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Lecture 18 Eigenvalue Problems II Shang-Hua Teng

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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

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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

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

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Random walks Transition Matrix 1 2 3 4 5 6

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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!!!!

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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

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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

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Example

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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.

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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

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2 by 2 Case Real Eigenvalues

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2 by 2 Case so

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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?

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

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

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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

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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 !!!!!

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