This definition can be given for matrices with entries from any field, but the most common case is the one of matrices with real entries, and only that case will be considered in the rest of this article.
A real square matrix is orthogonal if and only if its columns form an orthonormal basis of R^{n} with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of R^{n}.
Geometrically, orthogonal matrices describe linear transformations of R^{n} which preserve angles and lengths, such as rotations and reflections. They are compatible with the Euclidean inner product in the following sense: if G is orthogonal and x and y are vectors in R^{n}, then
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. This shows that the set of all nbyn orthogonal matrices forms a group. It is a Lie group of dimension n(n+1)/2 and is called the orthogonal group, denoted by O(n).
The determinant of any orthogonal matrix is 1 or 1. That can be shown as follows:
All eigenvalues of an orthogonal matrix, even the complex ones, have absolute value 1. Eigenvectors for different eigenvalues are orthogonal.
If Q is orthogonal, then one can always find an orthogonal matrix P such that
/R_{1} \  R_{2}   ...  P^{1}QP =  R_{k}   ±1   ...  \ ±1 /
where the matrices R_{1},...,R_{k} are 2by2 rotation matrices. Intuitively, this result means that every orthogonal matrix describes a combination of rotations and reflections. The matrices R_{1},...,R_{k} correspond to the nonreal eigenvalues of Q.
If A is an arbitrary mbyn matrix of rank n, we can always write
/ R \ A = Q \ 0 /
where Q is an orthogonal matrix mbym matrix and R is an upper triangular nbyn matrix with positive main diagonal entries. This is known as a QR decomposition of A and can be proven by applying the GramSchmidt Process to the columns of A. It is useful for numerically solving systems of linear equations and least squares problems.
The complex analog to orthogonal matrices are the unitary matrices.
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