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Unitary Matrix - Definition, Properties, Examples

TL;DR

A unitary matrix is a square matrix of complex numbers whose conjugate transpose equals its inverse — that is, U^H U = U U^H = I. This article covers the definition, the conjugate-transpose test, every property (orthonormal columns, |det U| = 1, eigenvalues on the unit circle), how unitary generalises the orthogonal matrix to complex numbers, and worked examples.

What Is A Unitary Matrix?

A unitary matrix is a square matrix U with complex entries whose conjugate transpose is also its inverse:

U^H U = U U^H = I

Here U^H (read "U-Hermitian," also written U* or U^\dagger) is the conjugate transpose: transpose the matrix, then replace every entry by its complex conjugate. I is the identity matrix.

Variable glossary.

A complex conjugate of a + bi is a - bi — flip the sign of the imaginary part. The conjugate transpose U^H combines that conjugation with the ordinary transpose. I is a square grid with 1s on the diagonal and 0s elsewhere.

Equivalently, U is unitary exactly when U^{-1} = U^H. The defining feature is that its inverse costs nothing to compute: you conjugate and transpose, and you are done.

How Do You Check If A Matrix Is Unitary?

The test is direct: compute U^H U, multiply it by U, and confirm the result is the identity.
The steps, each on its own line:

1. Transpose U
2. Conjugate every entry of the transpose
3. Multiply UH and U
4. If the product is I, the matrix is unitary.

What Are The Properties Of A Unitary Matrix?

The properties all flow from the condition U^H U = I.

• Always square and invertible.
• Inverse is the conjugate transpose: U^{-1} = U^H.
• Absolute value of the determinant equals 1: |det U| = 1.
• Eigenvalues lie on the unit circle: every eigenvalue λ satisfies |λ| = 1.
• Product of two unitary matrices is unitary.
• Preserves length and inner products: |Ux| = |x| for every vector x.

One caution: the sum of two unitary matrices is generally not unitary.

Is A Unitary Matrix The Same As An Orthogonal Matrix?

A unitary matrix is the complex-number version of an orthogonal matrix.

• If a unitary matrix has only real entries, it is orthogonal.

Examples Of A Unitary Matrix

Example 1

Show that U = [ 0 & i \newline i & 0 ] is unitary.

Example 2

Is U = \dfrac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \newline i & -i \end{bmatrix} unitary?

Example 3

Confirm [ \begin{bmatrix} i & 0 \newline 0 & i \end{bmatrix} ] is unitary.

Example 4

Verify a real unitary matrix is just orthogonal, using U = \begin{bmatrix} 0 & 1 \newline -1 & 0 \end{bmatrix}.

Example 5

Show the product of two unitary matrices is unitary for unitary U and V.

Example 6

Check that U = \dfrac{1}{2} \begin{bmatrix} 1+i & 1-i \newline 1-i & 1+i \end{bmatrix} is unitary.

Why Unitary Matrices Matter: "Transformations that never lose information"

A unitary matrix preserves length and angle in a complex vector space. Where that property carries real weight:

• Quantum mechanics.
• Signal processing.
• Numerical stability.

Key Takeaways

• A unitary matrix is a complex square matrix with U^H U = I.
• To test for unitarity, multiply by the original and check for the identity.
• A unitary matrix has |det U| = 1 and preserves vector length.