Define the orthogonal matrix.

An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors.

An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors. Orthonormal vectors are vectors that are both orthogonal (perpendicular) and normalized (have a length of 1). In other words, the dot product of any two columns or rows of an orthogonal matrix is 0 if they are different and 1 if they are the same.

The inverse of an orthogonal matrix is its transpose, which means that if A is an orthogonal matrix, then A^T is also an orthogonal matrix and A^(-1) = A^T. This property makes orthogonal matrices useful in many applications, such as in linear transformations, where they preserve distances and angles between vectors.

Another important property of orthogonal matrices is that they preserve the determinant of a matrix. If A is an orthogonal matrix, then |A| = ±1. This is because the determinant of a matrix is equal to the product of its eigenvalues, and the eigenvalues of an orthogonal matrix are either 1 or -1.

Orthogonal matrices are also useful in solving systems of linear equations, as they can be used to transform a system into an equivalent system with a simpler form. For example, if Ax = b is a system of linear equations, where A is an orthogonal matrix, then we can multiply both sides of the equation by A^T to obtain A^T Ax = A^T b. Since A^T A = I (the identity matrix), we have x = A^T b, which is a simpler form of the original equation.

Overall, orthogonal matrices have many useful properties and applications in mathematics and other fields, making them an important concept to understand in linear algebra.