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Found in: Page 395

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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 $${\mathbb{R}^n}$$.

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

See the step by step solution

Step 1: 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 $${\bf{u}} \cdot {\bf{v}} = {{\bf{u}}^T}{\bf{v}}$$.

Step 2: 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.