Linear Algebra & Linear Systems & Symmetric Matrices

Question

Let A = [1 -1 0 0 1 1 1 0 -1] 3x3 matrix. Find all symmetric matrices B such that AB is a symmetric matrix. How do I do this question I know B is same as it’s transpose

Answer 1

I assume here that A is the following 3 x 3 matrix:

1 -1 0

A = 0 1 1

1 0 -1

The fact that B is symmetric means that it is of the following form:

x₁ x₄ x₅

B = x₄ x₂ x₆

x₅ x₆ x₃

Here, I used x₁, x₂, and x₃ to denote the diagonal elements of B, and I used x₄, x₅, and x₆ to denote the upper (and lower) triangular elements of B (which are equal because B is symmetric).

We want to solve the system of equations:

(A·B)^T = A·B.

You want to write this matrix equation out in terms of x₁, ···, x₆ and equate upper triangular elements with the corresponding lower triangular elements. You will get three equations in six unknowns. (Just the upper triangular or lower triangular elements matter. The diagonal elements of a matrix are always equal to the diagonal elements of its transpose, so you don't get any conditions on the variables. You will get equations like x₁ - x₄ = x₁ - x₄, etc., which reduce to 0 = 0).

You can solve the super-diagonal elements of B in terms of the diagonal elements to get the following linear systems of equations.

x₄ = -2 x₁ - 4 x₂ + x₃

x₅ = -x₂ + x₃

x₆ = -x₁ - 2 x₂

We see that any choice of x₁, x₂, and x₃ will lead to a solution to this problem, so the solution set is a three-dimensional vector subspace of the space of 3 x 3 symmetric matrices parametrized by x₁, x₂, and x₃.