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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.