Singular value decomposition (economy) of matrix
Syntax: @svd(m1, v1, m2)
m1: matrix, sym
v1: vector
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m2: matrix, sym
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Return: matrix
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Performs an “economy” or “thin” singular value decomposition of the matrix m1, generating truncated results when m1 is not square (exploiting the reduced maximum rank of a non-square matrix).
The matrix
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is returned by the function, the vector
v1 will be filled (resized if necessary) with the singular values and the matrix
m2 will be assigned (resized if necessary) the other matrix,
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, of the decomposition. The singular value decomposition satisfies:
where
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is a diagonal matrix with the singular values along the diagonal. Singular values close to zero indicate that the matrix may not be of full rank. See the
@rank function for a related discussion.
Let r be the number of rows of m1 and s be the number of columns of m1 so that m1 has at most t = min(r, s) distinct singular values. Then
• m2 will be s-by-t
• v1 will be t-by-1
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will have dimensions
r-by-
t Examples
matrix x = @mnrnd(5, 7)
vector w
matrix v
matrix u = @svd(x, w, v)
performs the thin SVD of the matrix X. U is
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, W is a 5 element vector containing the singular values, and V is a
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matrix.
Alternately, if the rank is less than the number of rows,
matrix x = @mnrnd(7, 5)
matrix u = @svd(x, w, v)
U is
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, W is a 5 element vector containing the singular values, and V is a
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matrix.
The following demonstrate the properties of the decomposition:
sym i1 = @inner(u)
sym i2 = @inner(v)
matrix x1 = u * @makediagonal(w) * v.@t
where I1 and I2 and the identity matrix, and X1 is equal to X.
Cross-references