或る大学院生のメモ帳: Math

The support of a function is the set of points where the function is not zero-valued, or the closure of that set.

http://en.wikipedia.org/wiki/Support_%28mathematics%29

2012年12月6日木曜日

$\ \frac{d}{d\alpha}\int_{a(\alpha)}^{b(\alpha)} f(x,\alpha)\,dx = \frac{d b(\alpha)}{d \alpha}\,f(b(\alpha),\alpha)-\frac{d a(\alpha)}{d \alpha}\,f(a(\alpha),\alpha)+ \int_{a(\alpha)}^{b(\alpha)}\frac{\partial}{\partial \alpha}\,f(x,\alpha)\,dx\,$

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

2012年10月31日水曜日

固有値問題の解法

固有値は固有方程式(characteristic equation)
det (A − λI) = 0
の解となる。これから，固有方程式を解くことによって固有値を求めることが考えられる。これを直接解法という。行列Aの次元が大きくなると，固有方程式は高次元のλに関する多項式方程式となり，その係数を計算するのに手間がかかる，と同時に，．．．よって，固有方程式を解いて固有値を求めることは通常行われない。

固有値問題を解くには反復解法が普通用いられる。その基礎はシュール(Schur)分解と呼ばれるつぎの事実である．．．

大石「数値計算講義ノート 6　固有値問題の解法」
http://www.waseda.jp/ocw/ComputerScience/17-1004345-01NumericalComputationsSpring2003/StudyMaterials/lec6.pdf

http://www.oishi.info.waseda.ac.jp/~oishi/lec2003/num.htm

2012年10月20日土曜日

Lyapunov/ Asymptotic Stability

Lyapunov Stability
If all solutions of the dynamical system that start out near an equilibrium point x0 stay near x0 forever, then x0 is Lyapunov stable.

Asymptotic Stability
More strongly, if x0 is Lyapunov stable and all solutions that start out near x0 converge to x0, then x0 is asymptotically stable.

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

2012年10月3日水曜日

Partial Derivatives of Homogeneous Functions

if a function is homogeneous of degree k, its derivatives are homogeneous of degree k-1.

http://www.economics.utoronto.ca/osborne/MathTutorial/HOMF.HTM

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

Permutation Matrix

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8月26日日曜日

Propositional Logic

http://home.hiroshima-u.ac.jp/fujita/Class/Kisoron/logic.pdf

2012年8月19日日曜日

Laplace Transformation

http://www.ic.is.tohoku.ac.jp/~swk/lecture/yaruodsp/laplace.html

http://www-watt.mech.eng.osaka-u.ac.jp/~tasai/laplace.pdf

2012年5月17日木曜日

Inverse of A (matrix): Analytical Solution

$\mathbf{A}^{-1}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}^{\mathrm{T}}\right)_{ij}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}_{ji}\right)={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}} \begin{pmatrix} \mathbf{C}_{11} & \mathbf{C}_{21} & \cdots & \mathbf{C}_{n1} \\ \mathbf{C}_{12} & \mathbf{C}_{22} & \cdots & \mathbf{C}_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{C}_{1n} & \mathbf{C}_{2n} & \cdots & \mathbf{C}_{nn} \\ \end{pmatrix}$

where C(i,j) is the cofactor of A(i,j) such that

$C_{ij}=(-1)^{i+j} M_{ij} \,$

M(i,j) is defined to be the determinant of the submatrix obtained by removing from A its i-th row and j-th column. For example,

$M_{23} = \begin{vmatrix} b_{11} & b_{12} & \Box \\ \Box & \Box & \Box \\ b_{31} & b_{32} & \Box \\ \end{vmatrix}$

yields

$M_{23} = \begin{vmatrix} b_{11} & b_{12} \\ b_{31} & b_{32} \\ \end{vmatrix} = b_{11}b_{32} - b_{31}b_{12}$

http://en.wikipedia.org/wiki/Cramer%27s_rule
http://en.wikipedia.org/wiki/Invertible_matrix#Methods_of_matrix_inversion
http://en.wikipedia.org/wiki/Matrix_of_cofactors

Convex Function ⇔ Positive Semidefinite etc.

Let f (x) be a twice differentiable function in n variables defined on an open convex set S. Then we have:

1. f''(x) is positive semidefinite for all x ∈ S ⇔ f is convex in S
2. f''(x) is negative semidefinite for all x ∈ S ⇔ f is concave in S
3. f''(x) is positive definite for all x ∈ S ⇔ f is strictly convex in S
4. f''(x) is negative definite for all x ∈ S ⇔ f is strictly concave in S

http://home.bi.no/a0710194/Teaching/BI-Mathematics/GRA-6035/2010/lecture5-hand.pdf

Extremum Value Theorem

If an objective function is continuous and its domain is compact, there exists the global max and min points.

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