---
tag: CFB41AZX87
kind: statement
role: theorem
name: Reduction to diagonal form by transvections
msc: [15A23, 15A21]
uses: [0NMN00FNW1, 0Q9PGXD9E4, 0QA8CHH8MG, 076JGP6R3T]
about: [0DTCBCKEJP, 044JNC7ZAA]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_matrix
---

## Statement

Let $F$ be a [[field|04C5ZBG1WT]], let $n \ge 1$ and let $M$ be an $n \times n$ matrix over $F$.
Then there are finitely many [[transvection matrices|0DTCBCKEJP]] $L_1, \dots, L_r$ and $R_1, \dots, R_s$ and a [[diagonal matrix|044JNC7ZAA]] $D$ over $F$ such that
$$M = L_1 \cdots L_r \, D \, R_1 \cdots R_s.$$

## Remarks

Multiplying by a transvection on the left adds a multiple of one row to another row; on the right it adds a multiple of one column to another column.
The proof is an induction on $n$.
If the last row or the last column of $M$ has a nonzero entry, one row or column addition puts a nonzero entry at position $(n, n)$.
Further additions of multiples of row $n$ and column $n$ then clear the rest of the last row and the last column.
The upper left block of size $n - 1$ is treated by induction.
Example: $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ becomes $\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ after adding column $2$ to column $1$, and then $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ after subtracting column $1$ from column $2$.
Each transvection has [[determinant|00KDZRW4C0]] $1$, so $\det M = \det D$.
A refinement of the argument shows that an invertible $M$ is a product of transvections and the matrix $\operatorname{diag}(1, \dots, 1, \det M)$.
Hence the transvections generate the [[special linear group|0V9PMN0RBH]] $\mathrm{SL}_n(F)$.

## Explanations

### L1 human

Using only the move "add a multiple of one row (or column) to another", every square matrix over a field can be made diagonal.
No swaps and no scalings are needed.
These moves do not change the determinant, so the diagonal matrix keeps the determinant of the original matrix.

### L2 human

This is a sharper form of Gaussian elimination: [[elementary matrices|01REXW3WWE]] of the row-addition type already suffice, as long as both row and column operations are allowed.
The theorem is the main step in the proof that $\mathrm{SL}_n(F)$ is generated by transvections.
It is also a first step towards the Smith normal form, where the field is replaced by a principal ideal domain and more care is needed.