Matrix Problems

Problem. Given a matrix A = (aij)n×n, suppose one can find n vectors

V1, V2, . . . , Vn

such that AV1 = λ1V1, AV2 = λ2V2, . . . , AVn = λnVn

for some numbers λ1, λ2, . . ., λn. Also suppose the matrix whose column vectors are V1, . . ., Vn is an invertible matrix.

Under these assumptions, explain how to find an invertible matrix B and a diagonal matrix D so that

B−1AB = D.

(Note: the answer to this problem was explained in detail in lecture today.)

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