QR decomposition: A = QR Q is orthonormal R is upper triangular To find QR decomposition: 1.) Q: Use Gram-Schmidt to find orthonormal basis for column.

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Presentation transcript:

QR decomposition: A = QR Q is orthonormal R is upper triangular To find QR decomposition: 1.) Q: Use Gram-Schmidt to find orthonormal basis for column space of A 2.) Let R = Q T A

Find the QR decomposition of A = ) Use Gram-Schmidt to find orthonormal basis for column space of A

Find the QR decomposition of A = ) Use Gram-Schmidt to find orthonormal basis for column space of A col(A) = span, { }

Find the QR decomposition of A = ) Use Gram-Schmidt to find orthogonal basis for column space of A col(A) = span, { }

Find the QR decomposition of A = ) Use Gram-Schmidt to find orthogonal basis for column space of A col(A) = span, = span, { } { ?}?}

Find orthogonal projection of onto

Find orthogonal projection of onto (1, 2)  (4, 3) (1, 2)  (1, 2) proj = 1(4) + 2(3) =

Find orthogonal projection of onto (1, 2)  (4, 3) (1, 2)  (1, 2) proj = 1(4) + 2(3) =

Find orthogonal projection of onto (1, 2)  (4, 3) (1, 2)  (1, 2) proj = 1(4) + 2(3) = = 2 =

Find orthogonal projection of onto (1, 2)  (4, 3) (1, 2)  (1, 2) proj = =

Find the component of orthogonal to

Find the component of orthogonal to − =

Short-cut for R 2 case:

Find the QR decomposition of A = ) Use Gram-Schmidt to find orthogonal basis for column space of A col(A) = span, = span, { } { } 2

= √ = √ 5 = √ (-1) = √ 5 Find the length of each vector:

Divide each vector by its length: col(A) = span, = span, = span, { } { } 2 {}

{} col(A) = span, Q = A = QR

A = QR Q -1 A = Q -1 QR Q -1 A = R Q has orthonormal columns: Thus Q -1 = Q T Thus R = Q -1 A = Q T A

Find the QR decomposition of A = = QR R = Q -1 A = Q T A = = =

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