Dimension Rules For Block Matrix Products

Two Dimension Checks for a Block Product

Partitioning a matrix into blocks lets us treat rectangular groups of entries as units. But the blocks in two factors must fit together in a specific way before they can be multiplied block by block. There are two checks to keep separate: whether the full matrices can be multiplied at all, and whether the chosen block partitions line up for block multiplication.

Throughout, the matrices have real entries. A block is itself a matrix, so its size matters: a block with \(r\) rows and \(s\) columns is an \(r\times s\) matrix. Recall that an ordinary product \(\mathbf{A}\mathbf{B}\) is defined when the number of columns of \(\mathbf{A}\) equals the number of rows of \(\mathbf{B}\). Block multiplication imposes an additional requirement: the column widths of the blocks in \(\mathbf{A}\) must match the row heights of the blocks in \(\mathbf{B}\), in order.

The condition \(k=q\) says the two inner partitions have the same number of blocks. The conditions \(s_j=t_j\) say that the corresponding block dimensions match, not merely that the totals match. If the partitions are compatible, every block product used in forming the result has a defined matrix product. The output block in position \((i,\ell)\) has \(r_i\) rows and \(u_\ell\) columns.

A Dimension Rule for the Output Layout

Proof. Since the block column widths of \(\mathbf{A}\) are \(s_1,\dots,s_k\), its total number of columns is \(s_1+\cdots+s_k\). The block row heights of \(\mathbf{B}\) are \(t_1,\dots,t_k\), so its total number of rows is \(t_1+\cdots+t_k\). Compatibility gives \(s_j=t_j\) for each \(j\), and therefore the two totals are equal. Thus the ordinary product \(\mathbf{A}\mathbf{B}\) is defined. In that product, rows are grouped according to the row groups of \(\mathbf{A}\), so block row \(i\) has height \(r_i\). Columns are grouped according to the column groups of \(\mathbf{B}\), so block column \(\ell\) has width \(u_\ell\). Their intersection, the output block in position \((i,\ell)\), consequently has \(r_i\) rows and \(u_\ell\) columns. \(\square\)

This theorem gives the layout without requiring any entries of the product to be calculated. It also explains the central bookkeeping rule: the outer block partitions determine the output. The row partition comes from the first factor, and the column partition comes from the second.

When the Full Product Exists but the Blocks Do Not Line Up

An ordinary product can be defined even when the given block partitions are not compatible. In that situation, the obstacle is not the overall matrix dimensions. It is that the chosen internal boundaries do not match. Sometimes the partitions can be refined—that is, split into smaller blocks—until the boundaries agree.

Proof. Let the shared inner dimension be \(N\). Describe each partition by its boundary positions among \(0,1,\dots,N\), including \(0\) and \(N\). For example, block sizes \((2,3)\) give boundaries \(0,2,5\). Take the union of the boundary positions from the two partitions and list its distinct elements in increasing order. Consecutive positions in this list define positive block sizes, since distinct consecutive positions differ by at least \(1\). Every new interval lies within an original block of each partition: no original boundary lies strictly inside one of the intervals, because all original boundaries were included in the union. Thus each original block can be split at the new boundaries, and both partitions become the same sequence of interval lengths. These common block sizes give the required refinement. \(\square\)

Compatibility Is More Than Matching Block Counts

Having the same number of inner blocks is necessary, but not sufficient. The dimensions must match in corresponding positions. Conversely, when two inner partitions have the same total size, differing boundaries can be repaired by a common refinement; the refinement does not change either full matrix or the ordinary product.

A Common Mistake and a Reliable Check

Common mistake: checking only the number of inner blocks. For the partitions \((1,3)\) and \((2,2)\), a wrong calculation might say, “There are two blocks on each side, so the block product is defined.” That conclusion ignores the dimensions of the paired blocks: the first attempted block product would have inner dimensions \(1\) and \(2\), and the second would have inner dimensions \(3\) and \(2\). Neither pair matches. The correction is to compare each width with the height in the same position. If they differ but the total inner dimensions agree, find the union of boundary positions and refine before applying block multiplication.

For any proposed block product, use this order of checks:

1. Compare the total number of columns of the first matrix with the total number of rows of the second. If they differ, the ordinary product is not defined.
2. Compare the number of inner blocks. If the counts differ, the displayed partitions are not compatible.
3. Compare each corresponding block width and block height. Every pair must be equal.
4. Read the output layout from the outer partitions: first-factor block row heights, second-factor block column widths.

This separation prevents a common confusion. A failure of block compatibility does not by itself imply that the ordinary matrix product fails to exist. If the total inner dimensions agree, the ordinary product is defined; the given block divisions may simply need to be refined to make blockwise multiplication available.