Some of the things I've learned every day since Oct 10, 2016
56: Unitary Linear Operators
December 5, 2016Posted by on
In linear algebra, a unitary operator over a inner product space is one which satisfies
Thus it is a special kind of normal operator. The following conditions are equivalent to being unitary:
- preserves the inner product. That is, .
- is distance-preserving. That is, .
- is unitary.
- is invertible and .
- is a normal operator with eigenvalues on the complex unit circle.
and the following are additionally true of a unitary operator:
- is normal.
- The eigenspaces of are orthogonal.
- Every eigenvalue of has an absolute value of .
- for some unitary transformations , where is diagonal.
(When is over it is sometimes referred to as ‘orthogonal’ rather than ‘unitary’.)