Orthogonality

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

See also vector orthogonality.

Description

Orthogonality is an extension of perpendicularity to higher dimensions of Euclidean space, and also to arbitrary inner product spaces. To notate that x is orthogonal to y, use x ⊥ y.

Orthonormality is a further constraint: the orthogonal vectors are also unit vectors.

For a two vectors (which are not the zero vector) to be orthogonal, their inner product should be equal to 0. For a set of vectors to be orthogonal, every possible pair from the set should be orthogonal.

Matrices

A matrix is effectively a set of vectors. If the columns are orthonormal, then it can be called a matrix with orthonormal columns. These are usually denoted as Q.

Matrices with orthonormal columns 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 and only if such a matrix is square, it can be called an orthogonal matrix. These have several further 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).

Subspaces

For two subspaces to be orthogonal, every vector in the span of one should be orthogonal to every vector in the span of the other.

For example, consider a plane in R³. A plane is a subspace spanned by 2 vectors. The subspace that is orthogonal to a plane must be spanned by 1 vector, i.e. it is a line. A plane and a line can be checked for orthogonality by comparing each of the vectors spanning the plane for orthogonality with the single vector spanning the line.

For another example, null spaces are orthogonal by definiton. For any subspace A, the vectors spanning N(A) are precisely those that are not in R(A), therefore they are orthogonal. Similarly, the vectors spanning N(A^T) are precisely those that are not in C(A), therefore they are orthogonal.

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