A 4 by 5 matrix can be considered as a 2 by 2 matrix
|[||1||2||3||4||5||] = [||]|
with the matrices
|A = [||1||2||3||], B = [||4||5||], C = [||], D = [||]|
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||]|
One may perform the usual operations with partitioned matrices as usual, as long as we keep the following in mind:
|[||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||]|
|M = [||A||C||]|
is invertible. Assume
|M-1 = [||X||Y||]|
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||]|
The partitioned matrix also has interpretation as the linear transformation between direct sums of euclidean spaces.