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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

 y^^=(y·u_1)/(u_1·u_1)u_1+(y·u_2)/(u_2·u_2)u_2+...+(y·u_p)/(u_p·u_p)u_p,

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 Decomposition

This 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.

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Orthogonal Decomposition

Cite this as:

Bengtsson, Viktor. "Orthogonal Decomposition." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthogonalDecomposition.html

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