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^*, 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, P w〉 = 〈v_W, P w〉 = 〈P v, w〉.

idempotent | inner product | map projection | orthogonal set | projection | projection operator | pseudoinverse | symmetric matrix | vector space projection | vertical perspective projection