11. Hermitian Matrices#
We now know the definition of adjoints. Let’s now look at self-adjoint maps
Self-adjoint maps#
Consider \((H, \mathbb{F}, \langle\cdot,\cdot\rangle _H)\). Consider linear and continuous map \(\mathcal{A}:H \rightarrow H\). Meaning that \(\mathcal{A}^*:H\rightarrow H\). \(\mathcal{A}\) is self-adjoint if \(\mathcal{A} = \mathcal{A}^*\). This also means that:
Hermitian matrices#
Let \(\mathcal{A} \simeq A = (a_{ij}), i, j \in 1\dots n\), with \(\mathcal{A}\) the operator (map) and \(A\) the matrix representation \(\in \mathbb{F}^{n\times n}\).
The map \(\mathcal{A}\) is self-adjoint iff \(A\) is hermitian, i.e. \(A = A^*\). The \(A^*\) in matrix means complex conjugate transpose, i.e. \bar{a_{ji}}$.
Remark
Hermitian matrices always have real eigenvalues.
Unitary matrices#
Definition: Unitary matrix
A matrix \(U\) is unitary iff
Singular values#
Let us have the matrix \(A \in \mathbb{C}^{m\times n}\). We will have \(AA^* \in \mathbb{C}^{m\times m}\) a square matrix. Let \(\lambda_i , i=1\dots m\) be the eigenvalues of \(AA^*\). This also means that \(\lambda_i\) is real and non-negative.
If we have \(\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_r \dots \ge \lambda_m\). If \(r = \text{rank}(A) = \text{rank}(AA^*)\), then \(\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_r \ge 0\) and all the remaining eigenvalues are all zeros.
We define the singular values of the matrix to be the square root of the eigenvalues.
Definition: Singular values
The non-zero singular values of A are