Question

Prove that the characteristic roots of a diagonal matrix are the diagonal elements of the matrix.​

Answer 1

Step-by-step explanation:

First of all: what is the determinant of a triangular matrix? Developing along the first column you get a11det(A′11), where A′11 is the minor you get by crossing out the first row and column of A. But A′11 is also a triangular matrix. Developing along the first column you get a22detA′22, where A′22 is the minor you get by crossing out the first row and column of A′22. Going on in this fashion you see that det(A)=a11a22...an−1,n−1,ann.

Now the eigenvalues of A are given by the polynomial det(A−λI)=0. But this again is a triangular matrix, this time with diagonal elements (aii−λ). So your polynomial has been conveniently factored already:

(a11−λ)(a22−λ)...(ann−λ)=0

And the zeros are the diagonal elements of A.