Numerical Analysis The Power Method for Eigenvalues and Eigenvectors

The spectrum of a square matrix A, denoted by σ(A) is the set of all eigenvalues of A. The spectral radius of A, denoted by ρ(A) is defined as: ρ(A) = max{|λ| : λ ∈ σ(A)} An eigenvalue of A that is larger in absolute value than any other eigenvalue is called the dominant eigenvalue; a corresponding eigenvector is called a dominant eigenvector. The most commonly encountered vector norm (often simply called " the norm " of a vector) is the Euclidean norm, given by x 2 = x = x 2 1 + x 2 2 + ... + x 2 n. The max norm, also known as the infinity norm of a vector is the largest component of the vector in absolute value: An eigenvector V is said to be normalized if its coordinates are divided by its norm. If the max norm is used, then clearly, the coordinate of largest magnitude is equal to one. Theorem 1. If u is an eigenvector of a matrix A, then its corresponding eigenvalue is given by λ = u * · Au u * · u , where u * is the conjugate transpose of u. This quotient is called the Rayleigh quotient. Proof. Since u is an eigenvector of A, we know that Au = λu and we can write u * · Au u * · u = λu * · u u * · u = λ u * · u u * · u = λ × 1 = λ Theorem 2. Let λ 1 , λ 2 ,. .. λ m be the m eigenvalues (counted with multiplicity) of the n × n matrix A and let v 1 , v 2 ,. .. , v m be the corresponding eigenvectors. Suppose that λ 1 is the dominant eigenvalue, so that |λ 1 | > |λ 2 | ≥ · · · λ j ≥ · · · ≥ |λ m |.