babeeb0oL
2021-02-09
pattererX
Skilled2021-02-10Added 95 answers
Let M be the matrix representation of T with respect to the basis A and N be the matrix representation w.r.t. the basis B. To convert M to N, recall that you use
where P is the change of basis matrix for vectors expressed in the basis B to their expression in the basis A.
Let v be an eigenvector of N with eigenvalue λ. Then
Hence λ is also an eigenvalue of the matrix M for the eigenvector Pv. As this holds for any arbitrary bases A and B, we see that eigenvalues of one matrix representation are also eigenvalues of any other.
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a) .
b) .
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