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Consider the following matrix. -1 0 -1 0 -1 0 -1 A = -1 0 -1 0 -1 1 1 0 -1 0 -1 Use this list of theorems to answer the following questions. (a) Is A symmetric? Explain. O Yes, because A = AT. Yes, because A # AT. O No, because A = AT. O No, because A # AT. (b) Is A diagonalizable? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem O yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices no, by the Fundamental Theorem of Symmetric Matrices (c) Are the eigenvalues of A real? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices O no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices

Consider the following matrix. -1 0 -1 0 -1 0 -1 A = -1 0 -1 0 -1 1 1 0 -1 0 -1 Use this list of theorems to answer the following questions. (a) Is A symmetric? Explain. O Yes, because A = AT. Yes, because A # AT. O No, because A = AT. O No, because A # AT. (b) Is A diagonalizable? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem O yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices no, by the Fundamental Theorem of Symmetric Matrices (c) Are the eigenvalues of A real? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices O no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices

Advanced Engineering Mathematics
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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Consider the following matrix. -1 0 -1 0 -1 0 -1 A = -1 0 -1 0 -1 0 -1 1 1 0 -1 Use this list of theorems to answer the following questions. (a) Is A symmetric? Explain. O Yes, because A = AT. Yes, because A + AT. O No, because A = AT. O No, because A # AT. (b) Is A diagonalizable? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem O yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices no, by the Fundamental Theorem of Symmetric Matrices (c) Are the eigenvalues of A real? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices O no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices
Transcribed Image Text:Consider the following matrix. -1 0 -1 0 -1 0 -1 A = -1 0 -1 0 -1 0 -1 1 1 0 -1 Use this list of theorems to answer the following questions. (a) Is A symmetric? Explain. O Yes, because A = AT. Yes, because A + AT. O No, because A = AT. O No, because A # AT. (b) Is A diagonalizable? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem O yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices no, by the Fundamental Theorem of Symmetric Matrices (c) Are the eigenvalues of A real? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices O no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices
(d) The eigenvalues of A are distinct. What are the dimensions of the corresponding eigenspaces? Explain. O The multiplicity of each eigenvalue is 1, so the dimensions of the corresponding eigenspaces are 1. The multiplicity of each eigenvalue is 1, so the dimensions of the corresponding eigenspaces are 5. O The multiplicity of each eigenvalue is 5, so the dimensions of the corresponding eigenspaces are 1. O The multiplicity of each eigenvalue is 5, so the dimensions of the corresponding eigenspaces are 5. (e) Is A orthogonal? Explain. O Yes, because the columns form an orthonormal set. O No, because the columns do not form an orthonormal set. (f) For the eigenvalues of A, are the corresponding eigenvectors orthogonal? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices (9) Is A orthogonally diagonalizable? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem O yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices
Transcribed Image Text:(d) The eigenvalues of A are distinct. What are the dimensions of the corresponding eigenspaces? Explain. O The multiplicity of each eigenvalue is 1, so the dimensions of the corresponding eigenspaces are 1. The multiplicity of each eigenvalue is 1, so the dimensions of the corresponding eigenspaces are 5. O The multiplicity of each eigenvalue is 5, so the dimensions of the corresponding eigenspaces are 1. O The multiplicity of each eigenvalue is 5, so the dimensions of the corresponding eigenspaces are 5. (e) Is A orthogonal? Explain. O Yes, because the columns form an orthonormal set. O No, because the columns do not form an orthonormal set. (f) For the eigenvalues of A, are the corresponding eigenvectors orthogonal? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices (9) Is A orthogonally diagonalizable? Explain. (Select all that apply.) O yes, by the Real Spectral Theorem O yes, by the Property of Symmetric Matrices yes, by the Fundamental Theorem of Symmetric Matrices no, by the Real Spectral Theorem no, by the Property of Symmetric Matrices O no, by the Fundamental Theorem of Symmetric Matrices
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