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RealRootProbabilitySpecialPointProbability

12.1 Ordered Schur Decomposition

The function SchurDecompositionOrdered is an extension to the built-in function SchurDecomposition; it shares the same syntax and accepts the same options, with certain additions and exceptions.

Similar to the Schur decomposition of matrix , the ordered Schur decomposition finds a unitary matrix and triangular matrix , such that gives , where H denotes Hermitian transpose, with matrix being such that the eigenvalues of appear on the main diagonal of . In addition, the ordered Schur decomposition makes the eigenvalues appear on the diagonal in the prescribed order. The guidelines for selecting the ordering function in SchurDecompositionOrdered are the same as those for the built-in function Sort.

Ordered Schur decomposition.

Load the application.

In[1]:=

Here is a matrix with random elements.

In[2]:=

Out[2]=

This finds its Schur decomposition using the built-in Mathematica function.

In[3]:=

The eigenvalues on the main diagonal do not follow any particular order.

In[4]:=

Out[4]=

This finds the ordered Schur decomposition of the same matrix.

In[5]:=

Now the diagonal elements appear in canonical order.

In[6]:=

Out[6]=

The decomposition is still valid.

In[7]:=

Out[7]=

This is the ordered Schur decomposition in which the eigenvalues residing in the right half of the complex plane go first.

In[8]:=

This is the corresponding matrix t.

In[9]:=

Out[9]=

This sorts the eigenvalues in descending order of their real parts. For this particular matrix m, the result is the same as the previous one.

In[10]:=

In[11]:=

Out[11]=

The function SchurDecompositionOrdered accepts the option Pivoting, just as SchurDecomposition does. However, the option RealBlockForm in SchurDecompositionOrdered accepts only the default value False.

Option value specific to SchurDecompositionOrdered.

RealRootProbabilitySpecialPointProbability



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