Orthogonal Matrix

By Kardi Teknomo, PhD.

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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 non-orthogonal 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 (Schur’s 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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Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\