Scaling Matrix

76. Scaling Matrix

A scaling matrix stretches (or compresses) each coordinate independently. It’s a diagonal matrix:

𝑆 = [\begin{matrix} 𝑠_{1} & 0 & \dots & 0 \\ 0 & 𝑠_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 𝑠_{𝑛} \end{matrix}]

The $𝑖$ -th coordinate gets multiplied by $𝑠_{𝑖}$ :

𝑆 \vec{𝑥} = [\begin{matrix} 𝑠_{1} 𝑥_{1} \\ 𝑠_{2} 𝑥_{2} \\ ⋮ \\ 𝑠_{𝑛} 𝑥_{𝑛} \end{matrix}]

76.1. Uniform vs non-uniform

Uniform scaling: all $𝑠_{𝑖} = 𝑐$ , i.e. $𝑆 = 𝑐 𝐼$ — stretches every direction by the same factor $𝑐$ . Shapes are similar (same angles, scaled lengths).
Non-uniform scaling: $𝑠_{𝑖}$ ’s differ — distorts shapes. A circle becomes an ellipse.

Example

$𝑆 = [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]$ .

$𝑆 [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} 2 \\ 3 \end{matrix}]$ . The unit circle $𝑥^{2} + 𝑦^{2} = 1$ maps to the ellipse ${(\frac{𝑥}{2})}^{2} + {(\frac{𝑦}{3})}^{2} = 1$ .

76.2. Properties

Determinant: $det (𝑆) = 𝑠_{1} 𝑠_{2} \dots 𝑠_{𝑛}$ — total volume change factor
Inverse: $𝑆^{- 1} = diag (\frac{1}{𝑠_{1}}, \frac{1}{𝑠_{2}}, \dots, \frac{1}{𝑠_{𝑛}})$ (when all $𝑠_{𝑖} \neq 0$ )
Singular when any $𝑠_{𝑖} = 0$ : the corresponding direction is collapsed
Eigenvalues: the $𝑠_{𝑖}$ ’s themselves
Eigenvectors: the standard basis $𝑒_{1}, 𝑒_{2}, \dots, 𝑒_{𝑛}$
Already in diagonal form — diagonalization is trivial

76.3. Negative scaling

Scaling by $𝑠_{𝑖} < 0$ flips the $𝑖$ -th coordinate:

One negative entry: orientation reverses ( $det < 0$ ) — combination of a scaling and a reflection
Two negative entries: orientation preserved ( $det > 0$ ) — same as a 180° rotation by a scaling

76.4. Scaling in a general direction

To scale by factor $𝑐$ along a non-axis direction $\hat{𝑣}$ , conjugate by a rotation that aligns $\hat{𝑣}$ with a coordinate axis:

𝑆_{\hat{𝑣}, 𝑐} = 𝑅^{𝑇} [\begin{matrix} 𝑐 & 0 \\ 0 & 1 \end{matrix}] 𝑅

Or, more directly: $𝑆_{\hat{𝑣}, 𝑐} = 𝐼 + (𝑐 - 1) \hat{𝑣} {\hat{𝑣}}^{𝑇}$ — adds $𝑐 - 1$ extra of the projection onto $\hat{𝑣}$ .

76.5. Application to images / graphics

$𝑆 = diag (𝑠, 𝑠)$ — zoom in/out uniformly
$𝑆 = diag (2, 1)$ — stretch horizontally
$𝑆 = diag (1, - 1)$ — flip vertically (= reflection)
Singular value decomposition (SVD) writes every linear map as rotate–scale–rotate, so non-uniform scaling is the “shape-distorting” core of any matrix

76.6. See also

Diagonal Matrix
Rotation Matrix
Reflection Matrix
Shear Matrix
SVD — every matrix is rotate–scale–rotate