Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q29E

Expert-verified

Linear Algebra With Applications

Found in: Page 393

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Consider a symmetric matrix A. If the vector $\overset{⇀}{v}$ is in the image of A and $\overset{⇀}{w}$ is in the kernel of A, is $\overset{⇀}{v}$ necessarily orthogonal to $\overset{⇀}{w}$ ? Justify your answer.

The $\overset{⇀}{v}$ is orthogonal to $\overset{⇀}{w}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The matrix

A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
For example, is a two-row, three-column matrix

Step 2: Determine the orthogonal matrix

A theorem we studied in this book states that

${(im A)}^{-} = \ker (A^{T})$

And we know that A is symmetric $⇄ A = A^{T}$ , which implies

${(im A)}^{-} = \ker (A^{T}) \to {(im A)}^{-} = \ker (A)$

Hence $\overset{⇀}{v}$ is orthogonal to $\overset{⇀}{w}$ .

Most popular questions for Math Textbooks

Consider an SVD

$A = U \sum_{} V^{T}$

of an $n \times m matrix A$ . Show that the columns of U form an orthonormal eigenbasis for ${AA}^{T}$ . What are the associated eigenvalues? What does your answer tell you about the relationship between the eigenvalues of $A^{T} A and {AA}^{T}$ ? Compare this with Exercise 7.4.57.

We say that an $n \times n$ matrix A is triangulizable if is similar to an upper triangular $n \times n$ matrix B.

a. Give an example of a matrix with real entries that fails to be triangulizable over R .

b. Show that any $n \times n$ matrix with complex entries is triangulizable over C . Hint: Give a proof by induction analogous to the proof of Theorem 8.1.1.

Show that the diagonal elements of a positive definite matrix A are positive.

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

If A is an invertible symmetric matrix, then $A^{2}$ must be positive definite.

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free