Orthogonal Matrix

What is an Orthogonal Matrix?

An orthogonal matrix is a square matrix in which all of the vectors that make up the matrix are orthonormal to each other.  This must hold in terms of all rows and all columns. In terms of geometry, orthogonal means that two vectors are perpendicular to each other.

In terms of linear algebra, we say that two vectors are orthogonal if the dot product of the two vectors is equal to zero, and additionally if the magnitude (length) of each vector is equal to one, then they are said to be orthonormal. Therefore, if you select any two columns of an orthogonal matrix you will find that they are orthonormal and perpendicular to each other. Additionally you may select any two rows and find that the same property holds, as the transpose of an orthogonal matrix is itself an orthogonal matrix.

Properties of an Orthogonal Matrix

As mentioned above, the transpose of an orthogonal matrix is also orthogonal.  In fact its transpose is equal to its multiplicative inverse and therefore all orthogonal matrices are invertible.  Because the transpose preserves the determinant, it is easy to show that the determinant of an orthogonal matrix must be equal to 1 or -1.  In terms of linear transforms, an orthogonal matrix of rank n preserves an (real) inner product on n-dimentional space.  It can also be shown that any such transformation must be representable by an orthogonal matrix.  It immediately follows that operating on a vector space with an orthogonal matrix must preserve distances and angles.