Decompose any real square matrix in geometrically interpretable matrices

Question

Is it possible to decompose any real square matrix in a product of simple linear maps such as shear, reflection, squeeze, scale and rotation? I think that would provide great insights about the original matrix.

Thank you, Martin

Answer 1

The singular value decomposition is one such useful decomposition. Recall that the SVD proceeds as $\mathbf A=\mathbf U\mathbf \Sigma\mathbf V^\top$, where $\mathbf U$ and $\mathbf V$ are orthogonal, and $\mathbf \Sigma$ is diagonal. Geometrically, recall that orthogonal matrices can represent either rotations, reflections, or compositions thereof, and diagonal matrices represent scalings.