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# Matrix Factorizations: Singular Value Decomposition - PowerPoint PPT Presentation

Matrix Factorizations: Singular Value Decomposition. Presented by Nik Clark MTH 421. Introduction. In the exciting world of numerical analysis, one may wonder “Why? Why do I study matrices and their factorizations?”

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### Matrix Factorizations:Singular Value Decomposition

Presented by Nik Clark

MTH 421

• In the exciting world of numerical analysis, one may wonder “Why? Why do I study matrices and their factorizations?”

• Such a simple answer, “Because matrices and their decompositions can take you anywhere!”

Jordan

LU

Polar

Proper Orthogonal

QR

Schur

Singular Value

Spectral (Eigendecomposition)

Matrix Decompositions

Some interesting ways to decompose a matrix

We’d like to more formally introduce you to Singular Value Decomposition (SVD) and some of its applications.

• SVD is a type of factorization for a rectangular, real or complex, matrix.

• THEOREM:

All matrices Amxn have a singular value decomposition.

• All math is awesome.

• Also because SVD can be applied to several situations:

• Data Compression

• Analyzing DNA Gene Expression Data

• Solving Least Squares Problems

• Information Retrieval

• Image Processing

• De-blurring

• Seismology

• Digital Signal Processing

• Noise Reduction

• Data Hiding

• Cryptography

• Watermarks

• Researching Databases

• Each mxn matrix A decomposes into the product of three matrices A=UΣVT

• U is orthogonal and mxm. Its columns span col(A).

• V is also orthogonal, but is nxn. Its columns span row(A)

•  is a diagonal matrix where the singular values of A are along the main diagonal, and all other values are zero.

• The m columns of matrix U are the eigenvectors of AAT

• The n columns of matrix V are the eigenvectors of ATA

• The entries of the main diagonal of matrix  are the singular values of A, denoted by i

• An additional way to write A, apart from A=UΣVT :

A = 1u1vT1 +2u2vT2 + …+rurvTr ,

Where r = rank(A), and is defined to be the number of linearly independent columns of A.

• Each term of the expansion is already in order of importance.

• 1≥ 2 ≥ … ≥ k ≥ 0

• Ak=UkΣkVTk

• Where Ak is the first k terms of the SVD factorization of A

• i.e. A = 1u1vT1 +2u2vT2 + …+kukvTk

• A = 1u1vT1 +2u2vT2 + …+kukvTk

• Where each term of the expansion is a rank 1 matrix.

• Cipher- a method for encrypting a message.

• Cryptography – The art of creating a coded message using a cipher.

• Cryptanalysis – The art of breaking a code by finding its weakness.

• Cryptology – the study of the aforementioned definitions.

• Most commonly, a cryptogram is created by substituting one letter for another.

• When decoding a code (in english) it is easiest to first decode the vowels.

• Vowel pairs are less frequent than consonant-vowel pairs.

• More frequently, vowels follow consonants, vfc.

• When vowels follow consonants, there is a mathematical proportion:

number of vowel pairs

number of vowels

number consonant-vowel pairs

number of consonants

<

• We define matrix A to be a digram frequency of the letter combinations of our text.

• aij is the number of times the ith letter is followed by the jth letter.

• We define vector wi to equal 1 when the letter is a vowel and 0 otherwise.

• Vector ci is defined to equal 1 when the letter is a consonant and 0 otherwise.

• Our vectors w and c are orthogonal.

• wTAw is the number of vowel pairs.

• cTAw is the number of consonant vowel pairs.

• We can now rewrite our equation:

wTAw

wTA(w+c)

cTAw

cTA(w+c)

<

• A ~ A1 = 1u1vT1

• f = k1* u1

• f = k2* v1

• Where f represents the frequency of the words in the text.

• A ~ A2 = 1u1vT1 +2u2vT2

• Rank two is the frequency of vowel pairs.

• Each vowel (except for u) corresponds to a (+, -) vector pair.

• Each consonant corresponds to a (-, +) vector pair.

• There are some exceptions

• Those letters that correspond to a (+, +) vector pair or a (-, -) vector pair are called neuter.

• These letter patterns correspond to any text we consider.

• The following matrix is the digram matrix of an encrypted text (cryptogram) similar to the sentence below.

• What does this mean??

• Gsviv rh ml zkkorw nzgsvnzgrxk ru gsviv rh ml nzgsvnzgrxh gl zkkob.