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Lecture 18 Eigenvalue Problems II

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. L is the eigenvalue matrix. Matrix Power A k.

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Lecture 18 Eigenvalue Problems II

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  1. Lecture 18Eigenvalue Problems II Shang-Hua Teng

  2. Diagonalizing A Matrix • Suppose the n by n matrix A has n linearly independent eigenvectors x1, x2,…, xn. • Eigenvector matrix S: x1, x2,…, xn are columns of S. • Then L is the eigenvalue matrix

  3. Matrix Power Ak • S-1AS = L implies A = S LS-1 • implies A2 = S LS-1 S LS-1 = S L2S-1 • implies Ak = S LkS-1

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

  5. 1 2 6 3 4 5 Random walks Transition Matrix

  6. Matrix Powers • If A is diagonalizable as A = S LS-1 then for any vector u, we can compute Akuefficiently • Solve S c = u • Aku =S LkS-1 S c = S Lkc • As if A is a diagonal matrix!!!!

  7. Independent Eigenvectors from Different Eigenvalues • Eigenvectors x1, x2,…, xkthat correspond to distinct (all different) eigenvalues are linear independent. • An n by n matrix that has n different eigenvalues (no repeated l’s) must be diagonalizable Proof: Show that implies all ci = 0

  8. Addition, Multiplication, and Eigenvalues • If l is an eigenvalue of A and b is an eigenvalue of B, then in general lb is not an eigenvalue of AB • If l is an eigenvalue of A and b is an eigenvalue of B, then in general l+b is not an eigenvalue of A+B

  9. Example

  10. Spectral Analysis of Symmetric MatricesA = AT (what are special about them?) Spectral Theorem: Every symmetric matrix has the factorization A = QLQT with real eigenvalues in L and orthonormal eigenvectors in L: A =QLQ-1 = QLQT with Q-1= QT.

  11. 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

  12. 2 by 2 Case Real Eigenvalues

  13. 2 by 2 Case so

  14. 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?

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

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

  17. Spectral Theorem Every symmetric matrix has the factorization A = QLQT with real eigenvalues in L and orthonormal eigenvectors in L: A =QLQ-1 = QLQT with Q-1= QT.

  18. Spectral Theorem and Spectral Decomposition Every symmetric matrix has the factorization A = QLQT with real eigenvalues in L and orthonormal eigenvectors in L: A =QLQ-1 = QLQT with Q-1= QT. xi xiT is the projection matrix on to xi !!!!!

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