Unitary Matrix — Definition, Formula & Examples

A unitary matrix is a complex square matrix that, when multiplied by its conjugate transpose, produces the identity matrix. It is the complex-valued generalization of an orthogonal matrix.

A square matrix $U \in \mathbb{C}^{n \times n}$ is unitary if $U^*U = UU^* = I_n$ , where $U^* = \overline{U}^T$ denotes the conjugate transpose of $U$ and $I_n$ is the $n \times n$ identity matrix. Equivalently, $U^{-1} = U^*$ .

Key Formula

U^* U = U U^* = I

Where:

$U$ = A complex square matrix
$U^*$ = The conjugate transpose of U (also written U^\dagger)
$I$ = The identity matrix of the same size as U

How It Works

To check whether a complex matrix is unitary, compute its conjugate transpose

U^*

by taking the transpose and then replacing every entry with its complex conjugate. Then multiply

U^*U

(or

UU^*

). If the result is the identity matrix, the matrix is unitary. Key properties follow: every eigenvalue of a unitary matrix has absolute value 1, the columns (and rows) form an orthonormal set under the standard complex inner product, and

|\det(U)| = 1

Worked Example

Problem: Verify that the matrix U is unitary, where U = (1/√2) [[1, 1], [i, -i]].

Step 1: Write U explicitly: Express the matrix with its entries.

U = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ i & -i \end{bmatrix}

Step 2: Compute the conjugate transpose U*: Transpose U and take the complex conjugate of each entry. The conjugate of i is -i and vice versa.

U^* = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -i \\ 1 & i \end{bmatrix}

Step 3: Multiply U* U: Carry out the matrix multiplication and simplify.

U^*U = \frac{1}{2}\begin{bmatrix} 1\cdot1 + (-i)(i) & 1\cdot1 + (-i)(-i) \\ 1\cdot1 + i\cdot i & 1\cdot1 + i(-i) \end{bmatrix} = \frac{1}{2}\begin{bmatrix} 1+1 & 1-1 \\ 1-1 & 1+1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Answer: Since U*U = I, the matrix U is unitary.

Why It Matters

Unitary matrices preserve the length of complex vectors, making them essential in quantum mechanics where state transformations must conserve probability. They also appear in numerical linear algebra (QR factorization with complex matrices) and signal processing (the discrete Fourier transform matrix is unitary after proper scaling).

Common Mistakes

Mistake: Using the ordinary transpose instead of the conjugate transpose when checking unitarity.

Correction: For complex matrices, you must conjugate every entry after transposing. Forgetting the conjugation gives incorrect results whenever the matrix has non-real entries.