Cholesky Factorization

Functions > Vector and Matrix > Matrix Factorization > Cholesky Factorization

• Cholesky(M,[p,[u]])—Returns the Cholesky square root of matrix M.

By default, the function returns a vector of two nested matrices P and L such that PT. M . P = L . LT if matrix M is real or PT . M . P = L . conj(LT) if matrix M is complex. Matrix P represents the pivoting matrix and matrix L the lower factorization matrix.

Use arguments p and u to get the desired output matrix:

Pivoting		Upper/Lower		Default	M = Real	M = Complex Hermitian
Disabled	(p=0)	Lower	(u=0)	No	M = L . LT	M = L . conj(LT)
Disabled	(p=0)	Upper	(u=1)	No	M = UT . U	M = conj(UT) . U
Enabled	(p=1)	Lower	(u=0)	Yes	PT . M . P = L . LT	PT . M . P = L . conj(LT)
Enabled	(p=1)	Upper	(u=1)	No	PT . M . P = UT . U	PT . M . P = conj(UT) . U

Arguments

• M is a real positive definite square matrix or complex Hermitian definite square matrix.

◦ M must be a full-rank positive definite matrix.

◦ Use function eigenvals to ensure that the matrix is positive definite by verifying that the returned vector does not contain any negative values.

• p (optional) is an integer. A zero value disables pivoting. A non-zero value enables pivoting (default behavior).

• u (optional) is an integer. A zero value forms the lower factorization of M (default behavior). A non-zero value forms the upper factorization of M.

◦ You can set argument p by itself.

◦ If you set argument u then you must also set argument p.