The Theorem

Theorem 2.3 of Nielsen and Chuang is the polar decomposition, the operator analogue of writing a complex number as $z = e^{i\theta}|z|$ . Let $A$ be a linear operator on a finite-dimensional Hilbert space $V$ . Then $A$ is a positive operator $J$ followed by a unitary $U$ , and also the same unitary followed by a positive operator $K$ , so that $A = UJ = KU$ .^✦ The positive factors are fixed by $A$ , whatever unitary goes with them: $J$ is the positive square root of $A^\dagger A$ ^✦ and $K$ that of $AA^\dagger$ . When $A$ is invertible, the unitary is fixed as well.^✦

The Absolute Value

The library gives the positive factors as absolute values of operators, built from singular vectors. The right singular vectors $|v_1\rangle, \dots, |v_d\rangle$ of $A$ are an orthonormal basis of $V$ made of eigenvectors of $A^\dagger A$ , where $d$ is the dimension of $V$ .^✦ The absolute value $|A|$ of $A$ is the operator on $V$ that weights the projector onto the $i$ th right singular vector by the $i$ th singular value $\sigma_i$ of $A$ : $|A| = \sum_{i=1}^{d} \sigma_i \, |v_i\rangle\langle v_i| .$ (1)^✦ The same construction applied to the adjoint $A^\dagger$ gives $|A^\dagger|$ , from an orthonormal basis of eigenvectors of $AA^\dagger$ .

The Claim

Theorem 1. Let $A$ be a linear operator on a finite-dimensional Hilbert space $V$ . There are a unitary $U$ and positive operators $J$ and $K$ on $V$ such that $A = UJ = KU, \qquad J^2 = A^\dagger A, \qquad K^2 = AA^\dagger .$ (2)^✦ If $U'$ is a unitary and $J'$ a positive operator on $V$ with $A = U'J'$ , then $J' = |A| .$ (3)^✦ If $U'$ is a unitary and $K'$ a positive operator on $V$ with $A = K'U'$ , then $K' = |A^\dagger| .$ (4)^✦ If $A$ is invertible, and $U_1, U_2$ are unitaries and $J_1, J_2$ positive operators on $V$ with $A = U_1 J_1 = U_2 J_2$ , then $U_1 = U_2 .$ (5)^✦

The book’s $J$ and $K$ , the positive square roots of $A^\dagger A$ and $AA^\dagger$ , are here $|A|$ and $|A^\dagger|$ .

What Is Checked

The theorem is stated over the linear operators on a finite-dimensional Hilbert space $V$ , the unitary and the positive ones among them, and over the absolute value (1), taken of $A$ and of $A^\dagger$ .

Of these constructions, the library proves that $A^\dagger A$ is self-adjoint; the right singular vectors of $A$ are the orthonormal basis of eigenvectors this gives it.