1. Use the following definition of eigenvectors and eigenvalues of a matrix A to prove the fact below: Definition 1: An eigenvector of an n x n matrix A is a nonzero vector such that AA for some scalar X. A scalar A is called an eigenvalue of A if there is a nontrivial solution to Az = X. Prove: If matrix A, nxn, has two eigenvectors, w₁, 2 corresponding to two distinct eigenvalues A1, A2 then the set {₁, 2} is linearly independent. 2. Prove or disprove: (a) The eigenvalues of an triangular nxn matrix A are the entries on its main diagonal. (b) If an 3 x 3 matrix A has an eigenvalue of multiplicity three then A is singular.
1. Use the following definition of eigenvectors and eigenvalues of a matrix A to prove the fact below: Definition 1: An eigenvector of an n x n matrix A is a nonzero vector such that AA for some scalar X. A scalar A is called an eigenvalue of A if there is a nontrivial solution to Az = X. Prove: If matrix A, nxn, has two eigenvectors, w₁, 2 corresponding to two distinct eigenvalues A1, A2 then the set {₁, 2} is linearly independent. 2. Prove or disprove: (a) The eigenvalues of an triangular nxn matrix A are the entries on its main diagonal. (b) If an 3 x 3 matrix A has an eigenvalue of multiplicity three then A is singular.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Use the following definition of eigenvectors and eigenvalues of a matrix A to prove the fact below:
Definition 1: An eigenvector of an n x n matrix A is a nonzero vector such that AA for some scalar X.
A scalar A is called an eigenvalue of A if there is a nontrivial solution to Az = X.
Prove: If matrix A, nxn, has two eigenvectors, w₁, 2 corresponding to two distinct eigenvalues A1, A2 then the set
{₁, 2} is linearly independent.
2. Prove or disprove:
(a) The eigenvalues of an triangular nxn matrix A are the entries on its main diagonal.
(b) If an 3 x 3 matrix A has an eigenvalue of multiplicity three then A is singular.
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