Unitary Matrix - Properties

Properties

For any unitary matrix U, the following hold:

  • Given two complex vectors x and y, multiplication by U preserves their inner product; that is,
.
  • U is normal
  • U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus U has a decomposition of the form
where V is unitary and D is diagonal and unitary.
  • .
  • Its eigenspaces are orthogonal.
  • For any positive integer n, the set of all n by n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
  • Any square matrix with unit Euclidean norm is the average of two unitary matrices.

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