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 RationalCholeskyDecomposition

Source Notebook

Compute the rational Cholesky decomposition of a matrix

Contributed by: Jan Mangaldan

ResourceFunction["RationalCholeskyDecomposition"][m]

gives the rational Cholesky decomposition of a matrix m, given as a list {l,d} where l is a unit lower-triangular matrix and d is the diagonal of a diagonal matrix.

Details and Options

The rational Cholesky decomposition is also referred to as the decomposition.
The matrix m can be numerical or symbolic, but must be Hermitian and positive definite.
ResourceFunction["RationalCholeskyDecomposition"][m] yields a lower‐triangular matrix l and a list d so that l.DiagonalMatrix[d].ConjugateTranspose[l]⩵m.
With the setting TargetStructure→"Structured", ResourceFunction["RationalCholeskyDecomposition"][m] returns a list {l,d} where l is a LowerTriangularMatrix and d is a DiagonalMatrix.

Examples

Basic Examples (2) 

Perform a rational Cholesky decomposition on a matrix m:

In[1]:=
{lm, dm} = ResourceFunction["RationalCholeskyDecomposition"][{{2, 1}, {1, 2}}]
Out[1]=

Confirm that lm.DiagonalMatrix[dm].ConjugateTranspose[lm]⩵m:

In[2]:=
lm . DiagonalMatrix[dm] . ConjugateTranspose[lm]
Out[2]=

Scope (3) 

Hilbert matrices are symmetric and positive definite:

In[3]:=
m = HilbertMatrix[5]
Out[3]=

Compute the rational Cholesky decomposition with exact arithmetic:

In[4]:=
ResourceFunction["RationalCholeskyDecomposition"][m]
Out[4]=
In[5]:=
MatrixForm[First[%]]
Out[5]=

Compute the rational Cholesky decomposition with machine arithmetic:

In[6]:=
ResourceFunction["RationalCholeskyDecomposition"][N[m]]
Out[6]=

Compute the rational Cholesky decomposition with 24-digit precision arithmetic:

In[7]:=
ResourceFunction["RationalCholeskyDecomposition"][N[m, 24]]
Out[7]=

Compute the rational Cholesky decomposition of a random complex Hermitian matrix:

In[8]:=
r = RandomComplex[(1 + I) {-1, 1}, {4, 4}]; ResourceFunction["RationalCholeskyDecomposition"][ r . ConjugateTranspose[r] + IdentityMatrix[4]]
Out[9]=

Use symbolic matrices:

In[10]:=
ResourceFunction["RationalCholeskyDecomposition"][{{x, y}, {y, x}}]
Out[10]=

Options (1) 

TargetStructure (1) 

With TargetStructure→"Structured", a list containing a LowerTriangularMatrix and a DiagonalMatrix is returned:

In[11]:=
ResourceFunction["RationalCholeskyDecomposition"][\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "1", "1"}, {"1", "2", "2", "2"}, {"1", "2", "3", "3"}, {"1", "2", "3", "4"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), TargetStructure -> "Structured"]
Out[11]=

Applications (3) 

A symmetric Toeplitz matrix:

In[12]:=
tm = ToeplitzMatrix[{x, y, z}]
Out[12]=

Compute its rational Cholesky decomposition:

In[13]:=
ResourceFunction["RationalCholeskyDecomposition"][tm]
Out[13]=

Derive the conditions for the original matrix to be positive definite:

In[14]:=
Reduce[Thread[Last[%] > 0], {x, y, z}] // FullSimplify
Out[14]=

Properties and Relations (4) 

Create a symmetric positive definite matrix:

In[15]:=
m = {{4, 3, 2, 1}, {3, 4, 3, 2}, {2, 3, 4, 3}, {1, 2, 3, 4}};

Compute its rational Cholesky decomposition:

In[16]:=
{lm, dm} = ResourceFunction["RationalCholeskyDecomposition"][m]
Out[16]=

Compute the Cholesky decomposition from the result of RationalCholeskyDecomposition:

In[17]:=
DiagonalMatrix[Sqrt[dm]] . ConjugateTranspose[lm] // MatrixForm
Out[18]=

Compare with the result of CholeskyDecomposition:

In[19]:=
CholeskyDecomposition[m] // MatrixForm
Out[19]=

Version History

  • 2.0.0 – 27 March 2024
  • 1.0.0 – 06 January 2021

Source Metadata

  • Citation:
    • Golub, G.H., Van Loan, C.F., Matrix Computations, Third Edition. Baltimore, MD: Johns Hopkins University Press, 1996.

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