A Jordan Block is defined to be a square matrix of the form:
$\left[\begin{array}{ccccc}\mathrm{\λ}& 1& 0& \cdots & 0\\ 0& \mathrm{\λ}& 1& \cdots & 0\\ \vdots & \vdots & \ddots & & \vdots \\ 0& 0& 0& \mathrm{\λ}& 1\\ 0& 0& 0& \cdots & \mathrm{\λ}\end{array}\right]$
for some scalar l.
For example, choosing l = , click to display a Jordan block below.
Note: For simplicity, lambda can only be chosen to be an integer, otherwise it will not be displayed.



A Jordan Matrix is a matrix that has Jordan Blocks on its diagonal and the rest of the entries equal to 0:${}$${}$
$\left[\begin{array}{cccccccccc}\mathrm{\λ\_\_1}& {1}& {0}& 0& 0& 0& 0& 0& 0& 0\\ {0}& {\ddots}& {1}& 0& 0& 0& 0& 0& 0& 0\\ {0}& {0}& \mathrm{\λ\_\_1}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \mathrm{\λ\_\_2}& {1}& {0}& 0& 0& 0& 0\\ 0& 0& 0& {0}& {\ddots}& {1}& 0& 0& 0& 0\\ 0& 0& 0& {0}& {0}& \mathrm{\λ\_\_2}& & 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \ddots & 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \mathrm{\λ\_\_n}& {1}& {0}\\ 0& 0& 0& 0& 0& 0& 0& {0}& {\ddots}& {1}\\ 0& 0& 0& 0& 0& 0& 0& {0}& {0}& \mathrm{\λ\_\_n}\end{array}\right]$
where the colored regions are the Jordan Blocks of the matrix. Futhermore, note that the $\mathrm{\λ\_\_n}$ values in each Jordan block need not to be all equal.
Any square matrix M is similar to a Jordan matrix J, which is called the Jordan Canonical Form of M. For M, There exists an invertible Q such that:
M = Q · J · $Q{}^{\mathit{}\mathit{1}}$
As we can observe, J is upper triangular and almost diagonal, from this we know that M and J have the same:
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Eigenvalues with the same multiplicities

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Eigenspace dimension for each eigenvalue

As such, the calculations for these common features are simplified by working with the Jordan Canonical Form of M instead of M.
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