdot0 & 0\cdot0+0\cdot(-2a)+0\cdot0+1\cdot1\\ \end{array}\right]\\&= \left[\begin{array}{cccc} {\color{Green}\mathbf{1}} & 0 & 0 & 0\\ 0 & {\color{Green}\mathbf{1}} & 0 & 0\\ 0 & 0 & {\color{Green}\mathbf{1}} & 0\\ 0 & 0 & 0 & {\color{Green}\mathbf{1}} \end{array}\right]= \mathbf{I}_4 \end{align}
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