Orthogonality

Orthogonality is a generalization of perpendicularity. Orthonormality is a related concept, requiring that the components be unit vectors.

See also vector orthogonality.

Orthogonality

The notation for orthogonality is ⊥, as in x ⊥ y.

For a subspace S to be orthogonal to a subspace T, every vector in S must be orthogonal to every vector in T. Null spaces are a trivial example. For a given matrix A, its null space (i.e., N(A)) contains the vectors that are not in the row space (i.e., R(A)). Therefore it is orthogonal. Similarly, N(A^T) is orthogonal to the column space of A (i.e., C(A))

Orthonormality

If a matrix is composed of orthonormal columns, then it can be called a matrix with orthonormal columns. These are usually denoted as Q. These have several important properties:

• Q^TQ = I

• The projection matrix is given as P = QQ^T.

The second follows from the first. Recall that, when projecting b into C(A), the projection matrix is given as P = A(A^TA^-1)A^T. This comes from the linear system A^TAx̂ = A^Tb and requiring that p = Pb. For a matrix Q with orthonormal columns, the first property simplifies the linear system to x̂ = Q^Tb. Therefore, P = QQ^T.

If such a matrix with orthonormal columns is also square, then it can be called an orthogonal matrix. These have several important properties:

• Q^TQ = QQ^T = I

• Q^T = Q^-1

• The determinant is always 1 or -1

• The projection matrix is given as P = I, indicating that b must be in C(A).

CategoryRicottone