let A be an nxn matrix that is similar to an upper triangular matrix and has the distinct eigenvalues lambda_1, lambda_2,..., lambda_k with corresponding multiplicities m_1,m_2,...,m_k. Prove the following.

a. tr(A) = Sum(m_i*lambda_i)

b. det(A) = (lambda_1)^m_1 * ... * (lambda_k)^m_k

Question

let A be an nxn matrix that is similar to an upper triangular matrix and has the distinct eigenvalues lambda_1, lambda_2,..., lambda_k with corresponding multiplicities m_1,m_2,...,m_k. Prove the following.

a. tr(A) = Sum(m_i*lambda_i)

b. det(A) = (lambda_1)^m_1 * ... * (lambda_k)^m_k

jamesj · Answer

The determinant of an upper triangular matrix is the product of the diagonal.  Prove that / convince yourself of that.

Knowing this, it should make it easier for you to show the two results.