Today I Learned

Some of the things I've learned every day since Oct 10, 2016

56: Unitary Linear Operators

In linear algebra, a unitary operator T over a inner product space V is one which satisfies

TT^* = T^*T = I.

Thus it is a special kind of normal operator. The following conditions are equivalent to T being unitary:

  • T preserves the inner product. That is, \langle T(x), T(y) \rangle = \langle x, y \rangle.
  • T is distance-preserving. That is, ||T(x)|| = ||x||.
  • T^* is unitary.
  • T is invertible and T^{-1} = T^*.
  • T is a normal operator with eigenvalues on the complex unit circle.

and the following are additionally true of a unitary operator:

  • T is normal.
  • The eigenspaces of T are orthogonal.
  • Every eigenvalue of T has an absolute value of 1.
  • U = P^*DP for some unitary transformations D, P, where D is diagonal.

(When T is over \mathbb{R} it is sometimes referred to as ‘orthogonal’ rather than ‘unitary’.)


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