 TOPICS  # Projection Matrix

A projection matrix is an square matrix that gives a vector space projection from to a subspace . The columns of are the projections of the standard basis vectors, and is the image of . A square matrix is a projection matrix iff .

A projection matrix is orthogonal iff (1)

where denotes the adjoint matrix of . A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector can be written , so (2)

An example of a nonsymmetric projection matrix is (3)

which projects onto the line .

The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies (4)

where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.

Any vector in is fixed by the projection matrix for any in . Consequently, a projection matrix has norm equal to one, unless , (5)

Let be a -algebra. An element is called projection if and . For example, the real function defined by on and on is a projection in the -algebra , where is assumed to be disconnected with two components and .

Idempotent, Inner Product, Map Projection, Orthogonal Set, Projection, Projection Operator, Pseudoinverse, Symmetric Matrix, Vector Space Projection, Vertical Perspective Projection

Portions of this entry contributed by Mohammad Sal Moslehian

Portions of this entry contributed by Todd Rowland

## Explore with Wolfram|Alpha More things to try:

## 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 Matrix

## Cite this as:

Moslehian, Mohammad Sal; Rowland, Todd; and Weisstein, Eric W. "Projection Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProjectionMatrix.html