Unitary Matrix

A square matrix U is unitary if its conjugate transpose is equal to its inverse: U=U1

  • Properties:
    • In a real matrix, being unitary is equivalent to being orthogonal.
    • In a complex matrix, it satisfies UU=UU=I, where I is the identity matrix.
    • Unitary matrices preserve the inner product of vectors, maintaining lengths and angles between vectors.

Example of a Unitary Matrix:

U=12[1111]

In this case, U=U1, which makes it a unitary matrix.

Properties of Unitary Matrices:

  1. Conjugate Transpose: The conjugate transpose of a unitary matrix is its inverse: U=U1.

  2. Orthonormal Columns: The columns of a unitary matrix form an orthonormal set of vectors, meaning they are pairwise orthogonal and have unit length.

  3. Preservation of Inner Product: U preserves the inner product of vectors v and w: Uv,Uw=v,w.