# User talk:Oyz

## Welcome to the Wikipedia

Welcome, newcomer!

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[[User:ClockworkSoul|User:ClockworkSoul/sig]] 05:37, 1 Dec 2004 (UTC)

## Request for edit summary

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Oleg Alexandrov (talk) 17:08, 7 April 2006 (UTC)

${\displaystyle e^{j\pi }=1}$

## some notes

### Complex-conjugate multiplications with complex-swap

Efficient Implementation for Complex-Conjugate Multiplications with Complex-Swap

Coexistence of complex and complex-conjugate multiplications

### Householder transformation

• ${\displaystyle H=I-2\ v\ v^{*}}$
• ${\displaystyle v=e^{j\theta }\ e_{1}-w}$
• ${\displaystyle Hw=e_{1}}$
• ${\displaystyle He_{1}=w}$
• ${\displaystyle ||v||=1}$
• ${\displaystyle ||w||=1}$
• ${\displaystyle H^{*}H=I}$
• ${\displaystyle R=\Theta -W}$
• ${\displaystyle W^{*}W=I,\qquad \Theta ^{*}\Theta =I.}$
• ${\displaystyle \Gamma ^{*}R^{*}R\Gamma =I}$
• ${\displaystyle R+2R\Gamma \Gamma ^{*}R^{*}W=0}$
• ${\displaystyle R\Gamma +2R\Gamma \Gamma ^{*}R^{*}(\Theta -R)\Gamma =0}$
• ${\displaystyle R\Gamma +2R\Gamma \Gamma ^{*}R^{*}\Theta \Gamma -2R\Gamma \Gamma ^{*}R^{*}R\Gamma =0}$
• ${\displaystyle 2R\Gamma \Gamma ^{*}R^{*}\Theta \Gamma =R\Gamma }$
• ${\displaystyle R-2R\Gamma \Gamma ^{*}R^{*}\Theta =0}$
• ${\displaystyle R\Gamma \Gamma ^{*}R^{*}(W+\Theta )=0}$
• ${\displaystyle (\Theta -W)\Gamma \Gamma ^{*}(\Theta -W)^{*}(\Theta +W)=0}$
• ${\displaystyle \Gamma ^{*}(\Theta ^{*}W-W^{*}\Theta )=0}$
• It implies ${\displaystyle \Gamma }$ is not full-rank. It contradicts with ${\displaystyle \Gamma ^{*}R^{*}R\Gamma =I}$.
• Therefore, ${\displaystyle \Theta ^{*}W=W^{*}\Theta }$
• Since ${\displaystyle \Theta }$ or ${\displaystyle W}$ can not be Hermitian matrices, the failure of the generalization is proved.

The case of one-rank modification is the only possible one for the reflection with any desired hyperplane.

• But multiple-rank reflection transform can be used for finding the basis of the null space!

### Order-recursive calculation of SVD via column-wise augmentation

Low-latency SVD

applications to mimo detector, steering matrix gain ...

#### introduction

* motivation
* real-time or massive data application: small processing resource or high data volume.
* column- or row-wise data insertion: cache structure or memory limitation.
* need to update inovative column information...
* enabling ideas
* rank-one update formula: adding new column
* solving secular equation
* bi-digonalization for numerical stability


#### approach

• order-recursive formula:
${\displaystyle \mathbf {A} _{n+1}={\begin{pmatrix}\mathbf {A} _{n}&\mathbf {c} _{n+1}\\\end{pmatrix}}}$
• consider the SVD of A,,n,,is available: {{{#!latex
$$\mathbf A_n = \mathbf U_n \mathbf \Sigma_n \mathbf V_n^*$$


}}}

* uninary matrices can be used to obtain an almost diagonalized matrix: {{{#!latex
$$\mathbf U_n^* \ \mathbf A_{n+1} \begin{pmatrix} \mathbf V_n & \mathbf 0 \\ \mathbf 0^* & 1 \end{pmatrix} = \begin{pmatrix} \mathbf \Sigma_n[1:n,1:n] & \mathbf d_{n+1}[1:n] \\ \mathbf 0_{(m-n)\times n} & \mathbf d_{n+1}[n+1:m] \\ \end{pmatrix}$$
where $\mathbf d_{n+1} = \mathbf U_n^* \ \mathbf c_{n+1}$.


}}}

* Using Householder transformation, the upper-triangular form can be obtained (tall matrix assumed.): {{{#!latex
$$\begin{pmatrix} \mathbf I_n & \mathbf O \\ \mathbf O & \mathbf H_{m-n} \end{pmatrix} \begin{pmatrix} \mathbf \Sigma_n[1:n,1:n] & \mathbf d_{n+1}[1:n] \\ \mathbf 0_{(m-n)\times n} & \mathbf d_{n+1}[n+1:m] \\ \end{pmatrix} = \begin{pmatrix} \mathbf \Sigma_n[1:n,1:n] & \mathbf d_{n+1}[1:n] \\ \mathbf 0_{n}^* & f_{n+1} \\ \mathbf 0_{(m-n-1)\times n} & \mathbf 0_{m-n-1} \\ \end{pmatrix}$$
where $\mathbf d_{n+1} = \mathbf U_n^* \ \mathbf c_{n+1}$.


}}}

* The almost diagonal matrix can be diagonalized by means of the previous approaches.
* Among them, the secular equation solving is the best for rank-one update:
* it leads to finding simple zeros of polynomials.
* linear interplation/iterations are enough.


#### solving secular equation

• Summary:
1. move zero sigmas right-most: column-swap
2. move up zero d's: column-and-row swap
3. make a square part by householder transforming residual d.
4. apply secular equation for the square part of dimension r-q+1.
5. merge diagonal parts and unitary matrices: singular values are not ordered for calculation speed.
6. sort the diagonal
• re-visit formula:
${\displaystyle \mathbf {U} _{n}^{*}\ \mathbf {A} _{n+1}{\begin{pmatrix}\mathbf {V} _{n}&\mathbf {0} \\\mathbf {0} ^{*}&1\end{pmatrix}}={\begin{pmatrix}\mathbf {\Sigma } _{n}[1:r,1:r]&\mathbf {O} _{r\times (n-r)}&\mathbf {d} _{n+1}[1:r]\\\mathbf {O} _{(n-r)\times r}&\mathbf {O} _{n-r}&\mathbf {d} _{n+1}[r+1:n]\\\mathbf {O} _{(m-n)\times r}&\mathbf {O} _{(m-n)\times (n-r)}&\mathbf {d} _{n+1}[n+1:m]\\\end{pmatrix}}}$

where ${\displaystyle r}$ is rank of ${\displaystyle \Sigma _{n}}$. Note that ${\displaystyle \mathbf {d} _{n+1}[1:r]}$ may include zeros.

• more swapping rows and columns for zero singular values and diagonal parts.
1. move zero sigmas right-most: column-swap {{{#!latex
$$\mathbf U_n^* \ \mathbf A_{n+1} \begin{pmatrix} \mathbf V_n & \mathbf 0 \\ \mathbf 0^* & 1 \end{pmatrix} \mathbf P_{\mathbf\Sigma_n} = \begin{pmatrix} \mathbf \Sigma_n[1:r,1:r] & \mathbf d_{n+1}[1:r] & \mathbf O_{r\times(n-r)}\\ \mathbf O_{(n-r)\times r} & \mathbf d_{n+1}[r+1:n] & \mathbf O_{n-r} \\ \mathbf O_{(m-n)\times r} & \mathbf d_{n+1}[n+1:m] & \mathbf O_{(m-n)\times(n-r)} \\ \end{pmatrix}$$
where $\mathbf P_{\mathbf \Sigma_n}$ is a proper permutation matrix.


}}}

1. move up zero d's: column-and-row swap
${\displaystyle \mathbf {P} _{\mathbf {d} _{n+1}}^{*}\mathbf {U} _{n}^{*}\ \mathbf {A} _{n+1}{\begin{pmatrix}\mathbf {V} _{n}&\mathbf {0} \\\mathbf {0} ^{*}&1\end{pmatrix}}\mathbf {P} _{\mathbf {\Sigma } _{n}}\mathbf {P} _{\mathbf {d} _{n+1}}={\begin{pmatrix}\mathbf {\Sigma } _{n,0}&\mathbf {O} _{q\times (r-q)}&\mathbf {0} _{q}&\mathbf {O} _{q\times (n-r)}\\\mathbf {O} _{(r-q)\times q}&\mathbf {\Sigma } _{n,1}&\mathbf {f} _{n+1}[q+1:r]&\mathbf {O} _{(r-q)\times (n-r)}\\\mathbf {O} _{(m-r)\times q}&\mathbf {O} _{(m-r)\times (r-q)}&\mathbf {d} _{n+1}[r+1:m]&\mathbf {O} _{(m-r)\times (n-r)}\\\end{pmatrix}}}$

where ${\displaystyle \mathbf {P} _{\mathbf {d} _{n+1}}}$ is a proper permutation matrix s.t. the non-zero elements of ${\displaystyle \mathbf {d} _{n+1}[1:r]}$ form a new vector ${\displaystyle \mathbf {f} _{n+1}[q+1:r]}$.

1. make a square part by householder transforming residual d. {{{#!latex
\begin{*align}
\ &
\begin{pmatrix}
\mathbf I_r & \mathbf O \\
\mathbf O   & \mathbf H_{m-r}
\end{pmatrix}
\mathbf P_{\mathbf d_{n+1}}^*
\mathbf U_n^* \ \mathbf A_{n+1}
\begin{pmatrix}
\mathbf V_n & \mathbf 0 \\
\mathbf 0^* & 1
\end{pmatrix}
\mathbf P_{\mathbf \Sigma_n}
\mathbf P_{\mathbf d_{n+1}}
\\
&=
\begin{pmatrix}
\mathbf \Sigma_{n,0}      & \mathbf O_{q\times(r-q)}   &  \mathbf 0_q                & \mathbf O_{q\times(n-r)}\\
\mathbf O_{(r-q)\times q}  & \mathbf \Sigma_{n,1}    &  \mathbf f_{n+1}[q+1:r]     & \mathbf O_{(r-q)\times(n-r)}\\
\mathbf 0_{q}^*           &  \mathbf 0_{r-q}^*}   &  -\mathbf f_{n+1}[r+1] e^{j\angle \mathbf d_{n+1}[r+1]} & \mathbf 0_{n-r}^*        \\
\mathbf O_{(m-r-1)\times q} &  \mathbf O_{(m-r-1)\times(r-q)}    & \mathbf 0_{m-r-1}  & \mathbf O_{(m-r-1)\times(n-r)} \\
\end{pmatrix}
\end{*align}
\\
where $\mathbf f_{n+1}[r+1]=||\mathbf d_{n+1}[r+1:m]||$.


}}}

  1. apply secular equation for the square part of dimension r-q+1. {{{#!latex
$$\begin{pmatrix} \mathbf \Sigma_{n,1} & \mathbf f_{n+1}[q+1:r] \\ \mathbf 0_{r-q}^*} & \mathbf f_{n+1}[r+1] \\ \end{pmatrix} = \mathbf U_{n+\tfrac{1}{2}} \mathbf \Sigma_{n+\tfrac{1}{2}} \mathbf V_{n+\tfrac{1}{2}}^*$$


}}}

     Note that the coefficients of the secular equation will be non-zero. It leads to easy non-generic soluation.

1. merge diagonal parts and unitary matrices: singular values are not ordered for calculation speed.
${\displaystyle \mathbf {A} _{n+1}=\mathbf {U} _{n+1}{\begin{pmatrix}\mathbf {\Sigma } _{n+1}\\\mathbf {O} _{(m-n-1)\times (n+1)}\end{pmatrix}}\mathbf {V} _{n+1}^{*}}$

where the unordered diagonal matrix is

${\displaystyle \mathbf {\Sigma } _{n+1}={\begin{pmatrix}\left.{\begin{matrix}\mathbf {\Sigma } _{n,0}&\mathbf {O} _{q\times (r-q+1)}\\\mathbf {O} _{(r-q+1)\times q}&\mathbf {\Sigma } _{n+{\tfrac {1}{2}}}\\\end{matrix}}\right|&\mathbf {O} _{(n+1)\times (n-r)}\end{pmatrix}}}$

and the unitary matrices are calculated by multiplying the intermediate unitary matrices:

${\displaystyle \mathbf {U} _{n+1}=\mathbf {U} _{n}{\begin{pmatrix}\mathbf {I} _{r}&\mathbf {O} \\\mathbf {O} &\mathbf {H} _{m-r}\end{pmatrix}}\mathbf {P} _{\mathbf {d} _{n+1}}{\begin{pmatrix}\mathbf {I} _{r}&\mathbf {0} &\mathbf {O} \\\mathbf {0} ^{*}&-e^{j\angle \mathbf {d} _{n+1}[n+1]}&\mathbf {0} ^{*}\\\mathbf {O} &\mathbf {0} &\mathbf {I} _{m-r-1}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {I} _{q}&\mathbf {O} &\mathbf {O} \\\mathbf {O} &\mathbf {U} _{n+{\tfrac {1}{2}}}&\mathbf {O} _{(r-q+1)\times (m-r-1)}\\\mathbf {O} &\mathbf {O} _{(m-r-1)\times (r-q+1)}&\mathbf {I} _{m-r-1}\end{pmatrix}}}$

and

${\displaystyle \mathbf {V} _{n+1}={\begin{pmatrix}\mathbf {V} _{n}&\mathbf {0} \\\mathbf {0} ^{*}&1\end{pmatrix}}\mathbf {P} _{\mathbf {\Sigma } _{n}}\mathbf {P} _{\mathbf {d} _{n+1}}{\begin{pmatrix}\mathbf {I} _{q}&\mathbf {O} &\mathbf {O} \\\mathbf {O} &\mathbf {V} _{n+{\tfrac {1}{2}}}&\mathbf {O} \\\mathbf {O} &\mathbf {O} &\mathbf {I} _{n-r-1}\\\end{pmatrix}}.}$
1. sort the diagonal

#### example

* mx2 case
* formula: {{{#!latex
$$\mathbf A_{2} = \begin{pmatrix} \mathbf a_1 & \mathbf c_{2} \\ \end{pmatrix}$$


}}}

  * trivial SVD of a,,1,,: {{{#!latex
$$\mathbf a_1 = \mathbf u_1 \cdot \sigma_1 \cdot 1$$


}}}

  * almost digonalization: {{{#!latex
$$\mathbf U_n^* \ \mathbf A_{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \mathbf \sigma_1 & d_{1} \\ \mathbf 0_{m-1} & \mathbf d[2:m] \end{pmatrix}$$
where $d_{1} = \mathbf u_1^* \ \mathbf c_{2}$ and


$\mathbf d[2:m] = \mathbf U_1[2:m]^* \ \mathbf c_{2}$. }}}

  * upper-triangular form is good for numerical stability and compact calculation as well:
* mx3 case
* mx4 case


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ralphamale (talk) 22:02, 24 January 2012 (UTC)

## ArbCom elections are now open!

Hi,
You appear to be eligible to vote in the current Arbitration Committee election. The Arbitration Committee is the panel of editors responsible for conducting the Wikipedia arbitration process. It has the authority to enact binding solutions for disputes between editors, primarily related to serious behavioural issues that the community has been unable to resolve. This includes the ability to impose site bans, topic bans, editing restrictions, and other measures needed to maintain our editing environment. The arbitration policy describes the Committee's roles and responsibilities in greater detail. If you wish to participate, you are welcome to review the candidates' statements and submit your choices on the voting page. For the Election committee, MediaWiki message delivery (talk) 12:50, 23 November 2015 (UTC)

## ArbCom Elections 2016: Voting now open!

 Hello, Oyz. Voting in the 2016 Arbitration Committee elections is open from Monday, 00:00, 21 November through Sunday, 23:59, 4 December to all unblocked users who have registered an account before Wednesday, 00:00, 28 October 2016 and have made at least 150 mainspace edits before Sunday, 00:00, 1 November 2016. The Arbitration Committee is the panel of editors responsible for conducting the Wikipedia arbitration process. It has the authority to impose binding solutions to disputes between editors, primarily for serious conduct disputes the community has been unable to resolve. This includes the authority to impose site bans, topic bans, editing restrictions, and other measures needed to maintain our editing environment. The arbitration policy describes the Committee's roles and responsibilities in greater detail. If you wish to participate in the 2016 election, please review the candidates' statements and submit your choices on the voting page. Mdann52 (talk) 22:08, 21 November 2016 (UTC)