The relation between SVD and eigenvalue-decomposition

Usage notes

Please use Maple Notation

a: The SVD of an m x n - matrix A is given by

A = ___________________

where U is a orthogonal m x m - matrix, V is a orthogonal n x n - matrix and Sigma is a diagonal m x n - matrix with real, non-negative elements sigma_i ,i = 1..min(m,n) ,in descending order: sigma₁ >=..>=sigma_min(m,n)>=0

b: The eigenvalue / eigenvector decomposition of a symmetric n x n - matrix A^T.A is given by

A^T.A = ___________________

where Lambda is a diagonal matrix with entries lambda_i (i = 1..n), V is an orthogonal matrix (in row vector notation). Notice that lambda_i are the eigenvalues of Lambda and of A^T.A and that the columns of V are the eigenvectors of A^T.A.

A^T.A is symmetric because (A^T.A)^T = A^T.(A^T)^T = A^T.A. We assume that A is real. Then it holds that all eigenvalues are nonnegative.

c: Given the following equations:

x1			=	1
		x2	=	-1
2*x1	+	2*x2	=	1

We have an approximation problem since the equation system is overdetermined: we have 3 equations and 2 unknowns.

The system of equations could be written as A.x = b with

x = (x1, x2)^T

	___________________	___________________
A =	___________________	___________________
	___________________	___________________

	___________________
b =	___________________
	___________________

We would like to test the validity of the equation sigma_i = sqrt(lamdba_i), the relation between the SVD of A and the eigenvalue-decomposition of A^T.A, for the above given problem.

d: Compute according to the above equations

	___________________	___________________
A^T.A =	___________________	___________________

Eigenvalues lambda₁ = ___________________ lambda₂ = ___________________

	___________________		___________________
Eigenvector v₁ for eigenvalue lambda₁=	___________________	Eigenvector v₂ for eigenvalue lambda₂=	___________________

e: Sort the eigenvalues lambda_i in descending order lambda₁>=..>=lambda_n and those for v_i respectively , and normalize the matrix V so that ||V||=1.

	___________________	___________________
^{Lambda =}	___________________	___________________

	___________________	___________________
^{V =}	___________________	___________________

Proceeding in the same way for A.A^T = U.Lambda₂.U^T we get the following eigenvector/eigenvalue decomposition

f: Solve the equation A=U.Sigma.V^T in terms of Sigma using for U and V the values computed above.

	___________________	___________________
Sigma =	___________________	___________________
	___________________	___________________

As expected we get sigma_i = sqrt(lambda_i)

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