Exercise 1.6.15. For any square matrix B and matrices A and K where

prove that A is symmetric and K is skew-symmetric, i.e.,

For the case where

compute A and K and show that B can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Answer: Let . For all elements of A we then have

so that

and A is a symmetric matrix.

For all elements of K we have

so that

and K is a skew-symmetric matrix.

We then have

so that

and

Note that

so that

In our example we then have

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.