What Is an Orthogonal Matrix?

An orthogonal matrix is a square matrix that has a transpose identical to its inverse. The transpose of a matrix is found by swapping each element a(ij) of the matrix with the element a(ji), where i and j denote the row and column of the element, respectively.

All orthogonal matrices have inverses. The preferred method to find the inverse of an orthogonal matrix is to compute its transpose, as computing a transpose simplifies an equation that would otherwise compute the general inverse of a matrix. The set of n x n orthogonal matrices forms an orthogonal group, which is usually denoted O(n).