Skip to main content

Math Algebra Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Let A be a 5 × 5 nonsymmetric matrix with rank equal to 3, let B = A T A , and let C = e B . What, if anything, can you conclude about the nature of the eigenvalues of B ? Explain. What words best describe the type of matrix thatB is? What, if anything, can you conclude about the nature of the eigenvalues of C ? Explain. What words best describe the type of matrix that C is?

Let A be a 5 × 5 nonsymmetric matrix with rank equal to 3, let B = A T A , and let C = e B . What, if anything, can you conclude about the nature of the eigenvalues of B ? Explain. What words best describe the type of matrix thatB is? What, if anything, can you conclude about the nature of the eigenvalues of C ? Explain. What words best describe the type of matrix that C is?

Solution Summary: The author explains the nature of the eigenvalues of B and the type of matrix.

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

9th Edition

ISBN: 9780321962218

Author: Steven J. Leon

Publisher: PEARSON

See similar textbooks

Videos

Textbook Question

Chapter 6, Problem 10CTB

Let A be a 5 × 5 nonsymmetric matrix with rank equal to 3, let B = A T A , and let C = e B .

What, if anything, can you conclude about the nature of the eigenvalues of B? Explain. What words best describe the type of matrix thatB is?
What, if anything, can you conclude about the nature of the eigenvalues of C? Explain. What words best describe the type of matrix that C is?

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

Blurred answer

Chapter 6, Problem 9CTB

Chapter 6, Problem 11CTB

Chapter 6 Solutions

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 6.1 - Find the eigenvalues and the corresponding...Ch. 6.1 - Show that the eigenvalues of a triangular matrix...Ch. 6.1 - Let A be an nn matrix. Prove that A is singular if...Ch. 6.1 - Let A be a nonsingular matrix and let be an...Ch. 6.1 - Let A and B be nn matrices. Show that if none of...Ch. 6.1 - Let be an eigenvalue of A and let x be an...Ch. 6.1 - Let A bean nn matrix and let B=I2A+A2. Show that...Ch. 6.1 - An nn matrix A is said to be idempotent if A2=A....Ch. 6.1 - An nn matrix is said to be nilpotent if Ak=O for...Ch. 6.1 - Let A be an nn matrix and let B=AI for some scalar...

Ch. 6.1 - Let A be an nn matrix and let B=A+I. Is it...Ch. 6.1 - Show that A and AT have the same eigenvalues. Do...Ch. 6.1 - Show that the matrix A=( cos sin sin cos) will...Ch. 6.1 - Let A be a 22 matrix. If tr(A)=8 and det(A)=12,...Ch. 6.1 - Let A=(aij) be an nn matrix with eigenvalues...Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Let A be an nn matrix and let be an eigenvalue of...Ch. 6.1 - Prob. 19E Ch. 6.1 - Let =+bi and =c+di be complex scalars and let A...Ch. 6.1 - Let Q be an orthogonal matrix. Show that if is an...Ch. 6.1 - Let Q be an orthogonal matrix with an eigenvalue...Ch. 6.1 - Let Q be a 33 orthogonal matrix whose determinant...Ch. 6.1 - Let x1,...,xr be eigenvectors of an nn matrix A...Ch. 6.1 - Let A bean nn matrix and let be an eigenvalue of...Ch. 6.1 - Let B=S1AS and let x be an eigenvector of B...Ch. 6.1 - Let A be an nn matrix with an eigenvalue and let...Ch. 6.1 - Prob. 28E Ch. 6.1 - Let A be an nn matrix and let be a nonzero...Ch. 6.1 - Prob. 30E Ch. 6.1 - Let A be a matrix whose columns all add up to a...Ch. 6.1 - Let 1 and 2 be distinct eigenvalues of A. Let x be...Ch. 6.1 - Let A and B be nn matrices. Show that (a) If is a...Ch. 6.1 - Prove that there do not exist nn matrices A and B...Ch. 6.1 - Let p()=(1)n(nan1n1a1a0) be a polynomial of degree...Ch. 6.1 - The result given in Exercise 35(b) holds even if...Ch. 6.2 - Find the general solution of each of the following...Ch. 6.2 - Solve each of the following initial value...Ch. 6.2 - Given Y=c1e1tx1+c2e2tx2++cnentxn is the solution...Ch. 6.2 - Two tanks each contain 100 liters of a mixture....Ch. 6.2 - Prob. 5E Ch. 6.2 - Solve the initial value problem...Ch. 6.2 - In Application 2, assume that the solutions are of...Ch. 6.2 - Solve the the problem in Application 2, using the...Ch. 6.2 - Prob. 9E Ch. 6.2 - Three masses are connected by a series of springs...Ch. 6.2 - Transform the nth-order equation...Ch. 6.3 - In each of the following, factor the matrix A into...Ch. 6.3 - For each of the matrices in Exercise 1, use the...Ch. 6.3 - For each of the nonsingular matrices in Exercise...Ch. 6.3 - For each of the following, find a matrix B such...Ch. 6.3 - Let A be a nondefective nn matrix with...Ch. 6.3 - Let A be a diagonalizable matrix whose eigenvalues...Ch. 6.3 - Show that any 33 matrix of the form (a100a100b) is...Ch. 6.3 - For each of the following, find all possible...Ch. 6.3 - Let A be a 44 matrix and let be an eigenvalue of...Ch. 6.3 - Prob. 10E Ch. 6.3 - Let A be a nn matrix with real entries and let...Ch. 6.3 - Let A be an nn matrix with an eigenvalue of...Ch. 6.3 - Show that a nonzero nilpotent matrix is defective.Ch. 6.3 - Let A be a diagonalizable matrix and let X be the...Ch. 6.3 - It follows from Exercise 14 that for a...Ch. 6.3 - Prob. 16E Ch. 6.3 - Let x, y, be nonzero vectors in n,n2, and let...Ch. 6.3 - Let A be a diagonalizable nn matrix. Prove that if...Ch. 6.3 - Prob. 19E Ch. 6.3 - Let T be an upper triangular matrix with distinct...Ch. 6.3 - Each year, employees at a company are given the...Ch. 6.3 - The city of Mawtookit maintains a constant...Ch. 6.3 - Let A=( 1 2 1 3 1 5 1 4 1 3 2 5 1 4 1 3 2 5 ) be a...Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Consider a Web network consisting of only four...Ch. 6.3 - Prob. 27E Ch. 6.3 - The transition matrix in Example 5 has the...Ch. 6.3 - Let A be the PageRank transition matrix and let xk...Ch. 6.3 - Use the definition of the matrix exponential to...Ch. 6.3 - Compute eA for each of the following matrices: A=(...Ch. 6.3 - In each of the following, solve the initial value...Ch. 6.3 - Let X be an eigenvalue of an nn matrix A and let x...Ch. 6.3 - Show that eA is nonsingular for any diagonalizable...Ch. 6.3 - Let A be a diagonalizable matrix with...Ch. 6.4 - For each of the following pairs of vectors z and...Ch. 6.4 - Let z1=( 1+i 2 1i 2 ) and z2=( i 2 1 2 ) Show...Ch. 6.4 - Let {u1,u2} be an orthonormal basis for 2, and let...Ch. 6.4 - Which of the matrices that follow are Hermitian?...Ch. 6.4 - Find an orthogonal or unitary diagonalizing matrix...Ch. 6.4 - Prob. 6E Ch. 6.4 - Let A be an nn Hermitian matrix and let x be a...Ch. 6.4 - Let A be an Hermitian matrix and let B=iA. Show...Ch. 6.4 - Let A and C be matrices in mn and let Bnr. Prove...Ch. 6.4 - Prob. 10E Ch. 6.4 - Show that z,w=wHz defines an inner product on n.Ch. 6.4 - Let x, y, and z be vectors in n and let and be...Ch. 6.4 - Let {u1,...,un} be an orthonormal basis for a...Ch. 6.4 - Given that A=(40001i0 i1) find a matrix B such...Ch. 6.4 - Let U be a unitary matrix. Prove that U is normal....Ch. 6.4 - Let u be a unit vector in n and define U=I2uuH....Ch. 6.4 - Show that if a matrix U is both unitary and...Ch. 6.4 - Let A be a 22 matrix with Schur decomposition UTUH...Ch. 6.4 - Let A be a 55 matrix with real entries. Let A=QTQT...Ch. 6.4 - Let A be a nn matrix with Schur decomposition...Ch. 6.4 - Show that M=A+iB (where A and B are real matrices)...Ch. 6.4 - Prob. 22E Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - Prob. 25E Ch. 6.4 - Let A be a Hermitian matrix with eigenvalues...Ch. 6.4 - Let A=(0110) Write A as a sum 1u1u1T+2u2u2T, where...Ch. 6.4 - Let A be a Hermitian matrix with eigenvalues 12n...Ch. 6.4 - Given Amm,Bnn,Cmn, the equation AXXB=C(3) is known...Ch. 6.5 - Show that A and AT have the same nonzero singular...Ch. 6.5 - Use the method of Example 1 to find the singular...Ch. 6.5 - For each of the matrices in Exercise 2: determine...Ch. 6.5 - Let A=( 28 20 14 19 102 21)=( 3 5 4 5 0 4 5 3 5...Ch. 6.5 - The matrix A=(254630630254) has singular value...Ch. 6.5 - Prove that if A is a symmetric matrix with...Ch. 6.5 - Let A be an mn matrix with singular value...Ch. 6.5 - Let A be an nn matrix. Show that ATA and AAT are...Ch. 6.5 - Let A be an nn matrix with singular values...Ch. 6.5 - Let A be an nn matrix with singular value...Ch. 6.5 - Show that if is a singular value of A then there...Ch. 6.5 - Let A be an mn matrix of rank n with singular...Ch. 6.5 - Prob. 13E Ch. 6.6 - Find the matrix associated with each of the...Ch. 6.6 - Reorder the eigenvalues in Example 2 so that 1=4...Ch. 6.6 - Prob. 3E Ch. 6.6 - Let 1 and 2 be the eigenvalues of A=(abbc) What...Ch. 6.6 - Prob. 5E Ch. 6.6 - Which of the matrices that follow are positive...Ch. 6.6 - For each of the following functions, determine...Ch. 6.6 - Show that if A is symmetric positive definite,...Ch. 6.6 - Prob. 9E Ch. 6.6 - Prob. 10E Ch. 6.6 - Let A be a symmetric nn matrix with eigenvalues...Ch. 6.6 - Prob. 12E Ch. 6.6 - Prob. 13E Ch. 6.6 - Let A be a symmetric positive definite nn matrix....Ch. 6.7 - For each of the following matrices, compute the...Ch. 6.7 - Prob. 2E Ch. 6.7 - Let A=(2 100 12 100 12 100 12) Compute the LU...Ch. 6.7 - For each of the following, factor the given matrix...Ch. 6.7 - Find the Cholesky decomposition LLT for each of...Ch. 6.7 - Prob. 6E Ch. 6.7 - Prove each of the following: If U is a unit upper...Ch. 6.7 - Prob. 8E Ch. 6.7 - Let A be a symmetric positive definite matrix with...Ch. 6.7 - Let A be an mn matrix with rank n. Show that the...Ch. 6.7 - Prob. 11E Ch. 6.7 - Let A be a symmetric positive definite matrix and...Ch. 6.7 - Prob. 13E Ch. 6.7 - Prob. 14E Ch. 6.7 - Prob. 15E Ch. 6.7 - Let A be an nn symmetric negative definite matrix....Ch. 6.7 - Prob. 17E Ch. 6.8 - Find the eigenvalues of each of the following...Ch. 6.8 - Prob. 2E Ch. 6.8 - Find the output vector x in the open version of...Ch. 6.8 - Consider the closed version of the Leontief...Ch. 6.8 - Prob. 5E Ch. 6.8 - Prob. 6E Ch. 6.8 - Which of the matrices that follow are reducible?...Ch. 6.8 - Prob. 8E Ch. 6.8 - Prob. 9E Ch. 6.8 - Prove that a 22 matrix A is reducible if and only...Ch. 6.8 - Prove the Forbenius theorem in the case where A is...Ch. 6.8 - Prob. 12E Ch. 6.8 - Let A be an nn positive stochastic matrix with...Ch. 6.8 - Would the results of parts (c) and (d) in Exercise...Ch. 6.8 - A management student received fellowship offers...Ch. 6 - The top matrix on the menu is the diagonal matrix...Ch. 6 - Prob. 2E Ch. 6 - Prob. 3E Ch. 6 - Prob. 4E Ch. 6 - Prob. 5E Ch. 6 - Prob. 6E Ch. 6 - Prob. 7E Ch. 6 - The last item on the eigshow menu will generate a...Ch. 6 - Prob. 9E Ch. 6 - Prob. 10E Ch. 6 - Prob. 11E Ch. 6 - Consider the matrices A=(5 33 5) and B=(5 335)...Ch. 6 - Prob. 13E Ch. 6 - Prob. 14E Ch. 6 - Prob. 15E Ch. 6 - Prob. 16E Ch. 6 - Prob. 17E Ch. 6 - Prob. 18E Ch. 6 - Prob. 19E Ch. 6 - Let A be a nonsingular 22 matrix with singular...Ch. 6 - Set A=[1,1;0.5,0.5] and use MATLAB to verify each...Ch. 6 - Prob. 22E Ch. 6 - Prob. 23E Ch. 6 - Prob. 24E Ch. 6 - Prob. 25E Ch. 6 - If A is an nn matrix whose eigenvalues are all...Ch. 6 - If A is nn matrix, then A and AT have the same...Ch. 6 - If A and B are similar matrices, then they have...Ch. 6 - If A and B are nn matrices with the same...Ch. 6 - If A has eigenvalues of multiplicity greater than...Ch. 6 - If A is a 44 matrix of rank 3 and =0 is an...Ch. 6 - If A is a 44 matrix of rank 1 and =0 is an...Ch. 6 - The rank of an nn matrix A is equal to the number...Ch. 6 - The rank of an mn matrix A is equal to the number...Ch. 6 - If A is Hermitian and c is a complex scalar, then...Ch. 6 - If an nn matrix A has Schur decomposition A=UTUH,...Ch. 6 - If A is normal, but not Hermitian, then A must...Ch. 6 - Prob. 13CTA Ch. 6 - Prob. 14CTA Ch. 6 - If A is symmetric, then eA is symmetric positive...Ch. 6 - Let A=(10011 112 2) Find the eigenvalues of A. For...Ch. 6 - Let A be a 44 matrix with real entries that has...Ch. 6 - Let A be a nonsingular nn matrix and let be an...Ch. 6 - Show that if A is a matrix of the form...Ch. 6 - Let A=(4222 10 102 10 14) Without computing the...Ch. 6 - Prob. 6CTB Ch. 6 - Prob. 7CTB Ch. 6 - Let A be a 44 real symmetric matrix with...Ch. 6 - Let {u1,u2} be an orthonormal basis for 2 and...Ch. 6 - Let A be a 55 nonsymmetric matrix with rank equal...Ch. 6 - Let A and B be nn matrices. If A is real and...Ch. 6 - Let A be a matrix whose singular value...

Additional Math Textbook Solutions

Find more solutions based on key concepts

The given equation.

High School Math 2015 Common Core Algebra 1 Student Edition Grade 8/9

Fill in each blank so that the resulting statement is true. 1. The degree of the polynomial function is _____....

Algebra and Trigonometry (6th Edition)

Is there a relationship between wine consumption and deaths from heart disease? The table gives data from 19 de...

College Algebra Essentials (5th Edition)

Value of g(3p) .

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

Consider the damped spring-mass system whose motion is governed by d2ydt2+2dydt+5y=17sin2t, y(0)=2, dydt(0)=0. ...

Differential Equations and Linear Algebra (4th Edition)

If the run of a line is 10 and its rise is 6; then its slope is ______.

Elementary & Intermediate Algebra

Knowledge Booster

Background pattern image

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

CAPSTONE Explain how to determine whether an nn matrix A is diagonalizable using a similar matrices, b eigenvectors, and c distinct eigenvalues.
Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has two distinct real eigenvalues, one real eigenvalue, and no real eigenvalues.
a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning

Text book image

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:9781305658004

Author:Ron Larson

Publisher:Cengage Learning

Text book image

Linear Algebra: A Modern Introduction

Algebra

ISBN:9781285463247

Author:David Poole

Publisher:Cengage Learning

Text book image

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:9781133382119

Author:Swokowski

Publisher:Cengage

Text book image

College Algebra (MindTap Course List)

Algebra

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:Cengage Learning

Text book image

College Algebra

Algebra

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

Text book image

Algebra and Trigonometry (MindTap Course List)

Algebra

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

Lecture 46: Eigenvalues & Eigenvectors; Author: IIT Kharagpur July 2018;https://www.youtube.com/watch?v=h5urBuE4Xhg;License: Standard YouTube License, CC-BY

What is an Eigenvector?; Author: LeiosOS;https://www.youtube.com/watch?v=ue3yoeZvt8E;License: Standard YouTube License, CC-BY