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Prove that the eigenvalues of an upper triangular matrix M are the diagonal entries of M?
A brief overview would be great.
Sure, the proof is based on the fact that the determinant of an upper triangular matrix is equal to the product of its diagonal entries. Since the eigenvalues of a matrix are the roots of its characteristic polynomial, which can be expressed as the determinant of the matrix minus a multiple of the identity matrix, we can deduce that the characteristic polynomial of an upper triangular matrix is simply the product of its diagonal entries minus the eigenvalue. This implies that the eigenvalues of an upper triangular matrix are the diagonal entries of the matrix.