Projection Matrix
A projection matrix P is an n×n square matrix that gives a vector space projection from R^n to a subspace W. The columns of P are the projections of the standard basis vectors, and W is the image of P. A square matrix P is a projection matrix iff P^2=P.
A projection matrix P is orthogonal iff
| P=P^*, |
(1)
|
where P^* denotes the adjoint matrix of P. A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector v can be written v=v_W+v_(W^_|_), so
| <v,Pw>=<v_W,Pw>=<Pv,w>. |
(2)
|
An example of a nonsymmetric projection matrix is
| [画像: P=[0 1; 0 1], ] |
(3)
|
![画像: P=[0 1; 0 1],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ProjectionMatrix/NumberedEquation3.svg){kind=link}
which projects onto the line y=x.
The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies
| <v,Pw>=<v_W,Pw>=<Pv,w>, |
(4)
|
where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.
Any vector in W is fixed by the projection matrix Pw=w for any w in W. Consequently, a projection matrix P has norm equal to one, unless P=0,
| ||P||=sup_(|x|=1)|Px|>=1. |
(5)
|
Let A be a C^*-algebra. An element p in A is called projection if p^*=p and p^2=p. For example, the real function f defined by f(x)=0 on G_1 and f(x)=1 on G_2 is a projection in the C^*-algebra C(X), where X is assumed to be disconnected with two components G_1 and G_2.
See also
Idempotent, Inner Product, Map Projection, Orthogonal Set, Projection, Projection Operator, Pseudoinverse, Symmetric Matrix, Vector Space Projection, Vertical Perspective ProjectionPortions of this entry contributed by Mohammad Sal Moslehian
Portions of this entry contributed by Todd Rowland
Explore with Wolfram|Alpha
References
Kadison, R. V. and Ringrose, J. R. Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary Theory. Providence, RI: Amer. Math. Soc., 1997.Murphy, G. J. C-*-Algebras and Operator Theory. New York: Academic Press, 1990.Referenced on Wolfram|Alpha
Projection MatrixCite this as:
Moslehian, Mohammad Sal; Rowland, Todd; and Weisstein, Eric W. "Projection Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProjectionMatrix.html