Diagonalisable matrix

Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$. The matrix $A$ is diagonalisable over $F$ when there is an invertible $n \times n$ matrix $P$ over $F$ such that $P^{-1} A P$ is a diagonal matrix. An endomorphism of a finite-dimensional space is diagonalisable when its matrix in some basis is diagonal.

Remarks

So $A$ is diagonalisable if and only if it is similar to a diagonal matrix. The diagonal entries of $P^{-1} A P$ are the eigenvalues of $A$, and the columns of $P$ are eigenvectors. The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not diagonalisable over any field. Over $\R$, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not diagonalisable; over $\C$, it is.

Explanations

L1 human

A diagonalisable matrix becomes a diagonal matrix in a suitable basis. In that basis, the map just stretches each coordinate axis by a factor. This makes powers, exponentials and other functions of the matrix easy to compute.