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In linear algebra, an orthogonal matrix is a matrix having orthonormal rows or columns.

If the orthogonal matrix has more rows than columns, then the column vectors which make up the columns of the matrix are orthonormal. If the matrix has more columns than rows, then its rows are orthonormal. If the matrix has the same number of rows and columns (i.e. it's square), then both its rows and columns are orthonormal.

The columns or rows of a n×n (square) orthogonal matrix form a orthonormal basis for Rn. Multiplying a vector by a square orthogonal matrix serves as a change in the coordinate system used to represent the vector.

A m×n orthogonal matrix U has the following properties