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Q27E

Expert-verifiedFound in: Page 395

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.

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

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

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.

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