# Transfer matrix

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Transfer_matrix
> Markdown URL: https://mediated.wiki/source/Transfer_matrix.md
> Source: https://en.wikipedia.org/wiki/Transfer_matrix
> Source revision: 1340618272
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|In wavelet theory}}
{{about|the transfer matrix in wavelet theory|the transfer matrix in control systems|Transfer function matrix|the transfer matrix method in statistical mechanics|Transfer-matrix method (statistical mechanics)|the transfer matrix method in optics|Transfer-matrix method (optics)|the transfer matrix in dynamical systems theory|Transfer operator|the transfer matrix in combinatorics|Adjacency matrix|a single scalar|Transfer coefficient (disambiguation)}}

In [applied mathematics](/source/applied_mathematics), the '''transfer matrix''' is a formulation in terms of a [block-Toeplitz matrix](/source/block-Toeplitz_matrix) of the two-scale equation, which characterizes [refinable function](/source/refinable_function)s. Refinable functions play an important role in [wavelet](/source/wavelet) theory and [finite element](/source/finite_element) theory.

For the mask <math>h</math>, which is a vector with component indexes from <math>a</math> to <math>b</math>,
the transfer matrix of <math>h</math>, we call it <math>T_h</math> here, is defined as
:<math>
(T_h)_{j,k} = h_{2\cdot j-k}.
</math>
More verbosely
:<math>
T_h =
\begin{pmatrix}
h_{a  } &         &         &         &         &   \\
h_{a+2} & h_{a+1} & h_{a  } &         &         &   \\
h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a  } &   \\
\ddots  & \ddots  & \ddots  & \ddots  & \ddots  & \ddots \\
  & h_{b  } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\
  &         &         & h_{b  } & h_{b-1} & h_{b-2} \\
  &         &         &         &         & h_{b  }
\end{pmatrix}.
</math>
The effect of <math>T_h</math> can be expressed in terms of the [downsampling](/source/downsampling) operator "<math>\downarrow</math>":
:<math>T_h\cdot x = (h*x)\downarrow 2.</math>

==Properties==
{{unordered list
| <math>T_h\cdot x = T_x\cdot h</math>.
| 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](/source/Sylvester_matrix).
| The determinant of a transfer matrix is essentially a resultant.
{{pb}}
More precisely:
{{pb}}
Let <math>h_{\mathrm{e}}</math> be the even-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{e}})_k = h_{2k}</math>) and let <math>h_{\mathrm{o}}</math> be the odd-indexed coefficients of <math>h</math> (<math>(h_{\mathrm{o}})_k = h_{2k+1}</math>).
{{pb}}
Then <math>\det T_h = (-1)^{\lfloor\frac{b-a+1}{4}\rfloor}\cdot h_a\cdot h_b\cdot\mathrm{res}(h_{\mathrm{e}},h_{\mathrm{o}})</math>, where <math>\mathrm{res}</math> is the [resultant](/source/resultant).
{{pb}}
This connection allows for fast computation using the [Euclidean algorithm](/source/Euclidean_algorithm).
| For the [trace](/source/Trace_(linear_algebra)) of the transfer matrix of [convolved](/source/convolution) masks holds
{{pb}}
<math>\mathrm{tr}~T_{g*h} = \mathrm{tr}~T_g \cdot \mathrm{tr}~T_h</math>
| For the [determinant](/source/determinant) of the transfer matrix of convolved mask holds
{{pb}}
<math>\det T_{g*h} = \det T_g \cdot \det T_h \cdot \mathrm{res}(g_-,h)</math>
{{pb}}
where <math>g_-</math> denotes the mask with alternating signs, i.e. <math>(g_-)_k = (-1)^k \cdot g_k</math>.
| If <math>T_{h}\cdot x = 0</math>, then <math>T_{g*h}\cdot (g_-*x) = 0</math>.
{{pb}}
This is a concretion of the determinant property above. From the determinant property one knows that <math>T_{g*h}</math> is [singular](/source/Singular_matrix) whenever <math>T_{h}</math> is singular. This property also tells, how vectors from the [null space](/source/null_space) of <math>T_{h}</math> can be converted to null space vectors of <math>T_{g*h}</math>.
| If <math>x</math> is an eigenvector of <math>T_{h}</math> with respect to the eigenvalue <math>\lambda</math>, i.e.
{{pb}}
<math>T_{h}\cdot x = \lambda\cdot x</math>,
{{pb}}
then <math>x*(1,-1)</math> is an eigenvector of <math>T_{h*(1,1)}</math> with respect to the same eigenvalue, i.e.
{{pb}}
<math>T_{h*(1,1)}\cdot (x*(1,-1)) = \lambda\cdot (x*(1,-1))</math>.
| Let <math>\lambda_a,\dots,\lambda_b</math> be the eigenvalues of <math>T_h</math>, which implies <math>\lambda_a+\dots+\lambda_b = \mathrm{tr}~T_h</math> and more generally <math>\lambda_a^n+\dots+\lambda_b^n = \mathrm{tr}(T_h^n)</math>. This sum is useful for estimating the [spectral radius](/source/spectral_radius) of <math>T_h</math>. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small <math>n</math>.
{{pb}}
Let <math>C_k h</math> be the periodization of <math>h</math> with respect to period <math>2^k-1</math>. That is <math>C_k h</math> is a circular filter, which means that the component indexes are [residue class](/source/Modular_arithmetic)es with respect to the modulus <math>2^k-1</math>. Then with the [upsampling](/source/upsampling) operator <math>\uparrow</math> it holds
{{pb}}
<math>\mathrm{tr}(T_h^n) = \left(C_k h * (C_k h\uparrow 2) * (C_k h\uparrow 2^2) * \cdots * (C_k h\uparrow 2^{n-1})\right)_{[0]_{2^n-1}}</math>
{{pb}}
Actually not <math>n-2</math> convolutions are necessary, but only <math>2\cdot \log_2 n</math> ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the [Fast Fourier transform](/source/Fast_Fourier_transform).
| From the previous statement we can derive an estimate of the [spectral radius](/source/spectral_radius) of <math>\varrho(T_h)</math>. It holds
{{pb}}
<math>\varrho(T_h) \ge \frac{a}{\sqrt{\# h}} \ge \frac{1}{\sqrt{3\cdot \# h}}</math>
{{pb}}
where <math>\# h</math> is the size of the filter and if all eigenvalues are real, it is also true that
{{pb}}
<math>\varrho(T_h) \le a</math>,
{{pb}}
where <math>a = \Vert C_2 h \Vert_2</math>.
}}

==See also==
* [Hurwitz determinant](/source/Hurwitz_determinant)

==References==
* {{cite journal
|first=Gilbert|last=Strang
|author-link=Gilbert Strang
|title=Eigenvalues of <math>(\downarrow 2){H}</math> and convergence of the cascade algorithm
|journal=IEEE Transactions on Signal Processing
|volume=44
|pages=233–238
|year=1996
|doi=10.1109/78.485920
}}
* {{cite thesis
|first=Henning
|last=Thielemann
|url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131
|title=Optimally matched wavelets
|type=PhD thesis
|year=2006
}} (contains proofs of the above properties)

Category:Wavelets
Category:Numerical analysis

---
Adapted from the Wikipedia article [Transfer matrix](https://en.wikipedia.org/wiki/Transfer_matrix) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Transfer_matrix?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.