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Orthogonal Matrix


A n×n matrix A is an orthogonal matrix if

 AA^(T)=I,
(1)

where A^(T) is the transpose of A and I is the identity matrix. In particular, an orthogonal matrix is always invertible, and

 A^(-1)=A^(T).
(2)

In component form,

 (a^(-1))_(ij)=a_(ji).
(3)

This relation make orthogonal matrices particularly easy to compute with, since the transpose operation is much simpler than computing an inverse.

For example,

A=1/(sqrt(2))[1  1; 1 -1]
(4)
B=1/3[2 -2  1; 1  2  2; 2  1 -2]
(5)

are orthogonal matrices.

A matrix m can be tested to see if it is orthogonal in the Wolfram Language using OrthogonalMatrixQ[m].

The rows of an orthogonal matrix are an orthonormal basis. That is, each row has length one, and are mutually perpendicular. Similarly, the columns are also an orthonormal basis. In fact, given any orthonormal basis, the matrix whose rows are that basis is an orthogonal matrix. It is automatically the case that the columns are another orthonormal basis.

The orthogonal matrices are precisely those matrices which preserve the inner product

 <v,w>=<Av,Aw>.
(6)

Also, the determinant of A is either 1 or -1. As a subset of R^(n^2), the orthogonal matrices are not connected since the determinant is a continuous function. Instead, there are two components corresponding to whether the determinant is 1 or -1. The orthogonal matrices with detA=1 are rotations, and such a matrix is called a special orthogonal matrix.

The matrix product of two orthogonal matrices is another orthogonal matrix. In addition, the inverse of an orthogonal matrix is an orthogonal matrix, as is the identity matrix. Hence the set of orthogonal matrices form a group, called the orthogonal group O(n).


See also

Antisymmetric Matrix, Euler's Rotation Theorem, Improper Rotation, Inner Product, Orthogonal Group, Orthogonal Transformation, Orthogonality Condition, Orthonormal Basis, Rotation, Rotation Matrix, Special Orthogonal Matrix, Spin Group, Symmetric Matrix, Unitary Matrix

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Orthogonal Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthogonalMatrix.html

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