Module V·Article II·~1 min read
Self-Adjoint and Unitary Operators
Euclidean and Unitary Spaces
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Adjoint Operator
For a linear operator $A$ on a Euclidean space: the adjoint operator $A^*$ satisfies $(Av, w) = (v, A^*w)$ for all $v, w$.
In matrix form (orthonormal basis): $A^* = A^\mathsf{T}$ (the transposed matrix).
Self-Adjoint (Symmetric) Operators
$A = A^* \iff A = A^\mathsf{T}$ — a symmetric matrix.
Spectral theorem: A symmetric operator in a finite-dimensional Euclidean space has real eigenvalues and an orthonormal basis of eigenvectors (it can be diagonalized by an orthogonal transformation).
Proof: the eigenvalues are real (from $(Av, v) = \lambda(v, v) = (v, Av) = \bar{\lambda}(v, v)$); eigenvectors of different values are orthogonal.
Corollary: $A = QDQ^\mathsf{T}$ (spectral decomposition), where $Q$ is orthogonal, $D = \mathrm{diag}(\lambda_1, ..., \lambda_n)$.
$A$ is positive definite $\iff$ all $\lambda_i > 0 \iff A = BB^\mathsf{T}$ for some $B$ (Cholesky decomposition).
Unitary Spaces and Operators
A unitary space is a complex vector space with a Hermitian scalar product: $(u, v) = \bar{v}^\mathsf{T}u$, where $(v, u) = \overline{(u, v)}$.
Adjoint operator (Hermitian adjoint): $A^* = \bar{A}^\mathsf{T}$ (Hermitian conjugation, transposition + complex conjugation).
Hermitian operator: $A = A^*$ (a generalization of symmetric). Eigenvalues are real. In quantum mechanics observable quantities are Hermitian operators; their eigenvalues are measurement results.
Unitary operator: $UU^* = U^*U = I$ (generalization of orthogonal). Preserves norm and scalar product. Eigenvalues are on the unit circle.
Singular Value Decomposition (SVD)
Any matrix $A$ of size $m \times n$: $A = U \Sigma V^\mathsf{T}$, where $U$ ($m \times m$) and $V$ ($n \times n$) are orthogonal, $\Sigma$ is diagonal with $\sigma_1 \ge \sigma_2 \ge ... \ge 0$ (singular values).
SVD is a universal decomposition: rank = number of nonzero $\sigma$; the least squares solution via pseudoinverse; image compression (take the first $k$ singular values); PCA.
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