Question: Two n n matrices A, B are said to be simultaneously diagonalizable if there is a nonsingular matrix S such that both S-1 A

Two n × n matrices A, B are said to be simultaneously diagonalizable if there is a nonsingular matrix S such that both S-1 A S and S-1 B S are diagonal matrices.
(a) Show that simultaneously diagonalizable matrices commute: AB = BA.
(b) Prove that the converse is valid, provided that one of the matrices has no multiple eigenvalues.
(c) Is every pair of commuting matrices simultaneously diagonalizable?

Step by Step Solution

★★★★★

3.37 Rating (156 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a If A SS 1 and B SDS 1 where D are diagonal then AB SDS 1 SDS 1 BA since diagonal matrices ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

952-M-L-A-E (2654).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

A matrix A is orthogonally diagonalizable if there is an orthogonal matrix P that diagonalizes it. Show that the matrix is orthogonally diagonalizable. Synunetric matrices have orthogonal...

Show that each of the following matrices is nonsingular but not diagonalizable. a. b. c. d. 003 3 6-43 332 120 012 0

Two matrices A and B are said to be similar, written A ~ B, if there exists an invertible matrix S such that B = S-1 AS. Prove: (a) A ~ A (b) If A ~ B, then B ~ A. (c) If A ~ B ~ C, then A~ C.

Kindly see the photo of the question below: (a) Assume that A1 and 12 are distinct eigenvalues of the nxn matrix A. Prove that an nx1 vector x, which is not a null vector, cannot be an eigenvector of...

chapter 5 INTRODUCTION TO MATRIX ALGEBRA GOALS The purpose of this chapter is to introduce you to matrix algebra, which has many applications. You are already familiar with several algebras:...

Mark each of the following statements true or false: (a) For all square matrices A, det(-A) = -det A. (b) If A and B are n n matrices, then det(AB) = det (BA). (c) If A and B are n n matrices whose...

Math 636 Final - Quiz Component 1. The system of linear equations 2x2 x3 = 3 2x1 + x2 + 2x3 = 1 x1 + x2 + 3x3 = 2 (a) is consistent with a unique solution. (b) is consistent with infinitely many...

MATHEMATICS FOR MACHINE LEARNING Marc Peter Deisenroth A. Aldo Faisal Cheng Soon Ong Contents Foreword 1 Part I Mathematical Foundations 9 1 Introduction and Motivation 11 1.1 Finding Words for...

Consider the following. A= 17 60 P -3 -4 5 18 -1 -1 (a) Verify that A is diagonalizable by computing P~1AP. L I = )7 + U p~lap = (b) Use the result of part (a) and the theorem below to find the...

A porous formation has the following properties: area = 300 acres, thickness = 60 ft, net thickness = 45 ft, porosity = 21%, water saturation = 25%, gas saturation = 12 %, Bo = 1.26 bbl/STB, Bw =...

Prove that the set of intervals P= ( (Sk-4, 5k+1]:keZ is a partition of R. Piease upload your proof as a single PDF file relating to only this question. Maximum file size: 50MB, maximum number of...

What are the differences between the book balances and actual balances

Would you expect new and better machinery and equipment to be adopted more rapidly in a market economy or in a centrally planned economy? Briefly explain.

Consider Supplemental Exercise 10-72. Suppose that prior to collecting the data, you decide that you want the error in estimating 1 = 2 by x1 = x2 to be less than 1.5 psi. Specify the sample size for...

A random sample of 1500 residential telephones in Phoenix in 1990 found that 387 of the numbers were unlisted. x2 x1 = 88 = 91 A random sample in the same year of 1200 telephones in Scottsdale found...

In a random sample of 200 Phoenix residents who drive a domestic car, 165 reported wearing their seat belt regularly, while another sample of 250 Phoenix residents who drive a foreign car revealed...

The coupon rate on an issue of debt is 9% . The yield to maturity on this issue is 10% . The corporate tax rate is 35% . What would be the approximate after-tax cost of\ debt for a new issue of...

4. In relation to Protocols: (i) Explain (at a high level) the basic concept of a protocol. (3 marks) (ii) Using a box diagram describe the syntax (the structure) of the requests and responses...

What does the citation, "Reg. Sec. 1.83-2" refer to