BUILT-IN MATHEMATICA SYMBOL

JordanDecomposition

JordanDecomposition[m]
yields the Jordan decomposition of a square matrix m. The result is a list

where s is a similarity matrix and j is the Jordan canonical form of m.

Find the Jordan decomposition of a 3×3 matrix:

Out[1]=

Out[2]=

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 20-digit arithmetic:

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:

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:

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 machine-number 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: