A Jordan block (also canonical box matix) is a square matrix where each element along the main diagonal is λ and with ones along the superdiagonal. If **J** is an mxm Jordan block, this can be written as λ **I** + **N**, where **N** is the nilpotent matrix of order m. For example, if λ = 2:

|2 1 0 0 . . 0 0|
|0 2 1 0 . . 0 0|
|0 0 2 1 . .|
|0 0 0 2 . . .|
|. . . . . . .|
|. . . . 1 0|
|. . . 2 1|
|0 0 0 . . . 0 2|

A block matrix with Jordan blocks along the main diagonal is called a matrix of Jordan form. Such matrices are useful computationally. If a matrix **J** is an mxm Jordan block with λ = x, then since **I** and **N** commute:

**J**^{n} = (λ **I** + **N**)^{n} = _{n}C_{0}λ^{n}**I**^{n}**N**^{0} + _{n}C_{1}λ^{n-1}**I**^{n-1}**N**^{1} + _{n}C_{2}λ^{n-2}**I**^{n-2}**N**^{2} + ... + _{n}C_{n-1}λ^{1}**I**^{1}**N**^{n-1}
+ _{n}C_{n}λ^{0}**I**^{0}**N**^{n} =

|_{n}C_{0}x^{n} _{n}C_{1}x^{n-1} _{n}C_{2}x^{n-2} _{n}C_{3}x^{n-3} . . _{n}C_{m-2}x^{n-m+2} _{n}C_{m-1}x^{n-m+1}|
|_{ }0_{ } ^{ } _{n}C_{0}x^{n } _{n}C_{1}x^{n-1} _{n}C_{2}x^{n-2} . . _{n}C_{m-3}x^{n-m+3} _{n}C_{m-2}x^{n-m+2}|
|_{ }0_{ } ^{ } _{ }0_{ } ^{ } _{ n}C_{0}x^{n } _{n}C_{1}x^{n-1} _{ }._{ } ^{ } . |
|_{ }0_{ } ^{ } _{ }0_{ } ^{ } _{ }0_{ } _{n}C_{0}x^{n} . . . |
|_{ }._{ } ^{ } _{ }._{ } ^{ } _{ }._{ } ^{ } . . _{ }._{ } ^{ } _{ }._{ } ^{ } |
|_{ }._{ } ^{ } _{ }._{ } ^{ } _{ }._{ } ^{ } . _{n}C_{1}x^{n-1} _{n}C_{2}x^{n-2 } |
|_{ }._{ } ^{ } _{ }._{ } ^{ } _{ }._{ } ^{ } _{n}C_{0}x^{n} _{ n}C_{1}x^{n-1 } |
|_{ }0_{ } ^{ } _{ }0_{ } ^{ } _{ }0_{ } _{ }._{ } ^{ } _{ }._{ } ^{ } _{ }._{ } ^{ } 0 _{n}C_{0}x^{n}

Sources:

Eric W. Weisstein. "Jordan Block." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/JordanBlock.html