Q71E If is an indefinite n脳m聽matr... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q71E (page 403)

If is an indefinite $n 脳 m$ matrix, and R is any real $n 脳 m$ matrix, what can you say about the definiteness of the matrix role="math" localid="1659684209026" $R^{T} AR$ ?

Short Answer

Expert verified

Nothing can be said about the definiteness of $R^{T} AR$

Step by step solution

0f 2: Given information

Let $\overset{鈫�}{x} 鈭� {鈩�}^{m}$ .
Let us consider for symmetric matrix A of order n and any real matrix $R 鈭� {鈩�}^{n 脳 m}$ of rank X.
Thus we get ${\overset{鈫�}{x}}^{T} R^{T} AR \overset{鈫�}{x} = ({\overset{鈫�}{x}}^{T} R^{T}) (A) (R \overset{鈫�}{x}) (R \overset{鈫�}{x}) (A) (R \overset{鈫�}{x}) = {\overset{鈫�}{y}}^{T} A \overset{鈫�}{y} where \overset{鈫�}{y} = R \overset{鈫�}{x}$

Here R is any matrix of order $n 脳 m$ and thus, rank $(R) 鈮� n$ For R having rank $(R) = n$ from exercise $70, R^{T} AR$ , will be indefinite. But for R with rank (R) < n,

0f 2: Application

Let us consider the following examples
Let us have $A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ and $R = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
is indefinite while Rank (R) = 1 < 2 and $R^{T} AR = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ which is positive semi-definite matrix.
For $R = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}] rank = 2$ , we get $R^{T} AR = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{matrix}]$ which is indefinite.
Therefore, for indefinite matrix A and any matrix R nothing can be said about the definiteness of $R^{T} AR$

Result:

The definiteness of role="math" localid="1659683541132" $R^{T} AR$ cannot be said.

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91影视

If is an indefinite $n 脳 m$ matrix, and R is any real $n 脳 m$ matrix, what can you say about the definiteness of the matrix role="math" localid="1659684209026" $R^{T} AR$ ?

Short Answer

Step by step solution

0f 2: Given information

0f 2: Application

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91影视

If is an indefinite n脳mmatrix, and R is any real n脳mmatrix, what can you say about the definiteness of the matrix role="math" localid="1659684209026" RTAR ?

Short Answer

Step by step solution

0f 2: Given information

0f 2: Application

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

If is an indefinite $n 脳 m$ matrix, and R is any real $n 脳 m$ matrix, what can you say about the definiteness of the matrix role="math" localid="1659684209026" $R^{T} AR$ ?