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Prove that A and A' have the same eigenvalues. To prove that A and AT have the same eigenvalues it is sufficient to prove that their characteristic equations are the same. Start with the characteristic equation of AT then use the properties of martix ransposes and determinants to show it is equivalent to the characteristic equation of A. Olar-ATI (ar) - ATI = |(AIA)| = |AI - AI Ⓒlar+A¹1(ar) - Al = |(ar - A)| = |AI + Al Ⓒlar-AT-1(ar) + A| = |(ar + A)²| = 121 - Al Ⓒlar+ATI (az)T + A™| = |(AI + A)| = |AI + A Are the eigenspaces the same? O yes O no

Prove that A and A' have the same eigenvalues. To prove that A and AT have the same eigenvalues it is sufficient to prove that their characteristic equations are the same. Start with the characteristic equation of AT then use the properties of martix ransposes and determinants to show it is equivalent to the characteristic equation of A. Olar-ATI (ar) - ATI = |(AIA)| = |AI - AI Ⓒlar+A¹1(ar) - Al = |(ar - A)| = |AI + Al Ⓒlar-AT-1(ar) + A| = |(ar + A)²| = 121 - Al Ⓒlar+ATI (az)T + A™| = |(AI + A)| = |AI + A Are the eigenspaces the same? O yes O no

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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Prove that A and A' have the same eigenvalues. To prove that A and AT have the same eigenvalues it is sufficient to prove that their characteristic equations are the same. Start with the characteristic equation of AT then use the properties of martix transposes and determinants to show it is equivalent to the characteristic equation of A. Ⓒ|AI AT| = |(21)¹ – A²| = |(^I − A)¹| = |2 - A|| Ⓒ |a1 + A²| = |(21)¹ – A| = |(aI − A)²| = |21 + A| OAIAT|= |(az)T + A| = |(aI + A)| = |AI - A Ⓒ |λI + AT| = |(21)T + AT| = |(^I + A)¹| = |λ + A| Are the eigenspaces the same? O yes O no
Transcribed Image Text:Prove that A and A' have the same eigenvalues. To prove that A and AT have the same eigenvalues it is sufficient to prove that their characteristic equations are the same. Start with the characteristic equation of AT then use the properties of martix transposes and determinants to show it is equivalent to the characteristic equation of A. Ⓒ|AI AT| = |(21)¹ – A²| = |(^I − A)¹| = |2 - A|| Ⓒ |a1 + A²| = |(21)¹ – A| = |(aI − A)²| = |21 + A| OAIAT|= |(az)T + A| = |(aI + A)| = |AI - A Ⓒ |λI + AT| = |(21)T + AT| = |(^I + A)¹| = |λ + A| Are the eigenspaces the same? O yes O no
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