Orthogonal Matrix

An orthogonal matrix Q is a square matrix whose transpose is equal to its inverse: QT=Q1

  • Properties:
    • For real matrices, being orthogonal is equivalent to preserving lengths and angles, similar to a rotation or reflection.
    • For complex matrices, it satisfies QQ=QQ=I, where Q is the conjugate transpose of Q and I is the identity matrix.
    • Orthogonal matrices have orthonormal columns and rows.

Example of an Orthogonal Matrix:

Q=12[1111]

In this case, QT=Q1, which makes it an orthogonal matrix.

Properties of Orthogonal Matrices:

  1. Transpose and Inverse: The transpose of an orthogonal matrix is equal to its inverse: QT=Q1.

  2. Orthonormal Columns and Rows: The columns and rows of an orthogonal matrix form an orthonormal set of vectors, meaning they are pairwise orthogonal and have unit length.

  3. Preservation of Length and Angles: Orthogonal matrices preserve lengths of vectors and angles between vectors, maintaining orthogonality.