Lecture 18 Eigenvalue Problems II Shang-Hua Teng

Diagonalizing A Matrix Suppose the n by n matrix A has n linearly independent eigenvectors x 1, x 2,…, x n. Eigenvector matrix S: x 1, x 2,…, x n are columns of S. Then  is the eigenvalue matrix

Matrix Power A k S -1 AS =  implies A = S  S -1 implies A 2 = S  S -1 S  S -1 = S   S -1 implies A k = S  k S -1

Random walks How long does it take to get completely lost?

Random walks Transition Matrix

Matrix Powers If A is diagonalizable as A = S  S -1 then for any vector u, we can compute A k u efficiently –Solve S c = u –A k u = S  k S -1 S c = S  k c As if A is a diagonal matrix!!!!

Independent Eigenvectors from Different Eigenvalues Eigenvectors x 1, x 2,…, x k that correspond to distinct (all different) eigenvalues are linear independent. An n by n matrix that has n different eigenvalues (no repeated ’s) must be diagonalizable Proof: Show that implies all c i = 0

Addition, Multiplication, and Eigenvalues If is an eigenvalue of A and  is an eigenvalue of B, then in general  is not an eigenvalue of AB If is an eigenvalue of A and  is an eigenvalue of B, then in general  is not an eigenvalue of A+B

Example

Spectral Analysis of Symmetric Matrices A = A T (what are special about them?) Spectral Theorem: Every symmetric matrix has the factorization A = Q  Q T with real eigenvalues in  and orthonormal eigenvectors in Q: A =Q  Q -1 = Q  Q T with Q -1 = Q T.

Simply in English Symmetric matrix can always be diagonalized Their eigenvalues are always real One can choose n eigenvectors so that they are orthonormal. “Principal axis theorem” in geometry and physics

2 by 2 Case Real Eigenvalues

2 by 2 Case so

The eigenvalues of a real symmetric matrix are real Complex conjugate of a + i b is a - i b Law of complex conjugate : (a-i b) (c-i d) = (ac-bd) – i(bc+ad) which is the complex conjugate of (a+i b) (c+i d) = (ac-bd) + i(bc+ad) Claim: What can be?

Eigenvectors of a real symmetric matrix when they correspond to different ’s are always perpendicular What can the quantity be?

In general, so eigenvalues might be repeated Choose an orthogonal basis for each eigenvalue Normalize these vector to unit length

Every symmetric matrix has the factorization A = Q  Q T with real eigenvalues in  and orthonormal eigenvectors in  : A =Q  Q -1 = Q  Q T with Q -1 = Q T. Spectral Theorem

Every symmetric matrix has the factorization A = Q  Q T with real eigenvalues in  and orthonormal eigenvectors in  : A =Q  Q -1 = Q  Q T with Q -1 = Q T. Spectral Theorem and Spectral Decomposition x i x i T is the projection matrix on to x i !!!!!