The Exercise

The matrix $A$ of the exercise is the real $2 \times 2$ matrix $A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.$ (1)^✦ The exercise asks for its left and right polar decompositions: a unitary $U$ and positive matrices $J$ and $K$ such that $A = UJ$ , the left decomposition, and $A = KU$ , the right one.

The Three Factors

The unitary factor $U$ , common to both decompositions, is the real $2 \times 2$ matrix $U = \frac{1}{\sqrt{5}} \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}.$ (2)^✦

The factor $J$ of the left decomposition is the real $2 \times 2$ matrix $J = \frac{1}{\sqrt{5}} \begin{pmatrix} 3 & 1 \\ 1 & 2 \end{pmatrix},$ (3) and the factor $K$ of the right decomposition is the real $2 \times 2$ matrix $K = \frac{1}{\sqrt{5}} \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}.$ (4)

The Claim

Theorem 1. Let $A$ , $U$ , $J$ and $K$ be the real $2 \times 2$ matrices (1) to (4). Then $U$ is unitary, $U^\dagger U = U U^\dagger = I,$ (5)^✦ $J$ and $K$ are the positive square roots $J = \sqrt{A^\dagger A},$ (6)^✦ $K = \sqrt{A A^\dagger},$ (7)^✦ and $A$ has the left and right polar decompositions $A = UJ,$ (8)^✦ $A = KU.$ (9)^✦

What Is Checked

The claim is stated over the matrices (1) to (4).

The library proves of them that $U$ is unitary, (5), that $J$ and $K$ are the square roots (6) and (7), and that the products (8) and (9) both give back $A$ .