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If A and B are n x n matrices with the same eigenvalues, then they have they are similar. If A E Rxn has eigenvalues of multiplicity greater than 1, then A must be defective. If A E R4x4 is of rank 3 and λ = 0 is an eigenvalue of multiplicity 3, then A is diagonalizable.

If A and B are n x n matrices with the same eigenvalues, then they have they are similar. If A E Rxn has eigenvalues of multiplicity greater than 1, then A must be defective. If A E R4x4 is of rank 3 and λ = 0 is an eigenvalue of multiplicity 3, then A is diagonalizable.

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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4. If A and B are n x n matrices with the same eigenvalues, then they have they are similar. 5. If A E Rxn has eigenvalues of multiplicity greater than 1, then A must be defective. 6. If A E R4x4 is of rank 3 and λ = 0 is an eigenvalue of multiplicity 3, then A is diagonalizable. 7. If A ER4x4 is of rank 1 and λ = 0 is an eigenvalue of multiplicity 3, then A is defective. 8. The rank of A E Rmxn is equal to the number of nonzero singular values of A, where singular values are counted according to multiplicity.
Transcribed Image Text:4. If A and B are n x n matrices with the same eigenvalues, then they have they are similar. 5. If A E Rxn has eigenvalues of multiplicity greater than 1, then A must be defective. 6. If A E R4x4 is of rank 3 and λ = 0 is an eigenvalue of multiplicity 3, then A is diagonalizable. 7. If A ER4x4 is of rank 1 and λ = 0 is an eigenvalue of multiplicity 3, then A is defective. 8. The rank of A E Rmxn is equal to the number of nonzero singular values of A, where singular values are counted according to multiplicity.
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