Householder transformation | Householder Reflection with QR Decomposition
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Householder Reflection or Transformation is one the methods of decomposing a matrix into an Orthogonal Matrix (Q) and Right Upper Triangular Matrix (R). It helps to solve systems of equation using backward substitution.
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Householder transformation | Householder Reflection with QR Decomposition
- 1. Householder Transformation and QR Decomposition Isaac Amornortey Yowetu NIMS-GHANA January 4, 2021
- 2. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 3. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Introduction The Householder matrix (reflector) is a unitary (or orthogonal) matrix that is often used to decompose a matrix into an upper triangular matrix and orthogonal matrix. In particular, Householder matrices are usually used to destroy the entries below the main diagonal of a matrix. Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 4. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Householder Properties The Householder Transformation or Matrix has the following properties: It is symmetric H = HT It is also orthogonal HT = H−1 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 5. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Consider H to be the Householder Matrix: Proof: Symmetry H = HT (1) H = I − 2vvT (2) HT = (I − 2vvT )T (3) = IT − 2(vvT )T (4) = IT − 2(vT )T vT (5) = I − 2vvT (6) = H (7) Hence, proved! Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 6. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Consider H to be the Householder Matrix: Proof: Orthogonality HT = H−1 (8) HT H = H−1 H = I (9) HT H = (I − 2vvT )T (I − 2vvT ) (10) = (I − 2vvT )(I − 2vvT ) (11) = I − 2(vvT ) − 2(vvT ) + 4vvT vvT (12) = I − 4(vvT ) + 4v(vT v)vT (13) = I − 4(vvT ) + 4vvT (14) = I (15) Hence, proved! (16) Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 7. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Householder Reflection(Matrix) Derivation 1. Finding any vector u Considering n dimensional vector x = [x1, x2, ..., xn]T and e = [1, 0, ..., 0]T which a first standard basis vector. u = x − ||x||e1 2. Finding Householder vector v v = u ||u|| 3. Construction the Householder Matrix H = I − 2vvT = I − 2uuT ||u||2 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 8. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Remark When the Householder Matrix H is a applied to x, it obtains its reflection. Say, x = Hx Proof x = Hx = (I − 2vvT )x = I − 2uuT ||u||2 x (17) ||u||2 = (x − ||x||e1)T (x − ||x||e1) (18) = (xT − ||x||eT 1 )(x − ||x||e1) (19) = xT x − ||x||xT e1 − ||x||eT 1 x + ||x||2 eT 1 e1 (20) = ||x||2 − 2||x||x1 + ||x||2 (21) Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 9. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Proof continues... = 2||x||2 − 2||x||x1 (22) = 2(||x||2 − ||x||x1) (23) x = I − 2u(x − ||x||e1)T 2(||x||2 − ||x||x1) x (24) = x − 2u(xT x − ||x||eT 1 x) 2(||x||2 − ||x||x1) (25) = x − u(||x||2 − ||x||x1) (||x||2 − ||x||x1) (26) = x − u (27) = x − (x − ||x||e1) (28) = ||x||e1 (29) Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 10. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Question 1 Consider the matrix B = −1 −1 1 1 3 3 −1 −1 5 using Householder Reflection, determine the QR Factorization. Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 11. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution We consider x1 = [−1, 1 − 1]T and ||x1||2 = √ 3 u1 = x1 − ||x1||e1 = −1 1 −1 − √ 3 1 0 0 = −2.7321 1.0000 −1.0000 v1 = u1 ||u1|| = −0.8881 0.3251 −0.3251 H1 = I − 2vvT = −0.5774 0.5774 −0.5774 0.5774 0.7887 0.2113 −0.5774 0.2113 0.7887 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 12. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution Continues... H1B = −0.5774 0.5774 −0.5774 0.5774 0.7887 0.2113 −0.5774 0.2113 0.7887 −1 −1 1 1 3 3 −1 −1 5 R1 = 1.7321 2.8868 −1.7321 0 1.5774 4.0000 0 0.4226 4.0000 Q1 = H1 = −0.5774 0.5774 −0.5774 0.5774 0.7887 0.2113 −0.5774 0.2113 0.7887 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 13. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution continues... We consider x2 = [0, 1.5774, 0.4226]T and ||x2||2 = 1.6330 u2 = x2 − ||x2||e2 = 0 1.5774 0.4226 − 1.6330 0 1 0 = 0 −0.0556 0.4226 v2 = u2 ||u2|| = 0 −0.1305 0.9914 H2 = I − 2vvT = 1.0000 0 0 0 0.9659 0.2588 0 0.2588 −0.9659 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 14. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution Continues... H2R1 = 1.0000 0 0 0 0.9659 0.2588 0 0.2588 −0.9659 1.7321 2.8868 −1.7321 0 1.5774 4.0000 0 0.4226 4.0000 R2 = 1.7321 2.8868 −1.7321 0 1.6330 4.8990 0 0 −2.8284 Q2 = H1H2 = −0.5774 0.5774 −0.5774 0.5774 0.7887 0.2113 −0.5774 0.2113 0.7887 1.0000 0 0 0 0.9659 0.2588 0 0.2588 −0.9659 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 15. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution: Q = Q2 −0.5774 −0.4082 −0.7071 0.5774 −0.8165 0 −0.5774 −0.4082 0.7071 R = R2 1.7321 2.8868 −1.7321 0 1.6330 4.8990 0 0 −2.8284 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 16. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Question 2 Find QR Decomposition using Householder Method of the matrix B = 1 −1 4 1 4 −2 1 4 2 1 −1 0 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 17. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution We consider x1 = [1, 1, 1, 1]T and ||x1||2 = 2 u1 = x1 − ||x1||e1 = 1 1 1 1 − 2 1 0 0 0 = −1 1 1 1 v1 = u1 ||u1|| = −0.5000 0.5000 0.5000 0.5000 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 18. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition H1 = 0.50 0.50 0.50 0.50 0.50 0.50 −0.50 −0.50 0.50 −0.50 0.50 −0.50 0.50 −0.50 −0.50 0.50 H1B = 0.50 0.50 0.50 0.50 0.5000 0.5000 −0.50 −0.50 0.50 −0.50 0.50 −0.50 0.50 −0.50 −0.50 0.50 1 −1 4 1 4 −2 1 4 2 1 −1 0 R1 = 2 3 2 0 0 0 0 0 4 0 −5 2 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 19. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution Continues... Q1 = H1 = 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 −0.5000 −0.5000 0.5000 −0.5000 0.5000 −0.5000 0.5000 −0.5000 −0.5000 0.5000 We consider x2 = [0, 0, 0, −5]T and ||x2||2 = 5 u2 = x2 − ||x2||e2 = 0 0 0 −5 − 5 0 1 0 0 = 0 −5 0 −5 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 20. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution continues... v2 = u2 ||u2|| = 0 −0.7071 0 −0.7071 H2 = I − 2vvT = 1.00 0 0 0 0 0 0 −1 0 0 1 0 0 −1 0 0 H2R1 = 1.00 0 0 0 0 0 0 −1 0 0 1 0 0 −1 0 0 2 3 2 0 0 0 0 0 4 0 −5 2 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 21. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution Continues... R2 = 2 3 2 0 5 −2 0 0 4 0 0 0 Q2 = H1H2 = 0.50 0.50 0.50 0.50 0.50 0.50 −0.50 −0.50 0.50 −0.50 0.50 −0.50 0.50 −0.50 −0.50 0.50 1.00 0 0 0 0 0 0 −1 0 0 1 0 0 −1 0 0 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 22. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition Solution: Q = Q2 0.50 −0.50 0.50 0.50 0.50 0.50 −0.50 −0.50 0.50 0.50 0.50 −0.50 0.50 −0.50 −0.50 −0.50 R = R2 2 3 2 0 5 −2 0 0 4 0 0 0 Isaac Amornortey Yowetu Householder Transformation and QR Decomposition
- 23. Introduction Proof of Householder Transformation Properties Householder Reflection Derivation Application of Householder Reflection to QR Decomposition End THANK YOU Reference www.statlect.com/matrix-algebra/Householder-matrix Isaac Amornortey Yowetu Householder Transformation and QR Decomposition