PX153 - K13 - additional properties

$(i) :$ if $y$ is a column vector, $y^{T}$ is a row vector, and vice versa
$(i i) :$

(A B)^{T} = B^{T} A^{T}

- proof:

\begin{aligned} (A B)_{i k}^{T} & = \sum_{j} a_{k j} b_{j i} \\ = \sum_{j} a_{j k}^{T} b_{i j}^{T} \\ = (B^{T} A^{T})_{i k} \end{aligned}

( $i i i) :$ orthogonality for vectors
- vectors are orthogonal if

\begin{aligned} {\vec{y}}^{T} \cdot \vec{y} & = β \\ {\vec{y}}^{†} \cdot \vec{x} & = 0 \end{aligned}

- if vectors are normalized, $β = 1$
- for hermitian matrices ( $A^{†} = A$ ):
- eigenvalues are always real
- if $λ_{i} \neq λ_{j}, i \neq j :$ then eigenvectors are all orthogonal
- if $λ_{i} = λ_{j}, i \neq j :$ then orthogonal eigenvectors as long as $det A \neq 0$
- proof:
- consider a hermitian matrix, $A$ , with a distinct eigenvalue, $λ_{i}$ , such that $λ_{i} \neq λ_{j}, i \neq j$

A {\vec{x}}^{i} = λ_{i} {\vec{x}}^{i}

- taking the hermitian conjugate of the equation:

\begin{aligned} (A {\vec{x}}^{i})^{†} & = (λ_{i} {\vec{x}}^{i})^{†} \\ {\vec{x}}^{i †} A^{†} & = λ_{i}^{*} {\vec{x}}^{i †} \\ {\vec{x}}^{i †} A & = λ_{i}^{*} {\vec{x}}^{i †} \end{aligned}

- multiplying by ${\vec{x}}^{j}$ (from the right):

\begin{aligned} {\vec{x}}^{i †} A {\vec{x}}^{j} & = λ_{i}^{*} {\vec{x}}^{i †} {\vec{x}}^{j} \\ {\vec{x}}^{i †} λ_{j} {\vec{x}}^{j} & = λ_{i}^{*} {\vec{x}}^{i †} {\vec{x}}^{j} \\ (λ_{j} - λ_{i}^{*}) ({\vec{x}}^{i †} {\vec{x}}^{j}) & = 0 \end{aligned}

- if $i \neq j : {\vec{x}}^{i †} {\vec{x}}^{j} = 0$ , giving the definition of orthogonality
- if $i = j : λ_{i} = λ_{i}^{*} ⟹ λ_{i} \in R e$

- if the eigenvalues are degenerate, ie: $λ_{i} = λ_{j}, i \neq j$ , orthogonal eigenvectors can still be constructed

real symmetric and anti-symmetric matrices