Unitary Matrices Calculator

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Example

Created on 2024-06-20Asked by Benjamin Lee (Solvelet student)

Determine if the matrix

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

is unitary.

Solution

To determine if the matrix

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

is unitary: 1. **Compute the conjugate transpose

U^\dagger

:**

U^\dagger = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix}.

2. **Compute

U U^\dagger

:**

U U^\dagger = \left(\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}\right) \left(\frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix}\right) = \frac{1}{2} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix} \begin{pmatrix} 1 & -i \\ -i & 1 \end{pmatrix}.

3. **Perform the matrix multiplication:**

\frac{1}{2} \begin{pmatrix} 1 + i^2 & 1 \cdot (-i) + i \cdot 1 \\ i \cdot 1 + 1 \cdot (-i) & i^2 + 1 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 1 - 1 & 0 \\ 0 & -1 + 1 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.

4. **Result:** The matrix

U

is unitary since

U U^\dagger = I

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by William Nelson on Solvelet

1. Determine if the matrix

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

is unitary.2. Find the eigenvalues and eigenvectors of the unitary matrix

V = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}

DefinitionA unitary matrix U = (Uij) is a complex square matrix that fulfills U†U = I, where U† is the conjugate transpose of U and I is the identity matrix. A unitary matrix is one that preserves the inner product in a complex vector space. An example of a Unitary matrix is (1/21/2−1/21/2)

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