Prove that the eigenvalues of a triangular upper or lower ma

Prove that the eigenvalues of a triangular (upper- or lower-) matrix T are the entries _ii, and find the associated eigenvectors.

Solution

Suppose (v1,…,vn) is a basis of V with respect to which T has an upper-triangular matrix where the diagonal entries are 1,…,n

Let F

Then for matrix M(TI) where the diagonal entries are 1,…n. We can suppose we are dealing with complex vector spaces. T is not invertible if one of the k\'s equals 0. Hence TI is not invertible if and only if equals one of the j\'s. In other words, is an eigenvalue of T if and only if equals one of the js, as desired.

is an eigenvalue of T if and only if {1,…,n}

is an eigenvalue of T if and only if =1, or =2, ..., or =n

Thus, if I set equal to 1, the right side of the biconditional is true, so that is an eigenvalue of Twhen =1; and similarly with all of the diagonal entries 1,…,n.