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**Definitions:**A linear operator T on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis B for V such that [tex][T]_B[/tex] is a diagonal matrix. A square matrix A is called diagonalizable if [tex]L_A[/tex] is diagonalizable.

We want to determine when a linear operator T on a finite-dimensional vector space V is diagonalizable and, if so, how to obtain an ordered basis B = [tex]{v_1, v_2, ... , v_n}[/tex] for V such that [tex][T]_B[/tex] is a diagonal matrix. Note that, if D = [tex][T]_B[/tex] is a diagonal matrix, then for each vector [tex]v_j[/tex] in B, we have

[tex]T(v_j)[/tex] = [SUMMATION: from i = 1 to n][tex]D_i_jv_i[/tex] = [tex]D_j_jv_j[/tex] = [tex](lambda_j)v_j[/tex]

where (lambda_j) = Djj.

**Questions:**Could someone explain the following:

1. [tex]T(v_j)[/tex] = [SUMMATION: from i = 1 to n][tex]D_i_jv_i[/tex]

2. And maybe touch upon the other two equality relation in the line above.

Thanks,

JL