Diagonal Matrix¶

Definition¶

A diagonal matrix is a square matrix whose off-diagonal entries are all zero.

Interpretation¶

A diagonal matrix acts on each coordinate separately: $\boldsymbol{D}\boldsymbol{x}$ multiplies the $i$ -th coordinate of $\boldsymbol{x}$ by the $i$ -th diagonal entry and never mixes coordinates.

The unit square mapped by a diagonal matrix onto a rectangle twice as wide and half as tall, with e1 stretched by 2 and e2 shrunk by 1/2.

The matrix $\boldsymbol{D}=\begin{pmatrix}2&0\\0&\tfrac{1}{2}\end{pmatrix}$ stretches $\boldsymbol{e}_1$ to $2\boldsymbol{e}_1$ and shrinks $\boldsymbol{e}_2$ to $\tfrac{1}{2}\boldsymbol{e}_2$ , so the unit square becomes a rectangle twice as wide and half as tall. The axes keep their directions, and the area $2\cdot\tfrac{1}{2}=1$ is the product of the diagonal entries, the determinant.

A diagonal matrix scales each axis on its own: $\boldsymbol{A}\boldsymbol{e}_1=a_{11}\boldsymbol{e}_1$ and $\boldsymbol{A}\boldsymbol{e}_2=a_{22}\boldsymbol{e}_2$ $a_{22} e_{2}$ . Allow off-diagonal entries to see the axes turn.

Calculations reduce to calculations with numbers: products and powers are taken entry by entry on the diagonal, here $\boldsymbol{D}^k=\operatorname{diag}(2^k,2^{-k})$ $diag (2^{k}, 2^{- k})$ , and a diagonal matrix is invertible if and only if no diagonal entry is $0$ , with the reciprocals as inverse. Zeros on the diagonal are allowed, so the zero matrix is diagonal too. Later in the course this simplicity is the reason to bring a matrix into diagonal form where possible.

Key Results¶

The determinant of a diagonal matrix is the product of its diagonal entries.
The product and sum of diagonal matrices of the same size are diagonal.

Examples¶

$\begin{pmatrix}2&0\\0&-1\end{pmatrix}$ is diagonal.
$\boldsymbol{I}_n$ and $\boldsymbol{0}_n$ are diagonal.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 5

Last updated: 20 October 2026 03:02