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What is Singular Value Decomposition (SVD) ?
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any {\displaystyle m\times n}m\times n matrix. It is related to polar decomposition.
Let A be an m × n matrix. The Singular Value Decomposition (SVD) of A,
where U is m × m and orthogonal, V is n × n and orthogonal, and Σ is an m × n diagonal matrix with nonnegative diagonal entries
known as the singular values of A, is an extremely useful decomposition that yields much information about A, including its range, null space, rank, and 2-norm condition number. We now discuss a practical algorithm for computing the SVD of A, due to Golub and Kahan.
Let U and V have column partitions
From the relations
it follows that
That is, the squares of the singular values are the eigenvalues of AT A, which is a symmetric matrix.
It follows that one approach to computing the SVD of A is to apply the symmetric QR algorithm to AT A to obtain a decomposition AT A = V Σ T ΣV T . Then, the relations Avj = σjuj , j = 1, . . . , p, can be used in conjunction with the QR factorization with column pivoting to obtain U. However, this approach is not the most practical, because of the expense and loss of information incurred from computing AT A.
Instead, we can implicitly apply the symmetric QR algorithm to AT A. As the first step of the symmetric QR algorithm is to use Householder reflections to reduce the matrix to tridiagonal form, we can use Householder reflections to instead reduce A to upper bidiagonal form
It follows that T = BT B is symmetric and tridiagonal. We could then apply the symmetric QR algorithm directly to T, but, again, to avoid the loss of information from computing T explicitly, we implicitly apply the QR algorithm to T by performing the following steps during each iteration:
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