# Shared Flashcard Set

## Details

Linear Least Squares Problems
Orthogonalization methods; SVD
15
Computer Science
03/02/2014

Term
 Orthogonality
Definition
 matrix -- ATA = AAT = I two vectors -- = 0 functions -- ∫f(x)g(x)dx = 0
Term
 Idempotence
Definition
 P = P2
Term
 Orthogonal Projector
Definition
 Idempotent and Orthogonal Matrix
Term
 Existence/Uniqueness of Least Squares Solutions
Definition
 Always exist.  Unique when A is not rank deficient.
Term
 Normal Equations
Definition
 ATAx = ATb where ATA is an mxm matrix.  Can solve with LU factorization.
Term
 Householder Reflections
Definition
 Ha = a - 2(vTa/vTv)v v = a - alpha(ek) alpha = -sign(ak)||ak|| where ak is a vector from position k on
Term
 Givens Rotations
Definition
 [image]
Term
 QR Factorization
Definition
 Orthogonal transformation to triangular form A=QR [image]
Term
 Classical Gram-Schmidt
Definition
 Orthogonalize each successive vector against all the preceding ones. [image]
Term
 Modified Gram-Schmidt
Definition
 As soon as each new vector qk is computed, immediately orthogonalize all remaining vectors against it... can use column pivoting [image]
Term
 Euclidean Norm & SVD
Definition
 ||A||2 = σmax
Term
 Condition Number and SVD
Definition
 cond(A) = σmax/σmin
Term
 Rank and SVD
Definition
 The rank of a matrix is equal to the number of nonzero singular values that it has
Term
 Pseudoinverse and SVD
Definition
 A+ = VΣ+UT where the pseudo inverse of a scalar σ is 1/σ.
Term
 Orthonormal Bases and SVD
Definition
 The columns of U corresponding to nonzero singular values form an orthonoral basis for span(A) and remaining columns of U form orthonoral basis for its orthogonal complement.  Columns of V corresponding to zero singular values form orthonormal basis for null space of A, and remaining columns of V form orthonormal basis for orthogonal complement of null space.
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