# JordanDecomposition

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:

Format the results:

## Scope(10)

### Basic Uses(6)

Jordan decomposition of a machine-precision matrix:

Format the results:

Jordan decomposition of a complex matrix:

Jordan decomposition of an exact matrix with a deficient eigenspace:

The one in the third column of indicates that the eigenspace corresponding to 48 is deficient:

The third column of is therefore a generalized eigenvector where gives rather than :

Jordan decomposition of an arbitrary-precision matrix:

Jordan decomposition of a symbolic matrix:

The Jordan decomposition of large numerical matrices is computed efficiently:

### Special Matrices(4)

Jordan decomposition of sparse matrices:

Eigensystems of structured matrices:

The units of a QuantityArray object are in the Jordan form; the similarity matrix is dimensionless:

IdentityMatrix is a Jordan canonical form:

The associated similarity matrix is a square root of the identity:

Jordan decomposition of a HilbertMatrix:

## 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 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: