A 4 by 5 matrix can be considered as a 2 by 2 matrix
[ | 1 | 2 | 3 | 4 | 5 | ] = [ | ] | ||
6 | 7 | 8 | 9 | 10 | A | B | |||
11 | 12 | 13 | 14 | 15 | C | D | |||
16 | 17 | 18 | 19 | 20 |
with the matrices
A = [ | 1 | 2 | 3 | ], B = [ | 4 | 5 | ], C = [ | ], D = [ | ] | |||||
6 | 7 | 8 | 9 | 10 | 16 | 17 | 18 | 19 | 20 | |||||
11 | 12 | 13 | 14 | 15 |
as the entries. This way of dividing a matrix into compatible smaller matrices is a partition of the matrix.
We have partitioned matrices before. For example, for a system of m equations in n variables, the augmented matrix [A b] is partitioned into an m by n matrix A and an m by 1 matrix b. Moreover, when we think of the columns, a matrix A = [a_{1} a_{2} ... a_{n}] is partitioned into n m by 1 matrices.
Example The (m+n) by (m+n) identity matrix may be partitioned as follows
I_{m+n} = [ | IM | O_{n×m} | ] |
O_{m×n} | I_{n} |
One may perform the usual operations with partitioned matrices as usual, as long as we keep the following in mind:
Example For any two m by n matrices X and Y, we have
[ | IM | X | ] [ | IM | Y | ] = [ | II + XO | IY + XI | ] = [ | I | X + Y | ] |
O | I_{n} | O | I_{n} | OI + IO | OY + II | O | I |
In particular, we conclude that
[ | I | X | ]^{-1} = [ | I | - X | ] |
O | I | O | I |
Example Let A and B be square matrices, we would like to find out when
M = [ | A | C | ] |
O | B |
is invertible. Assume
M^{-1} = [ | X | Y | ] |
Z | W |
Then MM^{-1} = I means
AX + CZ = I, AY + CW = O, BZ = O, BW = I.
Since B is a square matrix, the equation BW = I means that B is invertible, and W = B^{-1}. Then by multiplying B^{-1} to the left of BZ = O, we find Z = O. Substituting Z = O into AX + CZ = I, we get AX = I. Since A is a square matrix, this means that A is invertible, and X = A^{-1}. Then substituting W = B^{-1} into AY + CW = O and solve the equation, we get Y = - A^{-1}CB^{-1}. Thus we conclude that
M is invertible ⇔ A and B are invertible
Moreover, the inverse is given by
M^{-1} = [ | A^{-1} | - A^{-1}CB^{-1} | ] |
O | B^{-1} |
The partitioned matrix also has interpretation as the linear transformation between direct sums of euclidean spaces.