Thread: Spectral Decomposition and Polynomials

Let T be a normal operator on a finite-dimensional complex inner product space V. Use the spectral decomposition to show that If g is a polynomial, then
g(T) = g(m_1)T_1 + g(m_2)T_2 + ... + g(m_k)T_k, where the m_i's are the eigenvalues of T and the T_i's are the orthogonal projection of T on the eigenspace, W_i corresponding to the eigenvalue m_i.


I know that from the spectral decomposition, we have T = (m_1)(T_1) + (m_2)(T_2) + ... + (m_k)(T_k). Hence, g(T) = g((m_1)(T_1)) + g((m_2)(T_2)) + ... + g((m_k)(T_k)). But how do i pull out the T_i's to obtain the above result?

Thank You.

Originally Posted by H12504106

Let T be a normal operator on a finite-dimensional complex inner product space V. Use the spectral decomposition to show that If g is a polynomial, then
g(T) = g(m_1)T_1 + g(m_2)T_2 + ... + g(m_k)T_k, where the m_i's are the eigenvalues of T and the T_i's are the orthogonal projection of T on the eigenspace, W_i corresponding to the eigenvalue m_i.


I know that from the spectral decomposition, we have T = (m_1)(T_1) + (m_2)(T_2) + ... + (m_k)(T_k).

Start by computing monomials (powers of T):



Multiply out the brackets, remembering that the are pairwise orthogonal projections so that and if Thus



Use induction to prove the analogous result for each power of T. Finally, use linearity to deduce the result for a polynomial.