Question

Hi, I know that if two matrices A and B are similar matrices then they must have the same eigenvalues with the same geometric multiplicities. However, I was wondering if that statement was equivalent. In order terms, if two matrices have the same eigenvalues with the same geometric multiplicities, must they be similar? If not, is it always false?

Answer #1

Hi,
I know that if two matrices A and B are similar matrices then
they must have the same eigenvalues with the same geometric and
algebraic multiplicities. However, I was wondering if that
statement was equivalent. In order terms, if two matrices have the
same eigenvalues with the same algebraic multiplicities, must they
be similar? What about geometric multiplicities?

We say two n × n matrices A and B are similar if there is an
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A = PBP^ -1.
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b) Show that if A and B are similar n × n matrices, then they
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c) Give an example to show that A and B can be...

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det(AB − λI) = det(ABAA−1 − λAA−1) = det(A(BA − λI)A−1)
= det(A) + det(BA − λI) − det(A)
= det(BA − λI)
Hence det(AB − λI) = det(BA − λI), and so...

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An infinite group must have an element of infinite order.
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Problem 30. Show that if two matrices A and B of the same size
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What condition should hold in order to multiply two
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A. The number of columns of the first matrix should be equal to
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The correlation between two variables A and B is .05 with a
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I am guessing that the correlation between A and B is not
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