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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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  • 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!”
matrix decompositions




Proper Orthogonal



Singular Value

Spectral (Eigendecomposition)

Matrix Decompositions

Some interesting ways to decompose a matrix

singular value decomposition
Singular Value Decomposition

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

what is svd
What is SVD?
  • SVD is a type of factorization for a rectangular, real or complex, matrix.

All matrices Amxn have a singular value decomposition.

why should we care
Why should we care?
  • 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
more caring
More Caring
  • Image Processing
    • De-blurring
  • Seismology
  • Digital Signal Processing
    • Noise Reduction
  • Data Hiding
    • Cryptography
    • Watermarks
  • Researching Databases
back to the svd
Back to the SVD
  • 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.
more on svd
More on SVD
  • 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
another way to write
Another Way to Write
  • 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
rank k approximation
Rank k Approximation
  • Ak=UkΣkVTk
  • Where Ak is the first k terms of the SVD factorization of A
  • i.e. A = 1u1vT1 +2u2vT2 + …+kukvTk
more on rank
More on Rank
  • A = 1u1vT1 +2u2vT2 + …+kukvTk
  • Where each term of the expansion is a rank 1 matrix.
an example code breaking
An Example: Code breaking
  • 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.
how do we decode the code
How do we decode the code?
  • 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


the matrix
The matrix
  • 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:






singular vectors
Singular Vectors
  • A ~ A1 = 1u1vT1
  • f = k1* u1
  • f = k2* v1
  • Where f represents the frequency of the words in the text.
rank two
Rank Two
  • 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
neuter letters
Neuter Letters
  • Those letters that correspond to a (+, +) vector pair or a (-, -) vector pair are called neuter.
  • These letter patterns correspond to any text we consider.
an encrypted example
An Encrypted Example
  • 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.