Chapter 8: Singular Value Decomposition (SVD). In the chapter mind map this icon labels Polar Decomp: $A = Q S$. The discussion below is excerpted and lightly edited from § Theorem: Polar Decomposition in Mathematics for AI and Machine Learning.

Any square matrix $A \in \mathbb{R}^{N \times N}$ can be decomposed as

• $Q$ is orthonormal
• $S$ is SPSD(symmetric positive semi-definite)

Given the SVD $A = U \Sigma V^\top$ (where $U$ contains the left singular vectors), we construct the polar decomposition as follows:

• $S = V \Sigma V^\top$ is SPSD because $\Sigma$ is diagonal with non-negative entries and $V$ is orthonormal
• $Q = U V^\top$ is orthonormal (product of two orthonormal matrices)
• $Q S = (U V^\top)(V \Sigma V^\top) = U (V^\top V) \Sigma V^\top = U \Sigma V^\top = A$ \checkmark

To understand the geometric meaning, we can draw an analogy with complex numbers. A complex number in polar form is $z = r e^{i\theta}$, where $e^{i\theta} = \cos\theta + i\sin\theta$ by Euler's formula (see the matrix chapter):

• $e^{i\theta}$ represents the direction (rotation), analogous to $Q$ in polar decomposition
• $r$ represents the magnitude (scaling), analogous to $S$ in polar decomposition