The inverse of a matrix A is a matrix that, when multiplied by A results in the identity. When working with numbers such as 3 or –5, there is a number called the multiplicative inverse that you can multiply each of these by to get the identity 1. In the case of 3, that inverse is 1/3, and in the case of –5, it is –1/5.

When two block matrices have the same shape and their diagonal blocks are square matrices, then they multiply similarly to matrix multiplication. For example, (7) Note that the usual rules of matrix multiplication hold even when the block matrices are not square (assuming that the block sizes correspond).

Block matrix multiplication

1. It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors.
2. and a matrix with row partitions and column partitions.
3. can be formed blockwise, yielding as an matrix with row partitions and column partitions.

Notice that the inverse of a block diagonal matrix is also block diagonal. Similarly, the inverse of a block secondary diagonal matrix is block secondary diagonal too, but in transposed partition so that there is a switch between B and C.

Block lower triangular matrix. In particular, all fill-in is confined to the blocks on the diagonal. Any row and column interchanges needed for the sake of stability and sparsity may be performed within the blocks on the diagonal and do not affect the block triangular structure.

If a 2×2 matrix A is invertible and is multiplied by its inverse (denoted by the symbol A−1), the resulting product is the Identity matrix which is denoted by I. To illustrate this concept, see the diagram below.

The only requirement is that the blocks be compatible. That is, the sizes of the blocks must be such that all matrix products of blocks that occur make sense. This means that the number of columns in each block of must equal the number of rows in the corresponding block of .

The inverse formula (1.1) of a 2 x 2 block matrix appears frequently in many subjects and has long been studied. Its inverse in terms of A -1 or D -1 can be found in standard textbooks on linear algebra, e.g., [1-3]. However, we give a complete treatment here.

This notation is particularly useful when we are multiplying the matrices A and B because the product A B can be computed in block form as follows: A B = [ I O P Q] [ X Y] = [ I X + O Y P X + Q Y] = [ X P X + Q Y] = [ 4 − 2 5 6 30 8 8 27] This is easily checked to be the product A B, computed in the conventional manner.

Real Jordan normal form. If a real matrix has multiple complex eigenvalues and is defective, then its Jordan form can be replaced with an upper block diagonal matrix in a way similar to the diagonal case illus- trated in §2.13.2, by replacing the generalized eigenvectors with their real and imaginary parts.