A unitary matrix is a complex square matrix whose conjugate transpose equals its inverse. Unitary matrices preserve inner products and have eigenvalues of unit magnitude.
⚠For real matrices, unitary matrices reduce to orthogonal matrices. The identity matrix is always unitary.
What is a Unitary Matrix?
A unitary matrix is a complex square matrix U satisfying U*U = UU* = I, where U* is the conjugate transpose of U. The columns (and rows) of a unitary matrix form an orthonormal basis for Cn.
Orthonormal Columns
Columns form orthonormal set: inner products are 0 or 1.
Preserves Norm
||Ux|| = ||x|| for all vectors x.
Eigenvalues
All eigenvalues have absolute value 1.
Determinant
|det(U)| = 1. Real orthogonal case: det = ±1.
Example: Identity Matrix I = [[1,0],[0,1]]
1. I* = [[1,0],[0,1]] = I
2. I*I = [[1,0],[0,1]] = I
3. II* = [[1,0],[0,1]] = I
4. I is unitary ✓
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