A matrix is orthogonal if the transpose is equal to its inverse , that is .
Since computing matrix inverse is rather difficult while computing matrix transpose is straightforward, orthogonal matrix make difficult operation easier. Orthogonal matrix is important in many applications because of its properties.
Example:
Is matrix an orthogonal matrix?
Answer:
To test whether a matrix is an orthogonal matrix, we multiply the matrix to its transpose. If the result is an identity matrix, then the input matrix is an orthogonal matrix.
Thus, matrix is an orthogonal matrix.
To create random orthogonal matrix as in the interactive program below, I created random symmetric matrix and compute the modal matrix from concatenation of the Eigen vectors .
The interactive program below is designed to answers the question whether the given input matrix is an orthogonal matrix. When you click Random Example button, it will create random input matrix to provide you with many examples of both orthogonal and nonorthogonal matrices. You can also try to input your own matrix to test whether it is an orthogonal matrix or not.
Properties
Some important properties of orthogonal matrix are
 Orthogonal matrix is always a square matrix

If
is an orthogonal matrix order
with real entries then
 The inner product of two row vectors or two column vectors of matrix is zero
 The row vector and the column vector of matrix is orthonormal (the Euclidean vector norm is one)
 The transpose matrix is also an orthogonal matrix
 The inverse matrix is also an orthogonal matrix
 The product to its transpose is identity matrix
 The absolute magnitude the determinant is one, that is
 The eigenvectors of are real and orthogonal
 The eigenvalues of are equal to +1 or 1 (note the theoretical of eigenvalue and eigenvectors of orthogonal matrix is sometimes difficult to obtain numerically due to round off error).

If
and
are vector of
dimensions, then
 Inner product with orthogonal matrix is preserved
 Euclidean norm is preserved
 For any square matrix , there exist a unitary matrix such that matrix is upper triangular (Schurs theorem).
 Unitary matrix is generalization of orthogonal matrix with entries of complex numbers
 Both Hermitian and Unitary matrix (including symmetric and orthogonal matrix) are called normal matrix because the eigen vectors form orthonormal set.
See also : Singular Value Decomposition , orthogonal vector , spectral decomposition
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