Consider the following matrices. -5 -2 11 2 -2 0 -2 5] [o -1 D = |1 Lo 01 -5 4 2 4 -3 0 L-5 -4 0] 11 1 4 a) Without computing the actual eigenvalues, can you tell if any of the matrices above must have all real eigenvalues? Explain why you make this judgement. b) Without computing the actual eigenvalues, can you tell if any of the matrices above must have eigenvalues that are all imaginary or zero? Explain why you make this judgement.
Consider the following matrices. -5 -2 11 2 -2 0 -2 5] [o -1 D = |1 Lo 01 -5 4 2 4 -3 0 L-5 -4 0] 11 1 4 a) Without computing the actual eigenvalues, can you tell if any of the matrices above must have all real eigenvalues? Explain why you make this judgement. b) Without computing the actual eigenvalues, can you tell if any of the matrices above must have eigenvalues that are all imaginary or zero? Explain why you make this judgement.
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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![s -4 ol
Consider the following matrices.
-5 -2
-6 21 -
-2 51
4
[o
|1
Lo
-1
01
2
4
à =|-5
B= T =
2
-2 0
-3
[3
1]
1
4
a) Without computing the actual eigenvalues, can you tell if any of the matrices above
must have all real eigenvalues? Explain why you make this judgement.
b) Without computing the actual eigenvalues, can you tell if any of the matrices above
must have eigenvalues that are all imaginary or zero? Explain why you make this
judgement.
c) Show that matrix D satisfies the condition D" = D-1. (Strictly real matrices that
satisfy this condition are called unitary.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b020904-503e-46f0-a83a-8c74ba785f01%2F3c083122-c8b2-4a48-8bcb-03f1efa72b62%2F94a3j5_processed.png&w=3840&q=75)
Transcribed Image Text:s -4 ol
Consider the following matrices.
-5 -2
-6 21 -
-2 51
4
[o
|1
Lo
-1
01
2
4
à =|-5
B= T =
2
-2 0
-3
[3
1]
1
4
a) Without computing the actual eigenvalues, can you tell if any of the matrices above
must have all real eigenvalues? Explain why you make this judgement.
b) Without computing the actual eigenvalues, can you tell if any of the matrices above
must have eigenvalues that are all imaginary or zero? Explain why you make this
judgement.
c) Show that matrix D satisfies the condition D" = D-1. (Strictly real matrices that
satisfy this condition are called unitary.)
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