Difference between revisions of "Decompose"

Latest revision as of 23:03, 8 March 2016

Decompose(C, I, J)

Returns the Cholesky decomposition (square root) matrix of matrix «C» along dimensions «I» and «J». Matrix «C» must be symmetric and positive-definite. (Positive-definite means that v*C*v > 0, for all vectors v.)

Cholesky decomposition computes a lower diagonal matrix L such that L*L' = C, where L' is the transpose of L.

Testing for positive definiteness

To avoid an error, the following user-defined function can be used to test for positive definiteness:

Declaration:

IsPosDefinite(A: Number[I, J]; I, J: Index)

Description:

Returns true (1) if «A» is positive-definite, 0 (false) otherwise.

Definition:

Min(EigenDecomp(A + Transpose(A, I, J), I, J)[.Item = 'value'], J) > 0

Complex numbers

When «C» contains complex numbers, then the matrix must be Hermitian and positive definite. The Hermitian condition is the same as the symmetric condition for real-valued matrices, but requires the transpose to be the complex conjugate of the matrix.

@@ Line 1: / Line 1: @@
 [[category:Matrix Functions]]
+[[Category:Functions that operate on complex numbers]]
 [[Category:Doc Status C]] <!-- For Lumina use, do not change -->
-'''Decompose(C,I,J)'''
+==Decompose(C, I, J)==
-Returns the Cholesky decomposition (square root) matrix of
+Returns the Cholesky decomposition (square root) matrix of matrix «C» along dimensions «I» and «J». Matrix «C» must be symmetric
-matrix C along dimensions I and J. Matrix C must be symmetric
+and positive-definite. (Positive-definite means that ''v*C*v > 0'', for all vectors ''v''.)
-and positive-definite. (Positive-definite means that v * C * v > 0,
-for all vectors v.)
-Cholesky decomposition computes a lower diagonal matrix L
+Cholesky decomposition computes a lower diagonal matrix ''L'' such that ''L*L' = C'', where ''L''' is the transpose of ''L''.
-such that L * L' = C, where L' is the transpose of L.
+== Testing for positive definiteness ==
+To avoid an error, the following [[User-Defined Functions|user-defined function]] can be used to test for positive definiteness:
+Declaration:
+:'''IsPosDefinite'''(A: Number[I, J]; I, J: Index)
+Description:
+:Returns true (1) if «A» is positive-definite, 0 (false) otherwise.
+Definition:
+::<code>Min(EigenDecomp(A + Transpose(A, I, J), I, J)[.Item = 'value'], J) > 0</code>
+== Complex numbers ==
+When «C» contains [[Complex Numbers|complex numbers]], then the matrix must be [http://en.wikipedia.org/wiki/Hermitian_matrix Hermitian] and positive definite. The Hermitian condition is the same as the symmetric condition for real-valued matrices, but requires the [[transpose]] to be the complex conjugate of the matrix.
+== See Also ==
+* [[SingularValueDecomp]]
+* [[EigenDecomp]]
+* [[Transpose]]
+* [[MatrixMultiply]]
+* [[Matrix functions]]
+* [[Complex Numbers]]