Question

Show that if A is an \(n \times n\) symmetric matrix, then \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\) for x, y in \({\mathbb{R}^n}\).

Short answer

The equation \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\) is true.

Step-by-step solution

Use the property of symmetric matrix for vector u and v

Since A is a symmetric matrix, so \({A^T} = A\).

If the vectors u and v are in \({\mathbb{R}^n}\), then \({\bf{u}} \cdot {\bf{v}} = {{\bf{u}}^T}{\bf{v}}\).

Prove the equation \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\)

Solve the expression \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}}\).

\(\begin{aligned}{}\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\left( {A{\bf{x}}} \right)^T}{\bf{y}}\\ = \left( {{{\bf{x}}^T}{A^T}} \right){\bf{y}}\\ = {{\bf{x}}^T}\left( {A{\bf{y}}} \right)\\ = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\end{aligned}\)

So, the equation \(\left( {A{\bf{x}}} \right) \cdot {\bf{y}} = {\bf{x}} \cdot \left( {A{\bf{y}}} \right)\) is true.