JordanDecomposition

JordanDecomposition[m]

yields the Jordan decomposition of a square matrix m. The result is a list {s,j} where s is a similarity matrix and j is the Jordan canonical form of m.

Details

  • The original matrix m is equal to s.j.Inverse[s]. »
  • The matrix m can be either numerical or symbolic.

Examples

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Basic Examples  (1)

Find the Jordan decomposition of a 3×3 matrix:

Scope  (1)

m is a 4×4 matrix:

Compute the Jordan decomposition using exact arithmetic:

Compute the Jordan decomposition using machine arithmetic:

Compute the Jordan decomposition using 20digit arithmetic:

Applications  (1)

Here is a function that tests diagonalizability of a square matrix:

Test if a particular matrix is diagonalizable:

Estimate the probability that a 4×4 matrix of ones and zeros will be diagonalizable:

Properties & Relations  (3)

m is a 4×4 matrix:

Find the Jordan decomposition:

m is equal to s.j.Inverse[s]:

The eigenvalues of m are on the diagonal of j:

m is a 3×3 matrix:

Find its Jordan decomposition:

Because of the canonical form of j, the n^(th) matrix power of j is given by:

Form the power series for the matrix exponential of j:

The matrix exponential of m is then given by:

This is equivalent to the value given by MatrixExp:

If m is diagonalizable, the Jordan decomposition is effectively the same as Eigensystem:

The ordering is different:

The eigenvalues are on the diagonal of j:

The eigenvectors are the columns of s:

Possible Issues  (1)

m is a 4×4 matrix with one small entry:

Find the Jordan decomposition using exact arithmetic:

This shows that m is diagonalizable:

Find the Jordan decomposition with machinenumber arithmetic:

Computation with machine-number arithmetic indicates that the matrix is not diagonalizable:

To machine precision, m is indistinguishable from a nearby non-diagonalizable matrix:

Introduced in 1996
 (3.0)
 |
Updated in 2010
 (8.0)