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 AasenDecomposition

Source Notebook

Compute the Aasen decomposition of a Hermitian matrix

Contributed by: Jan Mangaldan

ResourceFunction["AasenDecomposition"][m]

yields the Aasen decomposition of a matrix m.

Details and Options

The matrix m can be numerical or symbolic, but must be Hermitian.
The result of ResourceFunction["AasenDecomposition"] is a list {l,t,p}, where l is a unit lower triangular matrix, t is a SparseArray representing a Hermitian tridiagonal matrix, and p is a permutation matrix.
The matrices returned by ResourceFunction["AasenDecomposition"] satisfy the relation p.m.Transpose[p]=l.t.ConjugateTranspose[l].
With the setting TargetStructure→"Structured", ResourceFunction["AasenDecomposition"][m] returns a list {l,t,p} where l is a LowerTriangularMatrix, t is a SparseArray representing a Hermitian tridiagonal matrix, and p is a PermutationMatrix.

Examples

Basic Examples (3) 

Find the Aasen decomposition of a symmetric matrix:

In[1]:=
{l, t, p} = ResourceFunction["AasenDecomposition"][\!\(\* TagBox[ RowBox[{"m", "=", RowBox[{"(", "", GridBox[{ { RowBox[{"-", "1"}], "1", "1"}, {"1", "0", "2"}, {"1", "2", "1"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}]}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\)]
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View the results in matrix form:

In[2]:=
Map[MatrixForm, %]
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Verify the decomposition:

In[3]:=
p . m . Transpose[p] == l . t . ConjugateTranspose[l]
Out[3]=

Scope (3) 

A symmetric indefinite matrix:

In[4]:=
Sign[Eigenvalues[m]]
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Compute the Aasen decomposition with exact arithmetic:

In[5]:=
MatrixForm /@ ResourceFunction["AasenDecomposition"][m]
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Compute the Aasen decomposition with machine arithmetic:

In[6]:=
MatrixForm /@ ResourceFunction["AasenDecomposition"][N[m]]
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Compute the Aasen decomposition with 24-digit precision arithmetic:

In[7]:=
ResourceFunction["AasenDecomposition"][N[m, 24]]
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Generate a pseudorandom Hermitian matrix drawn from the Gaussian unitary ensemble:

In[8]:=
m = RandomVariate[GaussianUnitaryMatrixDistribution[4]]
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Compute its Aasen decomposition:

In[9]:=
{l, t, p} = ResourceFunction["AasenDecomposition"][m]
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Verify the decomposition:

In[10]:=
p . m . Transpose[p] - l . t . ConjugateTranspose[l] // Chop
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Compute the Aasen decomposition of a symbolic matrix:

In[11]:=
MatrixForm /@ ResourceFunction["AasenDecomposition"][ Symmetrize[Array[C, {3, 3}], Hermitian[{1, 2}]]]
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Options (1) 

TargetStructure (1) 

With TargetStructure→"Structured", a list containing a LowerTriangularMatrix, a SparseArray representing a Hermitian tridiagonal matrix, and a PermutationMatrix is returned:

In[12]:=
ResourceFunction["AasenDecomposition"][\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "1", "1"}, {"1", "2", "2", "2"}, {"1", "2", "0", "0"}, {"1", "2", "0", "0"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), TargetStructure -> "Structured"]
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Version History

  • 3.0.0 – 27 March 2024
  • 2.0.0 – 29 August 2022
  • 1.0.0 – 11 February 2021

Source Metadata

  • Citation:
    • Aasen J. O., "On the Reduction of a Symmetric Matrix to Tridiagonal Form." BIT Numerical Mathematics, Vol. 11, 233–242, 1971. DOI: 10.1007/BF01931804.

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