Lecture 8 : Eigenvalues and Eigenvectors

Hermitian Matrices It is simpler to begin with matrices with complex numbers. Let x = a + ib, where a, b are real numbers, and i = √ −1. Then, x∗ = a− ib is the complex conjugate of x. In the discussion below, all matrices and numbers are complex-valued unless stated otherwise. Let M be an n× n square matrix with complex entries. Then, λ is an eigenvalue of M if there is a non-zero vector ~v such that M~v = λ~v This implies (M − λI)~v = 0, which also means the determinant of M − λI is zero. Since the determinant is a degree n polynomial in λ, this shows that any M has n real or complex eigenvalues. A complex-valued matrix M is said to be Hermitian if for all i, j, we have Mij = M ∗ ji. If the entries are all real numbers, this reduces to the definition of symmetric matrix. In the discussion below, we will need the notion of inner product. Let ~v and ~ w be two vectors with complex entries. Define their inner product as