Polar Decomposition of a Matrix

Transcription

1 Polar Decomposition of a Matrix Garrett Buffington May 4, 2014

2 Table of Contents 1 The Polar Decomposition What is it? Square Root Matrix The Theorem

3 Table of Contents 1 The Polar Decomposition What is it? Square Root Matrix The Theorem 2 SVD and Polar Decomposition Polar Decomposition from SVD Example Using SVD

4 Table of Contents 1 The Polar Decomposition What is it? Square Root Matrix The Theorem 2 SVD and Polar Decomposition Polar Decomposition from SVD Example Using SVD 3 Geometric Concepts Motivating Example Rotation Matrices P and r

5 Table of Contents 1 The Polar Decomposition What is it? Square Root Matrix The Theorem 2 SVD and Polar Decomposition Polar Decomposition from SVD Example Using SVD 3 Geometric Concepts Motivating Example Rotation Matrices P and r 4 Applications Iterative methods for U

6 Table of Contents 1 The Polar Decomposition What is it? Square Root Matrix The Theorem 2 SVD and Polar Decomposition Polar Decomposition from SVD Example Using SVD 3 Geometric Concepts Motivating Example Rotation Matrices P and r 4 Applications Iterative methods for U 5 Conclusion

7 What is it? Definition (Right Polar Decomposition) The right polar decomposition of a matrix A C m n m n has the form A = UP where U C m n is a matrix with orthonormal columns and P C n n is positive semi-definite.

8 What is it? Definition (Right Polar Decomposition) The right polar decomposition of a matrix A C m n m n has the form A = UP where U C m n is a matrix with orthonormal columns and P C n n is positive semi-definite. Definition (Left Polar Decomposition) The left polar decomposition of a matrix A C n m m n has the form A = HU where H C n n is positive semi-definite and U C n m has orthonormal columns.

9 Square Root of a Matrix Theorem (The Square Root of a Matrix) If A is a normal matrix then there exists a positive semi-definite matrix P such that A = P 2.

10 Square Root of a Matrix Theorem (The Square Root of a Matrix) If A is a normal matrix then there exists a positive semi-definite matrix P such that A = P 2. Proof. Suppose you have a normal matrix A of size n. Then A is orthonormally diagonalizable. This means that there is a unitary matrix S and a diagonal matrix B whose diagonal entries are the eigenvalues of A so that A = SBS where S S = I n. Since A is normal the diagonal entries of B are all positive, making B positive semi-definite as well. Because B is diagonal with real, non-negative entries we can easily define a matrix C so that the diagonal entries of C are the square roots of the eigenvalues of A. This gives us the matrix equality C 2 = B. Define P with the equality P = SCS.

11 The Theorem Definition (P) The matrix P is defined as A A where A C m n.

12 The Theorem Definition (P) The matrix P is defined as A A where A C m n. Theorem (Right Polar Decomposition) For any matrix A C m n, where m n, there is a matrix U C m n with orthonormal columns and a positive semi-definite matrix P C n n so that A = UP.

13 Example A A = A A =

14 Example A A = A A = S, S 1, and C S = S 1 = C =

15 Example P P = A A = S CS 1 =

16 Example P P = A A = S CS 1 = U U =

17 Example P P = A A = S CS 1 = U U = A UP =

18 Polar Decomposition from SVD Theorem (SVD to Polar Decomposition) For any matrix A C m n, where m n, there is a matrix U C m n with orthonormal columns and a positive semi-definite matrix P C n n so that A = UP.

19 Polar Decomposition from SVD Theorem (SVD to Polar Decomposition) For any matrix A C m n, where m n, there is a matrix U C m n with orthonormal columns and a positive semi-definite matrix P C n n so that A = UP. Proof. A = U S SV = U S I n SV = U S V VSV = UP

20 Example Using SVD Give Sage our A and ask to find the SVD SVD A =

21 Example Using SVD Give Sage our A and ask to find the SVD SVD A = Components U S = S = V =

22 Example Using SVD U U = U S V = =

23 Example Using SVD U U = U S V = = P P = VSV = =

24 Geometry Concepts Matrices A = UP

25 Geometry Concepts Matrices A = UP Complex Numbers z = re iθ

26 Motivating Example 2 2 A = [ ]

27 Motivating Example 2 2 A = [ ] Polar Decomposition [ ] [ ] cos 30 sin 30 U = = sin 30 cos 30 [ ] P = = A A

28 P and r 2 2 [ ] cos θ sin θ R = sin θ cos θ r r = x 2 + y 2

29 P and r 2 2 [ ] cos θ sin θ R = sin θ cos θ r r = x 2 + y 2 r Vector r = r r

30 P and r 2 2 [ ] cos θ sin θ R = sin θ cos θ r r = x 2 + y 2 r Vector r = r r P P = A A

31 iitit Continuum Mechanics

32 iitit Continuum Mechanics ititit Computer Graphics

33 Iterative Methods for U Newton Iteration U k+1 = 1 2 (U k + U t k ), U 0 = A

34 Iterative Methods for U Newton Iteration U k+1 = 1 2 (U k + U t k ), U 0 = A Frobenius Norm Accelerator γ Fk = U 1 k 1 2 F U k 1 2 F

35 Iterative Methods for U Newton Iteration U k+1 = 1 2 (U k + U t k ), U 0 = A Frobenius Norm Accelerator γ Fk = U 1 k 1 2 F U k 1 2 F Spectral Norm Accelerator γ Sk = U 1 k 1 2 S U k 1 2 S

36

37 Rotation Matrices What s Up with U? U = R θ R ψ R κv cos ψ cos κ cos ψ sin κ sin ψ = sin θ sin ψ cos κ cos θ sin κ sin θ sin ψ sin κ + cos θ cos κ sin θ cos ψ V cos θ sin ψ cos κ + sin θ sin κ cos θ sin ψ sin κ sin θ cos κ cos θ cos ψ

38 P and r r r = x 2 + y 2

39 P and r r r = x 2 + y 2 r Vector r = r r

40 P and r r r = x 2 + y 2 r Vector r = r r P P = A A

41 Ideal Example U

42 Ideal Example U P

43 Ideal Example U P A = UP

44 Applications Use Continuum Mechanics

45 Applications Use Continuum Mechanics Another Use Computer Graphics

46 Iterative Methods for U Newton Iteration U k+1 = 1 2 (U k + U t k ), U 0 = A

47 Iterative Methods for U Newton Iteration U k+1 = 1 2 (U k + U t k ), U 0 = A Frobenius Norm Accelerator γ Fk = U 1 k 1 2 F U k 1 2 F

48 Iterative Methods for U Newton Iteration U k+1 = 1 2 (U k + U t k ), U 0 = A Frobenius Norm Accelerator γ Fk = U 1 k 1 2 F U k 1 2 F Spectral Norm Accelerator γ Sk = U 1 k 1 2 S U k 1 2 S

49

50 Conclusion

51 Conclusion

52 References 1. Beezer, Robert A. A Second Course in Linear Algebra.Web. 2. Beezer, Robert A. A First Course in Linear Algebra.Web. 3. Byers, Ralph and Hongguo Xu. A New Scaling For Newton s Iteration for the Polar Decomposition and Its Backward Stability Duff, Tom, Ken Shoemake. Matrix animation and polar decomposition. In Proceedings of the conference on Graphics interface(1992): Courses/838-s2002/Papers/polar-decomp.pdf. 5. Gavish, Matan. A Personal Interview with the Singular Value Decomposition Gruber, Diana. The Mathematics of the 3D Rotation Matrix McGinty, Bob. http: //