Orthogonal Decomposition
The orthogonal decomposition of a vector y in R^n is the sum of a vector in a subspace W of R^n and a vector in the orthogonal complement W^_|_ to W.
The orthogonal decomposition theorem states that if W is a subspace of R^n, then each vector y in R^n can be written uniquely in the form
| y=y^^+z, |
where y^^ is in W and z is in W^_|_. In fact, if {u_1,u_2,...,u_p} is any orthogonal basis of W, then
and z=y-y^^.
Geometrically, y^^ is the orthogonal projection of y onto the subspace W and z is a vector orthogonal to y^^
See also
Fredholm's Theorem, LU Decomposition, QR DecompositionThis entry contributed by Viktor Bengtsson
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References
Golub, G. and Van Loan, C. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.Referenced on Wolfram|Alpha
Orthogonal DecompositionCite this as:
Bengtsson, Viktor. "Orthogonal Decomposition." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthogonalDecomposition.html