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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク transfer matrix ： ウィキペディア英語版 transfer matrix In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory. For the mask $h$, which is a vector with component indexes from $a$ to $b$, the transfer matrix of $h$, we call it $T\_h$ here, is defined as :$$ (T_h)_ = h_. More verbosely :$$ T_h = \begin h_ & & & & & \\ h_ & h_ & h_ & & & \\ h_ & h_ & h_ & h_ & h_ & \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ & h_ & h_ & h_ & h_ & h_ \\ & & & h_ & h_ & h_ \\ & & & & & h_ \end. The effect of $T\_h$ can be expressed in terms of the downsampling operator "$\backslash downarrow$": :$T\_h\backslash cdot\; x\; =\; (h$ *x)\downarrow 2. ==Properties== * $T\_h\backslash cdot\; x\; =\; T\_x\backslash cdot\; h$. * If you drop the first and the last column and move the odd-indexed columns to the left and the even-indexed columns to the right, then you obtain a transposed Sylvester matrix. * The determinant of a transfer matrix is essentially a resultant. :More precisely: :Let $h\_\})\_k\; =\; h\_$) and let $h\_\})\_k\; =\; h\_$). :Then $\backslash det\; T\_h\; =\; (-1)^\backslash rfloor\}\backslash cdot\; h\_a\backslash cdot\; h\_b\backslash cdot\backslash mathrm(h\_\})$, where $\backslash mathrm$ is the resultant. :This connection allows for fast computation using the Euclidean algorithm. * For the trace of the transfer matrix of convolved masks holds :$\backslash mathrm~T\_\; =\; \backslash mathrm~T\_g\; \backslash cdot\; \backslash mathrm~T\_h$ * For the determinant of the transfer matrix of convolved mask holds :$\backslash det\; T\_\; =\; \backslash det\; T\_g\; \backslash cdot\; \backslash det\; T\_h\; \backslash cdot\; \backslash mathrm(g\_-,h)$ :where $g\_-$ denotes the mask with alternating signs, i.e. $(g\_-)\_k\; =\; (-1)^k\; \backslash cdot\; g\_k$. * If $T\_\backslash cdot\; x\; =\; 0$, then $T\_\backslash cdot\; (g\_-$ *x) = 0. : This is a concretion of the determinant property above. From the determinant property one knows that $T\_$ is singular whenever $T\_$ is singular. This property also tells, how vectors from the null space of $T\_$ can be converted to null space vectors of $T\_$. * If $x$ is an eigenvector of $T\_$ with respect to the eigenvalue $\backslash lambda$, i.e. : $T\_\backslash cdot\; x\; =\; \backslash lambda\backslash cdot\; x$, :then $x$ *(1,-1) is an eigenvector of $T\_$ with respect to the same eigenvalue, i.e. : $T\_\backslash cdot\; (x$ *(1,-1)) = \lambda\cdot (x *(1,-1)). * Let $\backslash lambda\_a,\backslash dots,\backslash lambda\_b$ be the eigenvalues of $T\_h$, which implies $\backslash lambda\_a+\backslash dots+\backslash lambda\_b\; =\; \backslash mathrm~T\_h$ and more generally $\backslash lambda\_a^n+\backslash dots+\backslash lambda\_b^n\; =\; \backslash mathrm(T\_h^n)$. This sum is useful for estimating the spectral radius of $T\_h$. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small $n$. :Let $C\_k\; h$ be the periodization of $h$ with respect to period $2^k-1$. That is $C\_k\; h$ is a circular filter, which means that the component indexes are residue classes with respect to the modulus $2^k-1$. Then with the upsampling operator $\backslash uparrow$ it holds :$\backslash mathrm(T\_h^n)\; =\; \backslash left(C\_k\; h$ * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^)\right)_{\sqrt{3\cdot \# h}} :where $\backslash \#\; h$ is the size of the filter and if all eigenvalues are real, it is also true that :$\backslash varrho(T\_h)\; \backslash le\; a$, :where $a\; =\; \backslash Vert\; C\_2\; h\; \backslash Vert\_2$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「transfer matrix」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.