5.8 Unitary transformations

What are the characteristics of linear transformations that preserve length? If L: V → V and it is always true that ‖L(v)‖ = ‖v‖, what can we say about the matrix of L?

A complex square matrix U is unitary if its adjoint U^† is its inverse U^–1. Hence, U U^† = U^† U = I. The columns of U are orthonormal, as are the rows, which follows from the definition of the complex inner product. matrix$unitary unitary$matrix

For a unitary matrix U, |det(U)| = 1 because

To prove that unitary matrices preserve length, we must do more transposition and conjugation math:

Since the lengths are nonnegative, ‖U v‖ = ‖v‖. Conversely, if this holds, we must have U^† U = I.

Rotations and reflections are unitary. An orthogonal matrix is unitary.

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