Difference between revisions of "Matrix inversion"
| Line 56: | Line 56: | ||
\end{array}\right]\\&= | \end{array}\right]\\&= | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} | ||
| − | 1 & 0 & 0 & 0\\ | + | {\color{Green}\mathbf{1}} & 0 & 0 & 0\\ |
| − | 0 & 1 & 0 & 0\\ | + | 0 & {\color{Green}\mathbf{1}} & 0 & 0\\ |
| − | 0 & 0 & 1 & 0\\ | + | 0 & 0 & {\color{Green}\mathbf{1}} & 0\\ |
| − | 0 & 0 & 0 & 1 | + | 0 & 0 & 0 & {\color{Green}\mathbf{1}} |
\end{array}\right]= | \end{array}\right]= | ||
\mathbf{I}_n | \mathbf{I}_n | ||
Revision as of 13:37, 22 May 2014
| ← Back: Determinant of a matrix | Overview: Matrix inversion | Next: Gauß-Jordan-Algorithm → |
The inverse of an n-by-n square matrix 
is denoted as 
and defined such that
where 
is the n-by-n identity matrix.
Prerequesite for the inversion is, that is an n-by-n square matrix and that is regular. Regular means that the row and column vectors are linearly independent and so the determinant is nonzero:
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.
 Example: inverse of a 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]](/wiki/robotics/images/math/2/6/9/26972fd789b662c263fb7ac640f9d040.png)
![\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}_n
\end{align}](/wiki/robotics/images/math/e/0/7/e07991c0592c85afa6e71877cea8b9d8.png)
