Function Repository Resource:

MatrixPencilSolve

Source Notebook

Find the null values and vectors for the pencil of a set of square matrices

Contributed by: Daniel Lichtblau

ResourceFunction["MatrixPencilSolve"][{m₀,m₁,…,m_n}]

finds the generalized eigenvalues λ and corresponding vectors v for which the matrix m₀+λm₁+λ²m₂+…λⁿm_n has a nontrivial null space.

Details and Options

The matrix inputs must be square and of the same dimensions.

The result is a list of lists of the form

.

Examples

Basic Examples (3)

Form some random real-valued matrices:

In[1]:=

n = 3;
m = 3;
SeedRandom[1234];
matrices = RandomReal[{-1, 1}, {m, n, n}];

Find the null-valued vectors for the pencil of these matrices:

In[2]:=

Out[2]=

Check that the first of these value/vector pairs satisfies the necessary equation:

In[3]:=

Out[3]=

Scope (1)

MatrixPencilSolve will handle exact as well as approximate input matrices:

In[4]:=

n = 2;
m = 3;
SeedRandom[1234];
matrices = RandomInteger[{-1, 1}, {m, n, n}]

Out[7]=

In[8]:=

Out[8]=

Properties and Relations (2)

MatrixPencilSolve with one argument is equivalent to obtaining vectors from NullSpace, with λ having no value:

In[9]:=

In[10]:=

Out[10]=

In[11]:=

Out[11]=

MatrixPencilSolve with two matrices is equivalent to the two-argument form of Eigensystem, with the second matrix negated and the result transposed:

In[12]:=

Out[13]=

In[14]:=

Out[14]=

In[15]:=

Out[15]=

Requirements

Wolfram Language 11.3 (March 2018) or above

Version History

1.0.0 – 28 February 2019

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License