BUILT-IN MATHEMATICA SYMBOL

Eigensystem

Eigensystem[m]
gives a list

of the eigenvalues and eigenvectors of the square matrix m.

Eigensystem

gives the generalized eigenvalues and eigenvectors of m with respect to a.

Eigensystem

gives the eigenvalues and eigenvectors for the first k eigenvalues of m.

Eigensystem

gives the first k generalized eigenvalues and eigenvectors.

MORE INFORMATION

Eigensystem finds numerical eigenvalues and eigenvectors if m contains approximate real or complex numbers.

For approximate numerical matrices m, the eigenvectors are normalized.

All the non-zero eigenvectors given are independent. If the number of eigenvectors is equal to the number of non-zero eigenvalues, then corresponding eigenvalues and eigenvectors are given in corresponding positions in their respective lists.

If there are more eigenvalues than independent eigenvectors, then each extra eigenvalue is paired with a vector of zeros. »

Eigensystem[m, ZeroTest->test] applies test to determine whether expressions should be assumed to be zero. The default setting is ZeroTest->Automatic.

The eigenvalues and eigenvectors satisfy the matrix equation m.Transpose[vectors]==Transpose[vectors].DiagonalMatrix[values]. »

Generalized eigenvalues and eigenvectors satisfy m.Transpose[vectors]==a.Transpose[vectors].DiagonalMatrix[values].

can be used to set vals and vecs to be the eigenvalues and eigenvectors respectively. »

Eigensystem is equivalent to applying Take to each element of Eigensystem[m].

The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.

SparseArray objects can be used in Eigensystem. »

EXAMPLES

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Basic Examples (3)

Symbolic eigenvalues and eigenvectors:

Exact eigenvalues and eigenvectors:

Numerical value:

Eigenvalues and eigenvectors computed with numerical methods:

Symbolic eigenvalues and eigenvectors:

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Exact eigenvalues and eigenvectors:

In[1]:=

Out[1]=

Numerical value:

In[2]:=

Out[2]=

Eigenvalues and eigenvectors computed with numerical methods:

In[1]:=

Out[1]=

Scope (6)

Eigenvalues and eigenvectors computed with numerical methods using 20-digit precision:

Set vals and vecs to be the eigenvalues and eigenvectors respectively:

Smallest two eigenvalues and eigenvectors:

Eigenvalues are listed according to multiplicity along with independent eigenvectors:

Zero vectors are used when there are more eigenvalues than independent eigenvectors:

Compute generalized eigenvalues and eigenvectors:

Generalizations & Extensions (2)

A block diagonal matrix:

Its eigenvalues and eigenvectors:

A tridiagonal matrix:

The three eigenvalues with largest magnitude and a plot of the corresponding eigenvectors:

Options (2)

A 3×3 Vandermonde matrix:

In general for exact 3×3 matrices the result will be given in terms of Root objects:

If you want the result in terms of radicals, you can use the Cubics option:

Note that the result with Root objects is better suited to subsequent numerical evaluation:

A 4×4 matrix:

In general, for a 4×4 matrix, the result will be given in terms of Root objects:

You can get the result in terms of radicals using the Cubics and Quartics options:

Applications (2)

Here is a quadratic form in 3 dimensions:

Plot the surface

:

Get the symmetric matrix for the quadratic form, using CoefficientArrays:

Numerically compute its eigenvalues and eigenvectors:

Show the principal axes of the ellipsoid:

The Lorenz equations:

Find the Jacobian for the right-hand side of the equations:

Find the equilibrium points:

Find the eigenvalues and eigenvectors of the Jacobian at one in the first octant:

A function that integrates backward from a small perturbation of pt in the direction dir:

Show the stable curve for the equilibrium point on the right:

Find the stable curve for the equilibrium point on the left:

Show the stable curves along with a solution of the Lorenz equations:

Properties & Relations (5)

Any square matrix satisfies its similarity relation:

Any pair of square matrices satisfies the generalized similarity relation:

The eigensystem for a circulant matrix:

Compute the matrix exponential using diagonalization:

Compute the matrix exponential using MatrixExp:

The eigenvalues and eigenvectors of a random symmetric matrix:

The eigenvectors are all real:

The numerical eigenvectors are orthonormal to the precision of the computation:

Not all matrices have a complete set of eigenvectors:

Use JordanDecomposition for exact computation:

Use SchurDecomposition for numeric computation:

Possible Issues (3)

The general symbolic case very quickly gets very complicated:

The expression sizes increase faster than exponentially:

Construct a 10,000×10,000 sparse matrix:

The eigenvector matrix is a dense matrix and too large to represent:

Computing the few largest or smallest eigenvalues is usually possible:

When eigenvalues are closely grouped the iterative method for sparse matrices may not converge:

The iteration has not converged well after 1000 iterations:

You can give the algorithm a shift near the expected value to speed up convergence: