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# 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.
• The matrix m can be either numerical or symbolic.
Find the Jordan decomposition of a 3×3 matrix:
Find the Jordan decomposition of a 3×3 matrix:
 Out[1]=
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 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 20-digit 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:
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:
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