It’s obvious that the trace of a diagonal matrix is the sum of its eigenvalues, and its determinant is their product (just look at Equation A.79). It follows (from Equations A.65 and A.68) that the same holds for any diagonalizable matrix. Prove that in fact

for any matrix. (The λ’s are the n solutions to the characteristic equation—in the case of multiple roots, there may be fewer linearly-independent eigenvectors than there are solutions, but we still count each λ as many times as it occurs.)

and use the result of Problem A.20.

Transcribed Image Text:

det (T) 12 An Tr(T) = 2₁+2₂+...+λ₂.. 9 (A.93)