---
tag: 0363JMQ42Y
kind: statement
role: theorem
name: Triangularisation
msc: [15A20, 15A21]
uses: [039HF58KQ5, 0MKABWDRNC, 027KDW2K72]
sources:
  - url: https://en.wikipedia.org/wiki/Triangular_matrix
    locator: Triangularisability
---

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[similar|0E89HQATM8]] to an [[upper triangular matrix|0N4XY7Z3NN]] over $F$ if and only if the [[characteristic polynomial|0VSSPAFM2C]] of $A$ splits over $F$.
Here, to split means to be a product of linear factors.
In particular, over an algebraically closed field, such as $\C$, each square matrix is similar to an upper triangular matrix.

## Remarks

The diagonal entries of the triangular matrix are the eigenvalues of $A$, with their algebraic multiplicities.
The proof is by induction: choose an eigenvector, extend it to a basis, and work on the quotient space.
Over $\C$, the matrix $P$ can even be chosen unitary; this is the Schur decomposition.

## Explanations

### L1 human

Not every matrix can be diagonalised, but over $\C$ every matrix can be made triangular by a change of basis.
A triangular form is almost as useful: its eigenvalues are on the diagonal.
Over $\R$, a rotation matrix cannot be triangularised, because it has no real eigenvalue.