Linear Algebra and Its Applications, Exercise 2.5.14

Exercise 2.5.14. Suppose we have the block matrices

$\begin{bmatrix} A&B \\ C&D \end{bmatrix} \begin{bmatrix} E&F \\ G&H \end{bmatrix}$

with the following properties:

• $A$ and $B$ have $m_1$ rows
• $C$ and $D$ have $m_2$ rows
• $E$ and $F$ have $n_1$ rows
• $G$ and $H$ have $n_2$ rows
• $E$ and $H$ are square matrices

What is the product of the two matrices, and what are the sizes of each of the product’s component submatrices?

Answer: Since $E$ is a square matrix with $n_1$ rows it must have $n_1$ columns. If $E$ has $n_1$ columns then $G$ must have $n_1$ columns also. Similarly, since $H$ is a square matrix with $n_2$ rows it must have $n_2$ columns, and $F$ must have $n_2$ columns as well.

In computing the (1, 1) element of  the product matrix we multiply $A$ by $E$ and $B$ by $G$. (See the post on multiplying block matrices.) This means that the number of columns of $A$ must equal the number of rows of $E$ or $n_1$, and the number of columns of $B$ must equal the number of rows of $G$ or $n_2$. The number of columns of $C$ and $D$ must then be $n_1$ and $n_2$ respectively.

The sizes of each of the elements $A$ through $H$ is thus as follows:

• $A$ is $m_1$ by $n_1$
• $B$ is $m_1$ by $n_2$
• $C$ is $m_2$ by $n_1$
• $D$ is $m_2$ by $n_2$
• $E$ is $n_1$ by $n_1$
• $F$ is $n_1$ by $n_2$
• $G$ is $n_2$ by $n_1$
• $H$ is $n_2$ by $n_2$

The product matrix is thus

$\begin{bmatrix} A&B \\ C&D \end{bmatrix} \begin{bmatrix} E&F \\ G&H \end{bmatrix} = \begin{bmatrix} AE+BG&AF+BH \\ CE+DG&CF+DH \end{bmatrix}$

with the size of each element as follows:

• $AE+BG$ is $m_1$ by $n_1$
• $AF+BH$ is $m_1$ by $n_2$
• $CE+DG$ is $m_2$ by $n_1$
• $CF+DH$ is $m_2$ by $n_2$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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