Matrix Inverses

Square full-rank matrices and their inverse
Full column rank matrices and left inverses
Full row rank matrices and right inverses

Square full rank matrices and their inverse

A square

matrix is said to be invertible if and only if its columns are independent. This is equivalent to the fact that its rows are independent as well. An equivalent definition states that a matrix is invertible if and only if its determinant is non-zero.

For invertible n times n

matrices

, there exist a unique matrix

such that

. The matrix

is denoted $A^{-1}$ and is called the inverse of

If a matrix

is square, invertible, and triangular, we can compute its inverse $R^{-1}$ simply, as follows. We solve

linear equations of the form Rx_i = e_i

, with

the

-th column of the n times n

identity matrix, using a process known as backwards substitution. Here is an example. At the outset we form the matrix $R^{-1} = [x_1,ldots,x_n]$ . By construction, $R cdot R^{-1} = I_n$ .

For general square, invertible matrices

, the QR decomposition allows to compute the inverse. For such matrices, the QR decomposition is of the form A = QR

, with

orthogonal matrix, and

is upper triangular. Then the inverse is $A^{-1} = R^{-1}Q^T$ .

Matlab syntax

Ainv = inv(A); % produces the inverse of a square, invertible matrix

A useful property is the expression of the inverse of a product of two square, invertible matrices A,B

: $(AB)^{-1} = B^{-1}A^{-1}$ . (Indeed, you can check that this inverse works.)

Full column rank matrices and left inverses

matrix is said to be full column rank if its columns are independent. This necessarily implies m ge n

A matrix

has full column rank if and only if there exist a n times m

matrix

such that

(here

is the small dimension). We say that

is a left-inverse of

. To find one left inverse of a matrix with independent columns

, we use the full QR decomposition of

to write

where

upper triangular and invertible, while

and orthogonal ( Q^TQ = I_m

). We can then set a left inverse

to be

$B = left( begin{array}{cc}R_1^{-1}&0 end{array}right) Q^T.$

Full row rank matrices and right inverses

matrix is said to be full row rank if its rows are independent. This necessarily implies m le n

A matrix

has full row rank if and only if there exist a n times m

matrix

such that

(here

is the small dimension). We say that

is a right-inverse of

. We can derive expressions of right inverses by noting that

is full row rank if and only if A^T

is full column rank. In particular, for a matrix with independent rows, the full QR decomposition (of A^T

) allows to write

$A = left( begin{array}{cc} R_1^T & 0 end{array}right)Q^T,$

where

upper triangular and invertible, while

and orthogonal ( Q^TQ = I_n

). We can then set a right inverse of

to be

$B = Qleft( begin{array}{c} R_1^{-1} 0 end{array}right).$