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

Using dot product(missing in the answer by ZACH), basically the same, but it may not be confusing.

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 1

Diagonalization of Symmetric Matrices

Introduction to Matrices

Missouri State University

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

02:02

Show that if $A$ is an $n …

01:31

Suppose that $\mathbf{A}$ …

02:26

Suppose $A$ is a symmetric…

02:11

An $n \times n$ matrix $\m…

01:13

Use properties of the …

03:10

Let $\mathbf{A}$ be an $m …

11:46

(a) Let $A$ be an $n \time…

02:50

Let $A$ be an $n \times n$…

03:42

Show that $A=B+C .$ Togeth…

Use properties of the inve…

Okay. The problems asking us if Matrix in this case is symmetric and is a skew symmetric. Did it happen then? It has to be a zero matrix or a no matrix. So I wrote the definition of each one, have a matrix very symmetric and is a skew symmetric. And what that really means isn't Here is a off i j is equal to a off J I Okay, so that's on this port on this board, we have a of i j equal to minus a off J I. So what I'd like to do here is multiply this by minus, So I get a minus a i j equal to a j. I basically multiply this whole thing. My minus one. Okay. And when I do that, as you can see, I have to equation that one side is equal to a J I. So if you have to equation, one side is equal. What I can do is I go to a different color now, have a my J equal to minus a Hi, Jay. And if I bring the minus on the other side, I get to a I J equal to zero. So what that means is that a I j has to be zero. So what that defines proves it is that this If that's the case that you have both is symmetric and skew symmetric, then all the elements off that matrix has to be zero. And that is a definition off zero matrix. Thank you.

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