Jordan DecompositionThe Jordan decomposition of a square matrix  involves finding the non-singular matrix  that can be used for a similarity transformation of  to generate a matrix  (known as the Jordan form) that has a particularly simple triangular structure. The Jordan decomposition always exists, but it is hard to compute with floating point arithmetic. However, computation with exact arithmetic avoids these problems. This demonstrates the Jordan form for a sample matrix. In[1]:=  |
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The Jordan form has the eigenvalues of the matrix along its diagonal. Any defective eigenvalues are grouped into blocks by 1s just above the diagonal. The Jordan form of the above matrix is shown below. In[4]:=  |
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The Jordan form shows that there are two eigenvalues: -1 and 2. The eigenvalue -1 is repeated twice and has a complete set of eigenvectors. The eigenvalue 2 is repeated four times. It appears once with its own eigenvector, and then three times with only one full eigenvector. This is demonstrated when the eigensystem for the matrix is computed. In[5]:=  |
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