Orthogonal Matrices
Definition of Orthogonal Matrices For some special matrices, its transpose equals its inverse. When an n x n matrix has all real entries and its transpose equals its inverse, the matrix is called an orthogonal matrix.
A real n x n matrix U is called an orthogonal matrix if the following is true:
UUT = UTU = I
Given the following matrix U:
To show it is an orthogonal matrix, verify it using one of the equations from the definition.
Since UUT = I, this matrix is orthogonal.
Given the following matrix U:
To show it is an orthogonal matrix, verify it using one of the equations from the definition.
Since UTU = I, this matrix is orthogonal.
Orthogonal Matrices and Summation Notation When the matrix U is orthogonal, the following is true:
The product of the ith row of U with the kth row of its transpose equals to 1 if i = k, or it equals 0 if i ≠ k. The same is true for the columns, because UTU = I.
The product of the ith column of U with the kth column of its transpose equals to 1 if i = k, or it equals 0 if i ≠ k.
If u₁, ..., un are the columns of an orthogonal matrix U, then the following is true:
The columns and rows form an orthonormal set of vectors. Therefore, a matrix is orthogonal if its rows or columns form an orthonormal set of vectors.
Note that the convention is to call such a matrix orthogonal instead of orthonormal.
Proposition of Orthonormal Basis The rows of an n x n orthogonal matrix form an orthonormal basis of ℝⁿ.
Any orthonormal basis of ℝⁿ can be used to construct an n x n orthogonal matrix.
Proposition of Determinant of Orthogonal Matrices Suppose U is an orthogonal matrix. Then its determinant is the following:
det(U) = ±1
The following is the proof:
Proper and Improper Orthogonal Matrices Orthogonal matrices are divided into two classes, proper and improper.
The proper orthogonal matrices are the orthogonal matrices with its determinant equal to 1.
The improper orthogonal matrices are the orthogonal matrices with its determinant equal to -1.
The reason for the classes is that the improper orthogonal matrices are sometimes considered to have no physical significance. These matrices cause a change in orientation, corresponding to material passing through itself in a non physical manner.
Geometrically, the linear transformations determined by the proper orthogonal matrices correspond to the composition of rotations.