User talk:Oyz

Welcome to the Wikipedia
Welcome, newcomer!

Here are some useful tips to ease you into the Wikipedia experience:


 * First, take a look at the Wikipedia Tutorial, and perhaps dabble a bit in the test area.
 * When you have some free time, take a look at the Manual of Style and Policies and Guidelines. They can come in very handy!
 * Remember to use a neutral point of view!
 * If you need any help, feel free to post a question at the Help Desk
 * Explore, be bold in editing pages, and, most importantly, have fun!

Also, here are some odds and ends that I find useful from time to time:


 * Policy Library
 * Utilities
 * Cite your sources
 * Verifiability
 * Wikiquette
 * Civility
 * Conflict resolution
 * Brilliant prose
 * Pages needing attention
 * Peer review
 * Bad jokes and other deleted nonsense
 * Village pump
 * Boilerplate text

Feel free to ask me anything the links and talk pages don't answer. You can most easily reach me by posting on my talk page.

You can sign your name on any page by typing 4 tildes, likes this: &#x7e;&#x7e;&#x7e;&#x7e;.

Best of luck, and have fun!

User:ClockworkSoul 05:37, 1 Dec 2004 (UTC)

Request for edit summary
When editing an article on Wikipedia there is a small field labelled "Edit summary" under the main edit-box. It looks like this: The text written here will appear on the Recent changes page, in the page revision history, on the diff page, and in the watchlists of users who are watching that article. See m:Help:Edit summary for full information on this feature. When you leave the edit summary blank, some of your edits could be mistaken for vandalism and may be reverted, so please always briefly summarize your edits, especially when you are making subtle but important changes, like changing dates or numbers. Thank you. – Oleg Alexandrov (talk) 17:08, 7 April 2006 (UTC)

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

Complex-conjugate multiplications with complex-swap
Efficient Implementation for Complex-Conjugate Multiplications with Complex-Swap

Coexistence of complex and complex-conjugate multiplications

Householder transformation

 * $$H = I - 2 \ v \ v^*$$
 * $$v = e^{j\theta} \ e_1 - w$$
 * $$H w = e_1 $$
 * $$H e_1 = w $$
 * $$||v|| = 1$$
 * $$||w|| = 1$$
 * $$H^* H = I$$


 * $$ R = \Theta - W $$
 * $$ W^*W=I, \qquad \Theta^*\Theta=I. $$
 * $$ \Gamma^* R^* R \Gamma = I $$
 * $$ R + 2 R \Gamma \Gamma^* R^* W =0 $$


 * $$ R\Gamma + 2 R \Gamma \Gamma^* R^* (\Theta - R)\Gamma =0 $$
 * $$ R\Gamma + 2 R \Gamma \Gamma^* R^* \Theta \Gamma - 2 R \Gamma \Gamma^* R^* R\Gamma =0 $$
 * $$ 2 R \Gamma \Gamma^* R^* \Theta \Gamma = R \Gamma $$


 * $$ R - 2 R \Gamma \Gamma^* R^* \Theta =0 $$


 * $$ R \Gamma \Gamma^* R^* (W + \Theta) =0 $$


 * $$ (\Theta - W) \Gamma \Gamma^* (\Theta - W)^* (\Theta + W) =0 $$


 * $$ \Gamma^* (\Theta^* W - W^* \Theta) =0 $$


 * It implies $$ \Gamma $$ is not full-rank. It contradicts with $$ \Gamma^* R^* R \Gamma = I $$.


 * Therefore, $$ \Theta^* W = W^* \Theta $$

The case of one-rank modification is the only possible one for the reflection with any desired hyperplane.
 * Since $$ \Theta $$ or $$ W $$ can not be Hermitian matrices, the failure of the generalization is proved.


 * 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:
 * $$ \mathbf A_{n+1} =

\begin{pmatrix} \mathbf A_n & \mathbf c_{n+1} \\ \end{pmatrix} $$

* uninary matrices can be used to obtain an almost diagonalized matrix: * Using Householder transformation, the upper-triangular form can be obtained (tall matrix assumed.): * 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.
 * consider the SVD of A,,n,,is available:

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:

\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} $$ where $$r$$ is rank of $$\Sigma_n$$. Note that $$\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
 * 1) move up zero d's: column-and-row swap

\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} $$ where $$\mathbf P_{\mathbf d_{n+1}}$$ is a proper permutation matrix s.t. the non-zero elements of $$\mathbf d_{n+1}[1:r]$$ form a new vector $$\mathbf f_{n+1}[q+1:r]$$. 1. apply secular equation for the square part of dimension r-q+1. }     Note that the coefficients of the secular equation will be non-zero. It leads to easy non-generic soluation.
 * 1) make a square part by householder transforming residual d. }


 * 1) merge diagonal parts and unitary matrices: singular values are not ordered for calculation speed.
 * $$ \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}^* $$ where the unordered diagonal matrix is
 * $$ \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:

\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

\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: * trivial SVD of a,,1,,: * almost digonalization: * upper-triangular form is good for numerical stability and compact calculation as well: * mx3 case * mx4 case

You are invited to join Stanford's WikiProject!
ralphamale (talk) 22:02, 24 January 2012 (UTC)

ArbCom elections are now open!
MediaWiki message delivery (talk) 12:50, 23 November 2015 (UTC)

ArbCom 2023 Elections voter message
 Hello! Voting in the 2023 Arbitration Committee elections is now open until 23:59 (UTC) on. All eligible users are allowed to vote. Users with alternate accounts may only vote once.

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 2023 election, please review the candidates and submit your choices on the voting page. If you no longer wish to receive these messages, you may add to your user talk page. MediaWiki message delivery (talk) 00:21, 28 November 2023 (UTC)

Kria SoM moved to draftspace
Thanks for your contributions to Kria SoM. Unfortunately, I do not think it is ready for publishing at this time because it needs more sources to establish notability. I have converted your article to a draft which you can improve, undisturbed for a while.

Please see more information at Help:Unreviewed new page. When the article is ready for publication, please click on the "Submit your draft for review!" button at the top of the page OR move the page back.  Waqar 💬 06:28, 9 July 2024 (UTC)