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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