MATH 235Diagonalization - 1Assignment 4aNot to be handed in.1. LetAbe ann×nmatrix. Also, letα, β, γbe three distinct eigenvalues ofAhavingcorresponding eigenvectorsx,y,z, respectively.Consider the vectorv=x+y+z.Canvbe an eigenvector ofAcorresponding to an eigenvalueλ(possibly different fromα, β, γ? Explain.2. Recall that the trace of ann×nmatrixA= [aij], denoted by tr(A), is the sum of thediagonal elements, that is, tr(A) =∑ni=1aii.(a) LetCandDbe any twon×nmatrices. Prove that tr(CD) = tr(DC).(b) Prove that if
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