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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 32E

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Determine a Sufficient Condition for Diagonalization In Exercises 23-26, find the eigenvalues of the matrix and determine there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. [432011002]
Proof Prove that if matrix A is diagonalizable with n real eigenvalues 1,2,,n, then |A|=12n.
Diagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1APis diagonal. Verify that P1APis a diagonal matrix with the eigenvalues on the main diagonal A=[223032012] See Exercise 20, section 7.1. Characteristic Equation, Eigenvalues, and Eigenvectors in Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [223032012]
Diagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal A=[122252663] See Exercise 23, section 7.1. Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [122252663]
True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a Geometrically, if is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to , then multiplying x by A produce a vector x parallel to x. b If A is nn matrix with an eigenvalue , then the set of all eigenvectors of is a subspace of Rn.
True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a An eigenvalue of a matrix A is a scalar such that det(IA)=0. b An eigenvector may be the zero vector 0. c A matrix A is orthogonally diagonalizable when there exists an orthogonal matrix P such that P1AP=D is diagonal.
True or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The scalar is an eigenvalue of an nn matrix A when there exists a vector x such that Ax=x. b To find the eigenvalues of an nn matrix A. you can solve the characteristic equation det(IA)=0.