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Let $A$ be an $n \times n$ invertible symmetric matrix. Show that if the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ is positive definite, then so is the quadratic form $\mathbf{x}^{T} A^{-1} \mathbf{x} .[\text { Hint: Consider eigenvalues. }]$

Thus, the eigenvalues of are all positive, and $A^{-1}$ is symmetric.Therefore, its quadratic form, is positive definite.

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

Missouri State University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

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01:25

Let $A$ and $B$ be symmetr…

01:31

Suppose that $\mathbf{A}$ …

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Let $A$ be an invertible m…

02:26

Suppose $A$ is a symmetric…

07:34

Deal with the eigenvalue/e…

11:46

(a) Let $A$ be an $n \time…

01:30

Let $A$ be an $n \times n$…

03:26

Let $A$ be an $m \times n$…

03:42

Use properties of the inve…

01:13

Use properties of the …

okay for a problem in 28 were given a matrix eight that is in my inn and symmetric and we want to show Oh, also the Matrix A is in vertical asymmetric and the inverse exists Inverse exists Now we want to show the quadratic form off. The inverse inverse matrix is also positive. Definite. If the key then we're ready. Form off the matrix. A is positive. Definite definite. All right, um two. To prove this to prove this statement, we have two parts too. Take care of. First part is to show that the inverse of a is symmetric, right? Because we if the if the inverse matrix is an awesome it's not symmetric, then we don't have any. We don't have any like way. Don't we can't. We cannot say that this court radical form off this off, Nancy Metric Matrix is positive. Definite right. We have to required this matrix to be symmetrical. Then we can talk about a positive definite or not. And the second, the second part we need to take care of is just a positive, definite thing To show X transpose times a times acts is positive. Definite definite. Okay, that's first and catch the first The first part. So first recall if we have you weigh have a convertible matrix. We always have this equality at this a inverse transpose is equal to hey transpose and in verse. And also we're given the Matrix A which he says mentally matrix And this is transpose is also equal to a so we have also a inverse. So these tells us that the inverse matrix he's symmetric because the transports off this inverse matrix is equal to the matrix itself. So symmetric and we are down for for sparks. Now, the second part, the thing we're looking to is the quadratic form off the inverse. Sorry, we missed negative. Why here? So we that the court uniform we need to consider is tthe e is for the members Inverse matrix. That's first, um, as first lookat the matrix A. Since this matrix ihsaa positive definite data means every single I can value is us too, right? This by our dear, I'm giving the textbook. Now if we consider the inverse matrix recall that the Eiken battery is just won over Lum di since London, guys always part of so one of alumni's definitely positive. So that means every single again battery off this inverse matrix is positive ends the inverse of a hiss. Positive. Definite, Definite. So we're done.

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