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Householder Transformations. Example:. DEF:. is called Householder matrix ( Reflection, Transformation). Householder Transformations. Example. DEF:. is called Householder matrix ( Reflection, Transformation). Rem :. They are rank-1 modifications of the identity.

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  1. Householder Transformations Example: DEF: is called Householder matrix ( Reflection, Transformation)

  2. Householder Transformations Example DEF: is called Householder matrix ( Reflection, Transformation) Rem: • They are rank-1 modifications of the identity 2) They are symmetric and orthogonal Example Example 3) They can be used to zero selected components of a vector

  3. QR factorization A = Q R Orthogonal = Upper triangular We begin with a QR factorization method that utilizes Householder transformations.

  4. QR factorization Upper triangular

  5. QR factorization Upper triangular Remark: Product of orthogonal matrices is an orthogonal

  6. Householder Transformations DEF: is called Householder matrix ( Reflection, Transformation) Example (left multiplication)

  7. Computing Householder Choice of sign: It is dangerous if x is close to a positive multiple of e1 because sever cancellation would occur. Solution:

  8. Applying Householder Householder matrices never formed explicitly

  9. Householder Transformations Matlab [Q,R] = qr(A), where A is m-by-n, produces an m-by-n upper triangular matrix R and an m-by-m unitary matrix Q so that A = Q*R.

  10. Reduced QR factorization Orthogonal = Upper triangular Orthogonal = Upper triangular

  11. Gram-Schmedit Gram-schmeditOrthogonalization

  12. Gram-Schmedit

  13. Gram-Schmedit Example:

  14. Givens Matrices Rotation Example:

  15. Givens Rotations To zero a specific entry (not all as Householder) Givens Rotations are of this form: We can force to be zero by setting: Givens Rotation are orthogonal

  16. Householder Transformations Example: Applying Givens Rotations # of operations = 6n Just effects two rows of A

  17. Householder Transformations Givens QR

  18. Householder Transformations Theorem: (QR Decomposition) Theorem: If A is real m-by-n matrix matrix of full rank, then A has a unique reduced QR factorization (QR Decomposition) If A is real m-by-n matrix, then there exist orthogonal matrix Q such that

  19. Householder Transformations function [v]=house(x) v=x; v(1)=sign(x(1))*norm(x)+x(1); function [Q,R]=myqr(A) [m,n]=size(A); for k=1:n x=A(k:m,k) [v]=house(x); A(k:m,k:n)= A(k:m,k:n) –( 2/v’*v) v*(v’**A(k:m,k:n)); end

  20. Householder Transformations

  21. Questions Orthogonal Matrices: 1) Two class of orthogonal matrices ( small modification from the identity ) Householder - Givens any others 2) Can we think of Q such that Q(col1) = multiple of e1 Q(col2) = multiple of e2

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