或る大学院生のメモ帳: Linear Algebra

ラベル Linear Algebra の投稿を表示しています。すべての投稿を表示

2012年9月22日土曜日

Householder Matrix

$H = I-\frac{2uu^\mathrm{T} }{||u||^2}$

Properties

$H^\mathrm{T}=H^{-1}=H$ (symmetric and orthogonal)

$H^2=I$ (involutary)

http://fussy.web.fc2.com/algo/algo11-1.htm

http://homepage3.nifty.com/rikei-index01/ouyoukaiseki/bekutorunaigai.html

http://en.wikipedia.org/wiki/Householder_transformation

2012年9月21日金曜日

Hermitian Matrix

A complex square matrix is Hermitian if

$A = A^*$

where A* is the conjugate transpose of A.

Example

$\begin{bmatrix}3&2+i\\ 2-i&1\end{bmatrix}$

http://en.wikipedia.org/wiki/Hermitian_matrix

Unitary Matrix

A complex square matrix U is unitary if

$U^* U = UU^* = I \,$

where I is the identity matrix and U* is the conjugate transpose of U.

http://en.wikipedia.org/wiki/Unitary_matrix

2012年9月20日木曜日

A permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere.

Example
The permutation matrix Pπ corresponding to the permutation :
$\pi=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 2 & 5 & 3 \end{pmatrix},$
is
$P_\pi = \begin{bmatrix} \mathbf{e}_{\pi(1)} \\ \mathbf{e}_{\pi(2)} \\ \mathbf{e}_{\pi(3)} \\ \mathbf{e}_{\pi(4)} \\ \mathbf{e}_{\pi(5)} \end{bmatrix} = \begin{bmatrix} \mathbf{e}_{1} \\ \mathbf{e}_{4} \\ \mathbf{e}_{2} \\ \mathbf{e}_{5} \\ \mathbf{e}_{3} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{bmatrix}.$

Properties
$P_{\pi}P_{\pi}^{T} = I$ (orthogonal matrices)

http://en.wikipedia.org/wiki/Permutation_matrix

2012年3月20日火曜日

Kronecker Products

1. (A ⊗ B)(C ⊗ D) = AC ⊗ BD.

2. (A ⊗ B)^(−1) = A^(−1) ⊗ B^(−1).

3. (A ⊗ B)′ = A′ ⊗ B′.

4. A ⊗ (B + C) = A ⊗ B + A ⊗ C.