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Matrix Factorizations: Singular Value Decomposition

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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!”

Choleski

Jordan

LU

Polar

Proper Orthogonal

QR

Schur

Singular Value

Spectral (Eigendecomposition)

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.