To install click the Add extension button. That's it.
The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.
How to transfigure the Wikipedia
Would you like Wikipedia to always look as professional and up-to-date? We have created a browser extension. It will enhance any encyclopedic page you visit with the magic of the WIKI 2 technology.
Try it — you can delete it anytime.
Install in 5 seconds
Yep, but later
4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
In mathematics, given a field , nonnegative integers , and a matrix, a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where and , where is the rank of .
YouTube Encyclopedic
1/5
Views:
330 336
59 755
430
14 451
105 723
Matrix Factorization - Numberphile
Lecture 16.5 — Recommender Systems | Vectorization Low Rank Matrix Factorization — [ Andrew Ng ]
Irène Waldspurger: Rank optimality for the Burer-Monteiro factorization
Lecture 30: Completing a Rank-One Matrix, Circulants!
Lecture 2: Multiplying and Factoring Matrices
Transcription
Existence
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are linearly independent columns in ; equivalently, the dimension of the column space of is . Let be any basis for the column space of and place them as column vectors to form the matrix . Therefore, every column vector of is a linear combination of the columns of . To be precise, if is an matrix with as the -th column, then
where 's are the scalar coefficients of in terms of the basis . This implies that , where is the -th element of .
Non-uniqueness
If is a rank factorization, taking and
gives another rank factorization for any invertible matrix of compatible dimensions.
Conversely, if are two rank factorizations of , then there exists an invertible matrix such that and .[1]
Construction
Rank factorization from reduced row echelon forms
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of . Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .
Then is obtained by removing the third column of , the only one which is not a pivot column, and by getting rid of the last row of zeroes from , so
It is straightforward to check that
Proof
Let be an permutation matrix such that in block partitioned form, where the columns of are the pivot columns of . Every column of is a linear combination of the columns of , so there is a matrix such that , where the columns of contain the coefficients of each of those linear combinations. So , being the identity matrix. We will show now that .
Transforming into its reduced row echelon form amounts to left-multiplying by a matrix which is a product of elementary matrices, so , where . We then can write , which allows us to identify , i.e. the nonzero rows of the reduced echelon form, with the same permutation on the columns as we did for . We thus have , and since is invertible this implies , and the proof is complete.
An immediate consequence of rank factorization is that the rank of is equal to the rank of its transpose . Since the columns of are the rows of , the column rank of equals its row rank.[2]
Proof: To see why this is true, let us first define rank to mean column rank. Since , it follows that . From the definition of matrix multiplication, this means that each column of is a linear combination of the columns of . Therefore, the column space of is contained within the column space of and, hence, .
Now, is , so there are columns in and, hence, . This proves that .
Now apply the result to to obtain the reverse inequality: since , we can write . This proves .
We have, therefore, proved and , so .
Notes
^Piziak, R.; Odell, P. L. (1 June 1999). "Full Rank Factorization of Matrices". Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882. JSTOR2690882.
^Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN978-1420095388
References
Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN978-1420095388
Lay, David C. (2005), Linear Algebra and its Applications (3rd ed.), Addison Wesley, ISBN978-0-201-70970-4
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations, Johns Hopkins Studies in Mathematical Sciences (3rd ed.), The Johns Hopkins University Press, ISBN978-0-8018-5414-9
Stewart, Gilbert W. (1998), Matrix Algorithms. I. Basic Decompositions, SIAM, ISBN978-0-89871-414-2
Piziak, R.; Odell, P. L. (1 June 1999). "Full Rank Factorization of Matrices". Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882. JSTOR2690882.
This page was last edited on 16 October 2022, at 18:49