Polar Decomposition of a Matrix

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Polar Decomposition of a Matrix Garrett Buffington May 4, 2014

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

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

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

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

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

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.

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.

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.

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.

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

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.

Example A 3 8 2 14 38 26 A = 2 5 7 A A = 38 105 75 1 4 6 25 76 89

Example A 3 8 2 14 38 26 A = 2 5 7 A A = 38 105 75 1 4 6 25 76 89 S, S 1, and C 1 1 1 S = 0.3868 2.3196 2.8017 0.0339 3.0376 2.4687 0.8690 0.3361 0.0294 S 1 = 0.0641 0.1486 0.1946 0.0669 0.1875 0.1652 0.4281 0 0 C = 0 4.8132 0 0 0 13.5886

Example P P = 1.5897 3.1191 1.3206 A A = S CS 1 = 3.1191 8.8526 4.1114 1.3206 4.1114 8.3876

Example P P = 1.5897 3.1191 1.3206 A A = S CS 1 = 3.1191 8.8526 4.1114 1.3206 4.1114 8.3876 U 0.3019 0.9175 0.2588 U = 0.6774 0.0154 0.7355 0.6708 0.3974 0.6262

Example P P = 1.5897 3.1191 1.3206 A A = S CS 1 = 3.1191 8.8526 4.1114 1.3206 4.1114 8.3876 U 0.3019 0.9175 0.2588 U = 0.6774 0.0154 0.7355 0.6708 0.3974 0.6262 A 1.5897 3.1191 1.3206 0.3019 0.9175 0.2588 UP = 3.1191 8.8526 4.1114 0.6774 0.0154 0.7355 1.3206 4.1114 8.3876 0.6708 0.3974 0.6262

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.

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

Example Using SVD Give Sage our A and ask to find the SVD SVD 3 8 2 A = 2 5 7 1 4 6

Example Using SVD Give Sage our A and ask to find the SVD SVD 3 8 2 A = 2 5 7 1 4 6 Components 0.5778 0.8142 0.0575 U S = 0.6337 0.4031 0.6602 0.5144 0.4179 0.7489 13.5886 0 0 S = 0 4.8132 0 0 0 0.4281 0.2587 0.2531 0.9322 V = 0.7248 0.5871 0.3605 0.6386 0.7689 0.0316

Example Using SVD U U = U S V 0.5778 0.8142 0.0575 0.2587 0.7248 0.6386 = 0.6337 0.4031 0.6602 0.2531 0.5871 0.7689 0.5144 0.4179 0.7489 0.9322 0.3605 0.0316 0.3019 0.9175 0.2588 = 0.6774 0.0154 0.7355 0.6708 0.3974 0.6262

Example Using SVD U U = U S V 0.5778 0.8142 0.0575 0.2587 0.7248 0.6386 = 0.6337 0.4031 0.6602 0.2531 0.5871 0.7689 0.5144 0.4179 0.7489 0.9322 0.3605 0.0316 0.3019 0.9175 0.2588 = 0.6774 0.0154 0.7355 0.6708 0.3974 0.6262 P P = VSV 0.2587 0.2531 0.9322 = 0.7248 0.5871 0.3605 13.5886 0 0 0.2587 0.7248 0.6386 0 4.8132 0 0.2531 0.5871 0.7689 0.6386 0.7689 0.0316 0 0 0.4281 0.9322 0.3605 0.0316 1.5897 3.1191 1.3206 = 3.1191 8.8526 4.1114 1.3206 4.1114 8.3876

Geometry Concepts Matrices A = UP

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

Motivating Example 2 2 A = [ 1.300 ].375.750.650

Motivating Example 2 2 A = [ 1.300 ].375.750.650 Polar Decomposition [ ] [ ] 0.866 0.500 cos 30 sin 30 U = = 0.500 0.866 sin 30 cos 30 [ ] 1.50 0.0 P = = A 0.0 0.75 A

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

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 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

iitit Continuum Mechanics

iitit Continuum Mechanics ititit Computer Graphics

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

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

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

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

P and r r r = x 2 + y 2

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

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

Ideal Example U.5.5.7071.1464.8536.5.8536.1768.5

Ideal Example U.5.5.7071.1464.8536.5.8536.1768.5 P 1 0 0 0.5 0 0 0.33

Ideal Example U.5.5.7071.1464.8536.5.8536.1768.5 P 1 0 0 0.5 0 0 0.33 A = UP.5.25.2355.1464.4268.1665.8536.0884.1665

Applications Use Continuum Mechanics

Applications Use Continuum Mechanics Another Use Computer Graphics

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

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

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

Conclusion

Conclusion

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. http://www.math.ku.edu/~xu/arch/bx1-07r2.pdf. 4. Duff, Tom, Ken Shoemake. Matrix animation and polar decomposition. In Proceedings of the conference on Graphics interface(1992): 258-264.http://research.cs.wisc.edu/graphics/ Courses/838-s2002/Papers/polar-decomp.pdf. 5. Gavish, Matan. A Personal Interview with the Singular Value Decomposition. http://www.stanford.edu/~gavish/documents/svd_ans_you.pdf. 6. Gruber, Diana. The Mathematics of the 3D Rotation Matrix. http://www.fastgraph.com/makegames/3drotation/. 7. McGinty, Bob. http: //www.continuummechanics.org/cm/polardecomposition.html.

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