Short Answer

If A is a positive definite $n \times n$ matrix, and R is any real $n \times n$ matrix, what can you say about the definiteness of the matrix $R^{T} AR$ ? For which matrices R is $R^{T} AR$ positive definite?

$R^{T} A R$ will be positive definite for all R with $k e r (R) = \{0\} .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1 0f 2: Given information

Let $\vec{x} \in ℝ^{m}$ .
Let us consider for symmetric matrix A of order n and any real matrix

$R \in ℝ^{n \times m}, {\vec{x}}^{T} R^{T} A R \vec{x} = ({\vec{x}}^{T} R^{T}) (A) (R \vec{x}) = {(R \vec{x})}^{T} (A) (R \vec{x}) = {\vec{y}}^{T} A \vec{y}$

Here, $\vec{y} = R \vec{x}$

Step 2 0f 2: Application

Now, for $\vec{x} \in k e r (R), R \vec{x} = \vec{0} = \vec{y}$ and ${\vec{y}}^{T} A \vec{y} = 0$ thus . For $\vec{x} \notin k e r (R)$ and $\vec{x} \neq 0, \vec{y} = R \vec{x} \neq \vec{0}$ .
As A is a positive defining matrix, and therefore ${\vec{y}}^{T} A \vec{y} \geq 0 .$
Thus for $\vec{x} \in ℝ^{m}$ , we get ${\vec{x}}^{T} R^{T} A R \vec{x} \geq 0$ where $R^{T} A R$ is positive semi-definite for any $R \in ℝ^{n \times m}$ and symmetric positive definite matrix A .
When all $\vec{x} \in ℝ^{m}, \vec{x} \neq 0, {\vec{x}}^{T} R^{T} A R \vec{x} > 0$ that is, no $\vec{x} \neq 0$ is in $k e r (R)$ then $R^{T} A R$ will be positive definite and $k e r (R) = \{0\}$ .
Thus, for all R with $k e r (R) = \{0\}, R^{T} A R$ will be positive definite.

Result:

$R^{T} A R$ Is positive semi-definite for any R and it is Proved using definiteness of A and properties of kernel. Hence, $R^{T} A R$ will be positive definite for all with $k e r (R) = \{0\}$

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Short Answer

If A is a positive definite $n \times n$ matrix, and R is any real $n \times n$ matrix, what can you say about the definiteness of the matrix $R^{T} AR$ ? For which matrices R is $R^{T} AR$ positive definite?

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Linear Algebra With Applications

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If A is a positive definite $n \times n$ matrix, and R is any real $n \times n$ matrix, what can you say about the definiteness of the matrix $R^{T} AR$ ? For which matrices R is $R^{T} AR$ positive definite?