76. Scaling Matrix

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

The -th coordinate gets multiplied by :

76.1. Uniform vs non-uniform

• Uniform scaling: all , i.e. — stretches every direction by the same factor . Shapes are similar (same angles, scaled lengths).
• Non-uniform scaling: ‘s differ — distorts shapes. A circle becomes an ellipse.

76.2. Properties

• Determinant: — total volume change factor
• Inverse: (when all )
• Singular when any : the corresponding direction is collapsed
• Eigenvalues: the ‘s themselves
• Eigenvectors: the standard basis
• Already in diagonal form — diagonalization is trivial

76.3. Negative scaling

Scaling by flips the -th coordinate:

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

76.4. Scaling in a general direction

To scale by factor along a non-axis direction , conjugate by a rotation that aligns with a coordinate axis:

Or, more directly: — adds extra of the projection onto .

76.5. Application to images / graphics

• — zoom in/out uniformly
• — stretch horizontally
• — 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