An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors (i.e., unit vectors that are mutually perpendicular). This leads to several important properties and applications.
Definition:
A square matrix Q is orthogonal if its transpose is also its inverse, meaning:
where Q<sup>T</sup> is the transpose of Q, and I is the identity matrix.
Key Properties:
Orthonormal Columns and Rows: The columns and rows of an orthogonal matrix form an orthonormal basis. This means each column (or row) is a unit vector (length 1), and any two distinct columns (or rows) are orthogonal (their dot product is zero).
Preservation of Length and Angle: Orthogonal matrices preserve the length of vectors and the angles between them when used in transformations. This makes them essential in areas like rotations and reflections.
Determinant: The determinant of an orthogonal matrix is either +1 or -1. If the determinant is +1, the orthogonal matrix represents a rotation (proper rotation). If the determinant is -1, it represents a reflection or roto-inversion (improper rotation).
Inverse: As defined, the inverse of an orthogonal matrix is simply its transpose: Q<sup>-1</sup> = Q<sup>T</sup>.
Product of Orthogonal Matrices: The product of two orthogonal matrices is also an orthogonal matrix.
Examples:
Applications:
Linear Transformations: Used for rotations, reflections, and other transformations that preserve geometric properties.
Numerical Stability: Utilized in numerical algorithms (e.g., QR decomposition) to enhance stability and reduce errors.
Signal Processing: Applied in areas like image and audio processing for various transformations and representations.
Coordinate System Rotations: Facilitating changes of basis in vector spaces.
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