1 Matrix Inverse
The inverse of an matrix is denoted by . The inverse is defined so that
where is the identity matrix.
It should be stressed that only square matrices have inverses proper– however, a matrix of any size may have “left” and “right” inverses (which will not be discussed here).
The general form of the inverse of a matrix is
where is the adjugate of (the matrix formed by the cofactors of , i.e. with ).11Some other sources call the adjugate the adjoint; however on PM the adjoint is reserved for the conjugate transpose. This can also be thought of as a generalization of the formula given in the next section. However, due to the inclusion of the determinant in the expression, it is impractical to actually use this to calculate inverses.
This general form also explains why the determinant must be nonzero for invertibility; as we are dividing through by its value.
An invertible matrix is also said to be nonsingular.
2 Calculating the Inverse By Hand
An easy way to calculate the inverse of a matrix by hand is to form an augmented matrix from and , then use Gaussian elimination to transform the left half into . At the end of this procedure, the right half of the augmented matrix will be (that is, you will be left with ).
One can calculate the th element of the inverse by using the general formula; i.e.
where is the th cofactor expansion of the matrix .
Note that the indices on the left-hand side are swapped relative to the right-hand side.
2.1 2-by-2 case:
For the case, the general formula reduces to a memorable shortcut. For the matrix
The inverse is always
where is simply .
3 Avoiding the Inverse and Numerical Calculation
The need to find the matrix inverse depends on the situation– whether done by hand or by computer, and whether the matrix is simply a part of some equation or expression or not.
Instead of computing the matrix as part of an equation or expression, it is nearly always better to use a matrix factorization instead. For example, when solving the system , actually calculating to get is discouraged. LU-factorization is typically used instead.
We can even use this fact to speed up our calculation of the inverse by itself. We can cast the problem as finding in
For matrices , , and (where and ). To solve this, we first find the decomposition of , then iterate over the columns, solving and each time (). The resulting values for are then the columns of .
4 Elements of Invertible Matrices
- 1 Golub and Van Loan, “Matrix Computations,” Johns Hopkins Univ. Press, 1996.
- 2 “Matrix Math,” http://easyweb.easynet.co.uk/ mrmeanie/matrix/matrices.htmhttp://easyweb.easynet.co.uk/ mrmeanie/matrix/matrices.htm
|Date of creation||2013-03-22 13:04:25|
|Last modified on||2013-03-22 13:04:25|
|Last modified by||akrowne (2)|
|Defines||adjugate of a matrix|