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A matrix U is unitary if its inverse equals its Hermitian transpose:

Unitary matrices are normal: Furthermore, given two vectors, multiplication by U preserves inner products . Unitary matrices are important in quantum mechanics because they preserve norms and thus probability amplitudes. See the Appendix in this chapter.

A real square matrix Q is orthogonal if:

(5.17)

Orthogonal matrices are important in numerical linear algebra applications because of their numeric stability properties. Some application areas are matrix decomposition methods (Golub and van Loan (1996)) such as:

 QR decomposition orthogonal, upper triangular.

 Singular Value Decomposition (SVD) and V orthogonal, diagonal matrix.

 Eigendecomposition symmetric, Q orthogonal, diagonal.

 Polar decomposition where is orthogonal, symmetric positive-semidefinite.

A particular application area is solving overdetermined systems of linear equations and solving ill-posed linear systems.