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:

Jn = (λ I + N)n = nC0λnInN0 + nC1λn-1In-1N1 + nC2λn-2In-2N2 + ... + nCn-1λ1I1Nn-1 + nCnλ0I0Nn =

```|nC0xn nC1xn-1 nC2xn-2 nC3xn-3 . . nCm-2xn-m+2 nCm-1xn-m+1|
| 0    nC0xn   nC1xn-1 nC2xn-2 . . nCm-3xn-m+3 nCm-2xn-m+2|
| 0     0      nC0xn   nC1xn-1        .          .      |
| 0     0      0      nC0xn  .       .          .      |
| .     .      .            .   .    .          .      |
| .     .      .                .  nC1xn-1    nC2xn-2   |
| .     .      .                   nC0xn      nC1xn-1   |
| 0     0      0   .     .      .     0        nC0xn    ‌
```

Sources:
Eric W. Weisstein. "Jordan Block." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/JordanBlock.html