Orthogonal Matrices
An orthogonal matrix is a square matrix with real entries whose rows and columns are orthonormal unit vectors. Orthonormal vectors are vectors with a length of 1 and are orthogonal to each other, meaning their dot product is zero.
Table of Contents
Preservation of Length
If you have a vector x and multiply it by an orthogonal matrix Q, the length of the resulting vector Qx will be the same as the length of the original vector x. Mathematically, |Qx| = |x|.
Preservation of Angles
Orthogonal matrices also preserve angles between vectors. If you have two vectors u and v, the angle between them is the same as the angle between the vectors Qu and Qv after multiplication by an orthogonal matrix.
Rotation and Reflection
Orthogonal matrices are often used to represent rotations and reflections in linear transformations. Since the columns of an orthogonal matrix are orthonormal vectors, they can be thought of as forming a set of basis vectors that preserve lengths and angles during transformations.
Solving Linear Systems
Orthogonal matrices are useful in solving linear systems of equations.
Numerical Stability
Orthogonal matrices play a role in numerical algorithms, particularly in numerical linear algebra, where they can help in maintaining numerical stability during computations.
Determinant
The determinant of an orthogonal matrix is either +1 or -1. This is because the product of the eigenvalues of an orthogonal matrix is either 1 or -1, and the determinant is the product of these eigenvalues.
The eigenvalues of an orthogonal matrix have a magnitude of 1.
Transpose
The inverse of an orthogonal matrix is equal to its transpose.
Orthogonal Diagonalization
An orthogonal matrix can be diagonalized by another orthogonal matrix.
Orthogonal Complement
The columns of an orthogonal matrix form an orthonormal basis for the column space of the matrix. The orthogonal complement of the column space is the null space, and the rows of the matrix form an orthonormal basis for the row space.