Difference between revisions of "Decompose"

Revision as of 00:56, 22 January 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:

Function

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 3: / Line 3: @@
 [[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 =
+== Testing for positive definiteness ==
+To avoid an error, the following [[User-Defined Functions|user-defined function]] can be used to test for positive definiteness:
+:Function
+::'''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>
-To avoid an error, the following [[User-Defined Functions|UDF]] can be used to test for positive definiteness:
+== Complex numbers ==
- Function IsPosDefinite( A : Number[I,J] ; I,J : Index )
+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.
- 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 =
-''new in [[Analytica 4.5]]''.
-When «C» contains 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 =
+== See Also ==
 * [[SingularValueDecomp]]
 * [[EigenDecomp]]
+* [[Transpose]]
+* [[Matrix functions]]
+* [[Complex Numbers]]