Block Matrix¶

Definition¶

A block matrix is a matrix viewed as being partitioned into smaller matrices called blocks or submatrices.

A block diagonal matrix is a square block matrix of the form

\begin{pmatrix} \boldsymbol{A}_1&\boldsymbol{0}&\cdots&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{A}_2&\cdots&\boldsymbol{0}\\ \vdots&\vdots&\ddots&\vdots\\ \boldsymbol{0}&\boldsymbol{0}&\cdots&\boldsymbol{A}_k \end{pmatrix},

where $\boldsymbol{A}_1,\ldots,\boldsymbol{A}_k$ are square matrices.

Interpretation¶

Block notation treats a large matrix as a small matrix whose entries are themselves matrices. As long as the sizes fit, calculations can then be done with the blocks as if they were numbers, which keeps the structure of the matrix visible.

A 3 by 5 matrix divided by one horizontal and one vertical line into four blocks, shown equal to a 2 by 2 matrix of blocks A11, A12, A21 and A22.

Cutting the $3\times5$ matrix after row $2$ and after column $2$ gives four blocks: $\boldsymbol{A}_{11}$ is $2\times2$ $2$ , $\boldsymbol{A}_{12}$ is $2\times3$ $3$ , $\boldsymbol{A}_{21}$ is $1\times2$ and $\boldsymbol{A}_{22}$ is $1\times3$ $3$ . Blocks in the same block row have the same number of rows, and blocks in the same block column have the same number of columns.

When the column partition of $\boldsymbol{A}$ matches the row partition of $\boldsymbol{B}$ , block multiplication uses the same row-by-column rule as matrix multiplication; for example, the top-left block of $\boldsymbol{A}\boldsymbol{B}$ is $\boldsymbol{A}_{11}\boldsymbol{B}_{11}+\boldsymbol{A}_{12}\boldsymbol{B}_{21}$ $A_{12} B_{21}$ . The order of the factors must be kept, since the blocks are matrices and $\boldsymbol{A}_{11}\boldsymbol{B}_{11}$ differs in general from $\boldsymbol{B}_{11}\boldsymbol{A}_{11}$ .

Block diagonal matrices describe parts that do not interact. If $\boldsymbol{A}$ is block diagonal with blocks $\boldsymbol{A}_1,\boldsymbol{A}_2$ and $\boldsymbol{x}$ is split accordingly into $\boldsymbol{x}_1,\boldsymbol{x}_2$ , then $\boldsymbol{A}\boldsymbol{x}$ consists of $\boldsymbol{A}_1\boldsymbol{x}_1$ and $\boldsymbol{A}_2\boldsymbol{x}_2$ , so the system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ splits into two smaller systems. Powers, and inverses when every block is invertible, are likewise computed block by block, and the determinant is the product of the blocks' determinants. A diagonal matrix is the special case with $1\times1$ blocks.

Key Results¶

Compatible block matrices can be added and multiplied block-by-block using the usual matrix rules.
If $\boldsymbol{A}$ is block diagonal with diagonal blocks $\boldsymbol{A}_1,\ldots,\boldsymbol{A}_k$ , then

\det(\boldsymbol{A})=\det(\boldsymbol{A}_1)\det(\boldsymbol{A}_2)\cdots\det(\boldsymbol{A}_k).

Examples¶

$\begin{pmatrix}\boldsymbol{A}_{11}&\boldsymbol{A}_{12}\\\boldsymbol{A}_{21}&\boldsymbol{A}_{22}\end{pmatrix}$ is a matrix written in four blocks.
$\begin{pmatrix}\boldsymbol{A}_1&\boldsymbol{0}\\\boldsymbol{0}&\boldsymbol{A}_2\end{pmatrix}$ is block diagonal.

Prerequisites¶

matrix

Sources¶

Math II: Linear Algebra · Week 5
Math V: Advanced Linear Algebra · Week 6 (revisited)

Aliases: partitioned matrix, block matrices, submatrices, block diagonal matrices Keywords: submatrix, block diagonal matrix

Last updated: 20 October 2026 03:02