### Partitioned Matrix

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 = [a1 a2 ... an] is partitioned into n m by 1 matrices.

Example The (m+n) by (m+n) identity matrix may be partitioned as follows

 Im+n = [ IM On×m ] Om×n In

One may perform the usual operations with partitioned matrices as usual, as long as we keep the following in mind:

1. sizes must match,
2. matrices must be multiplied in the correct order.

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 In O In 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-1CB-1. Thus we conclude that

M is invertible ⇔ A and B are invertible

Moreover, the inverse is given by

 M-1 = [ A-1 - A-1CB-1 ] O B-1

The partitioned matrix also has interpretation as the linear transformation between direct sums of euclidean spaces.