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An orthogonal matrix is a square matrix whose rows and columns are orthogonal unit vectors. This means that the dot product of any two rows or columns is equal to 0, and the magnitude of each row or column is equal to 1.
An orthogonal matrix differs from a regular matrix in that its rows and columns are not only perpendicular to each other, but also have a magnitude of 1. This property makes orthogonal matrices useful for transformations, as they preserve the length and angle between vectors.
Some examples of orthogonal matrices include rotation matrices, reflection matrices, and permutation matrices. A rotation matrix, for example, is an orthogonal matrix that represents a rotation in 2D or 3D space.
The inverse of an orthogonal matrix is equal to its transpose. This means that if we multiply an orthogonal matrix by its transpose, we will get the identity matrix. In other words, an orthogonal matrix is its own inverse.
Orthogonal matrices are commonly used in computer graphics, robotics, and physics to represent and manipulate rotations, transformations, and reflections. They are also used in statistics and data analysis, as they can be used to perform orthogonal transformations on data sets.