Orthogonality A matrix A is orthogonal if AAt = I. If A is the matrix of a coordinate transformation X = AY from variables yi to variables xi, then if A is orthogonal, X tX = Y tY, or:

The equation:

Ax = λx

where A is a square matrix, x a column vector, and λ is a number (in general complex), has at most n solutions (x, λ). The values of λ are eigenvalues and those of x are eigenvectors of the matrix A. The relation may be written:

(A - lI )x = 0

so that if x π 0, the equation A - ll = 0 gives the eigenvalues. If A is symmetric and real, the eigenvalues are real. If A is symmetric, the eigenvectors are orthogonal. If A is not symmetric, the eigenvalues are complex and the eigenvectors are not orthogonal.