Singular Value Decomposition

Question 1

How does A, a square matrix, decompose with the help of eigenvalues?

Answer 1

A = XVX^(-1)
Where A is the matrix to be decomposed,
V (usually lambda) is a diagonal matrix whose elements are the eigenvalues of A,
And X is a matrix whose columns are the eigenvectors of A

Question 2

What are the singular values of A?

Answer 2

The singular value o_i = sqrt(L_i), 1 =< i =< r
Where L _i is the i^th eigenvalue of A^tA, and r is the rank of the matrix (and therefore the number of non-zero eigenvalues)

Question 3

How do we generalise our idea of eigenvalues (Ax = Lx) for singular values?

Answer 3

We demand that Aq_i = o_iq^{^}_i
Where q_i is a column of Q, where the columns of Q are the eigenvectors of A^tA, and q^{^}_i becomes an orthogonal set of vectors

Question 4

if state = (going to fail):
	don’t

Answer 4

you’re welcome

Question 5

How is AQ = Q^{^}E formed?

Answer 5

A is the original matrix
Q is composed of the eigenvectors
Q^{^} is formed from the column vectors q^{^}
E is of size m x n, and is entirely zero apart from containing o_i along the first section of the leading diagonal, where o_i are the singular values (roots of the eigenvalues)

Question 6

What is the singular decomposition of A?

Answer 6

A = Q^{^}EQ^t