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matrix -- ATA = AAT = I
two vectors -- <u,v> = 0
functions -- ∫f(x)g(x)dx = 0 |
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| Idempotent and Orthogonal Matrix |
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| Existence/Uniqueness of Least Squares Solutions |
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| Always exist. Unique when A is not rank deficient. |
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| ATAx = ATb where ATA is an mxm matrix. Can solve with LU factorization. |
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Ha = a - 2(vTa/vTv)v
v = a - alpha(ek)
alpha = -sign(ak)||ak|| where ak is a vector from position k on |
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Orthogonal transformation to triangular form A=QR
[image] |
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Orthogonalize each successive vector against all the preceding ones.
[image] |
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As soon as each new vector qk is computed, immediately orthogonalize all remaining vectors against it... can use column pivoting
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| The rank of a matrix is equal to the number of nonzero singular values that it has |
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| A+ = VΣ+UT where the pseudo inverse of a scalar σ is 1/σ. |
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| Orthonormal Bases and SVD |
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| 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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