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Description |
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CholeskyDecomposition
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Cholesky Decomposition.
For a symmetric, positive definite matrix A, the Cholesky decomposition
is an lower triangular matrix L so that A = L*L'.
If the matrix is not symmetric or positive definite, the constructor
returns a partial decomposition and sets an internal flag that may
be queried by the isSPD() method.
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EigenvalueDecomposition
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Eigenvalues and eigenvectors of a real matrix.
If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
diagonal and the eigenvector matrix V is orthogonal.
I.e. A = V.Multiply(D.Multiply(V.Transpose())) and
V.Multiply(V.Transpose()) equals the identity matrix.
If A is not symmetric, then the eigenvalue matrix D is block diagonal
with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The
columns of V represent the eigenvectors in the sense that A*V = V*D,
i.e. A.Multiply(V) equals V.Multiply(D). The matrix V may be badly
conditioned, or even singular, so the validity of the equation
A = V*D*Inverse(V) depends upon V.cond().
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GeneralMatrix
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LUDecomposition
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Maths
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QRDecomposition
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QR Decomposition.
For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n
orthogonal matrix Q and an n-by-n upper triangular matrix R so that
A = Q*R.
The QR decompostion always exists, even if the matrix does not have
full rank, so the constructor will never fail. The primary use of the
QR decomposition is in the least squares solution of nonsquare systems
of simultaneous linear equations. This will fail if IsFullRank()
returns false.
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SingularValueDecomposition
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