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Definition
matrix  A^{T}A = AA^{T} = 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 

Definition
Always exist. Unique when A is not rank deficient. 


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A^{T}Ax = A^{T}b where A^{T}A is an mxm matrix. Can solve with LU factorization. 


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Definition
Ha = a  2(v^{T}a/v^{T}v)v
v = a  alpha(e_{k})
alpha = sign(a_{k})a_{k} where a_{k }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 q_{k} is computed, immediately orthogonalize all remaining vectors against it... can use column pivoting
[image] 


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cond(A) = σ_{max}/σ_{min} 


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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^{+}U^{T} where the pseudo inverse of a scalar σ is 1/σ. 


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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. 

