It’s often useful to partition a matrix, thinking of it as a matrix whose entries are matrices. For example, you could look at the matrix 6 × 6

M = begin{bmatrix} 3 & 1 & 4 & 0 & 0 & 0 \ 2 & 7 & 1 & 0 & 0 & 0 \ 5 & 0 & 6 & 0 & 0 & 0 \ 0 & 0 & 0 & 2 & 4 & 8 \ 0 & 0 & 0 & 1 & 7 & 6\ 0 & 0 & 0 & 3 & 5 & 1 \ end{bmatrix}

as a 2 × 2 matrix whose entries are 3 × 3 matrices.

begin{bmatrix} begin{bmatrix} 3 & 1 & 4 \ 2 & 7 & 1 \ 5 & 0 & 6 \ end{bmatrix} & begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \ end{bmatrix} \ & \ begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \ end{bmatrix} & begin{bmatrix} 2 & 4 & 8 \ 1 & 7 & 6\ 3 & 5 & 1 \ end{bmatrix} end{bmatrix}

M is not a diagonal matrix of real numbers, but its block structure is a diagonal matrix of blocks. It might be possible in some application to take advantage of the fact that the off-diagonal blocks are zero matrices.

You can multiply two partitioned matrices by multiplying the blocks as if they were numbers. The only requirement is that the blocks be of compatible sizes for multiplication: the block widths on the left have to match the corresponding block heights on the right.

For example, if you have two partitioned matrices

A = begin{bmatrix} A_{11} & A_{12} \ A_{21} & A_{22} end{bmatrix}
and
B = begin{bmatrix} B_{11} & B_{12} \ B_{21} & B_{22} end{bmatrix}

then their product is the partitioned matrix

C = begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} end{bmatrix}

The only requirement is that the matrix products above make sense. The number of columns in A11 must equal the number of rows in B11, and the number of columns in A12 must equal the number of rows in B21. (The rest of the compatibility conditions follow because the blocks in a column of a partitioned matrix have the same number of columns, and the blocks in a row of a partitioned matrix have the same number of rows.)

You can partition matrices into any number of rows and columns of blocks; the examples above have been 2 × 2 arrays of blocks only for convenience.

It’s natural to oscillate a bit between thinking “Of course you can do that” and “Really? That’s incredible.” If you haven’t gone through at least one cycle perhaps you haven’t thought about it long enough.

What could go wrong?

Manipulation of partitioned matrices works so smoothly that you have to wonder where things could go wrong.

For the most part, if a calculation makes logical sense then it’s likely to be correct. If you add two blocks, they have to be compatible for addition. If you “divide” by a block, i.e, multiply by its inverse, then the block must have an inverse. Etc.

One subtle pitfall is that matrix multiplication does not commute in general, and so you have to watch out for implicitly assuming that AB = BA where A and B are blocks. For example, theorems about the determinant of a partitioned matrix are not as simple as you might hope. The determinant of the partitioned matrix

begin{vmatrix} A & B \ C & D end{vmatrix}

is not simply det(AD − BC) in general, though it is true if all the blocks are square matrices of the same size and the blocks in one row or one column commute, i.e. if AB = BA, CD = DC, AC = CA, or BD = DB.

More linear algebra posts

The post Matrices of Matrices first appeared on John D. Cook.

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