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Unitary Matrix

Unitary Matrix is a square matrix of complex numbers. The product of the conjugate transpose of a unitary matrix, with the unitary matrix, gives an identity matrix. From this, we can also understand that a unitary matrix is a nonsingular matrix, and is invertible.

Let us learn more about the properties, and examples of the unitary matrix.

1.	What is a Unitary Matrix?
2.	Properties of Unitary Matrix
3.	Terms Related to Unitary Matrix
4.	Examples on Unitary Matrix
5.	Practice Questions on Unitary Matrix
6.	FAQs on Unitary Matrix

What is Unitary Matrix?

A unitary matrix is a square matrix of complex numbers, whose inverse is equal to its conjugate transpose. Alternatively, the product of the unitary matrix and the conjugate transpose of a unitary matrix is equal to the identity matrix. i.e., if U is a unitary matrix and U^H is its complex transpose (which is sometimes denoted as U^*) then one /both of the following conditions is satisfied.

U^H = U^-1
U^H U = U U^H = I

where I is the identity matrix whose order is the same as U.

Unitary matrix

Also, a unitary matrix is a nonsingular matrix. Or the determinant of a unitary matrix is not equal to zero. The columns and rows of a unitary matrix are orthonormal.

Properties of Unitary Matrix

The properties of a unitary matrix are as follows.

The unitary matrix is a non-singular matrix.
The unitary matrix is an invertible matrix
The product of two unitary matrices is a unitary matrix.
The inverse of a unitary matrix is another unitary matrix.
A matrix is unitary, if and only if its transpose is unitary.
A matrix is unitary if its rows are orthonormal, and the columns are orthonormal.
The unitary matrices can also be non-square matrices but have orthonormal columns and rows.

Note: The sum or difference of two unitary matrices doesn NOT need to be a unitary matrix. For example, if A is a unitary matrix, then A - A = O (null matrix), which is NOT unitary.

The following terms related to matrices are helpful for a better understanding of this concept of unitary matrix.

Non-Singular Matrix:The determinant of a non singular matrix is a a non zero value. For a square matrix A = \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), the condition of it being a non singular matrix is|A| =ad - bc ≠ 0.
Invertible Matrix: The matrix whose inverse matrix can be computed, is called an invertible matrix. The inverse of a matrix A is A^-1 = Adj A/|A|.
Conjugate Matrix: The conjugate matrix of a given matrix is obtained by replacing the corresponding elements of the given matrix, with their complex conjugates.
Transpose Matrix: The transpose of a matrix A is represented as A^T, and the transpose of a matrix is obtained by changing the rows into columns and columns into rows for a given matrix.
Orthogonal Matrix: If the product of a matrix and its transpose is an identity matrix, then it is called an orthogonal matrix. A.A^T = I.
Hermitian Matrix: A hermitian matrix is a square matrix, which is equal to its conjugate transpose matrix. The non-diagonal elements of a hermitian matrix are all complex numbers. \(A = \bar A^T\).

☛ Related Topics:

The following topics are helpful for a better understanding of the non-unitary matrix.

Examples on Unitary Matrix

Example 1: Show that the matrix A = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} &\frac{-1i}{\sqrt 2}\end{bmatrix}\) is a unitary matrix.

Solution:

The given matrix is A = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} &\frac{-1i}{\sqrt 2}\end{bmatrix}\)

Conjugate of matrix A = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{-1i}{\sqrt 2} &\frac{1i}{\sqrt 2}\end{bmatrix}\)

Conjugate transpose of matrix A = A*= \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{-1i}{\sqrt 2}\\ \frac{1}{\sqrt 2} &\frac{1i}{\sqrt 2}\end{bmatrix}\)

A*.A = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{-1i}{\sqrt 2}\\ \frac{1}{\sqrt 2} &\frac{1i}{\sqrt 2}\end{bmatrix}\).\(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} &\frac{-1i}{\sqrt 2}\end{bmatrix}\)

A*.A = \(\begin{bmatrix}1 & 0\\ 0 &1\end{bmatrix}\) = I

Answer: Therefore, the matrix A is a unitary matrix.
Example 2: Prove that the columns of the matrix \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} &\frac{-1i}{\sqrt 2}\end{bmatrix}\) are orthonormal.

Solution:

The given matrix is A = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} &\frac{-1i}{\sqrt 2}\end{bmatrix}\)

Let us segregate each of the columns of this matrix.

\(C_1\) = \(\begin{bmatrix}\frac{1}{\sqrt 2} \\ \frac{1i}{\sqrt 2} \end{bmatrix}\), and \(C_2\) = \(\begin{bmatrix}\frac{1}{\sqrt 2} \\ \frac{-1i}{\sqrt 2} \end{bmatrix}\).

The product of two orthonormal matrices is equal to 1. \(C_1.C_2\) = 1.

\(C_1.C_2\) = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1i}{\sqrt 2}\end{bmatrix}\).\(\begin{bmatrix}\frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} \end{bmatrix}\)

= \(\frac{1}{\sqrt2}.\frac{1}{\sqrt2} - \frac{1i}{\sqrt2}\frac{1i}{\sqrt2}\)

= \(\frac{1}{2}- \frac{1}{2}i^2\) = \(\frac{1}{2}+ \frac{1}{2}\) = 1.

Also, the magnitude of each column is equal to 1.

Answer: Hence the two columns of the unitary matrix are orthonormal.
Example 3: Is A = \(\left[\begin{array}{ll}
1 & 0 \\
0 & i
\end{array}\right]\) a unitary matrix? Use the condition A^H = A^-1 to verify it.

Solution:

The given matrix is, A = \(\left[\begin{array}{ll}
1 & 0 \\
0 & i
\end{array}\right]\).

Its conjugate matrix = \(\left[\begin{array}{ll}
1 & 0 \\
0 & -i
\end{array}\right]\).

Its transpose is A^H = \(\left[\begin{array}{ll}
1 & 0 \\
0 & -i
\end{array}\right]\) ... (1)

The inverse of A is, A^-1 = 1 / (i - 0) = \(\left[\begin{array}{ll}
i & 0 \\
0 & 1
\end{array}\right]\)

= \(\left[\begin{array}{ll}
1 & 0 \\
0 & -i
\end{array}\right]\) ... (2)

From (1) and (2), A^H = A^-1.

Answer: Thus, A is a unitary matrix.

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Practice Questions on Unitary Matrix

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FAQs on Unitary Matrix

What is the Definition of a Unitary Matrix?

A unitary matrix is a square matrix of complex numbers. A unitary matrix is a matrix, whose inverse is equal to its conjugate transpose. Its product with its conjugate transpose is equal to the identity matrix. i.e., a square matrix is unitary if either U^H = U^-1 (or) U^H U = U U^H = I, where U^H is the conjugate transpose of U.

How to Find the Complex Transpose Matrix?

The complex conjugate of a matrix can be found in two steps:

First, replace all elements with their complex conjugates.
Then take the transpose of the resultant matrix.

How Do You Know If a Matrix is Unitary Matrix?

The given matrix can be identified as a unitary matrix if the product of its conjugate transpose, with the given matrix gives the identity matrix. Also a unitary matrix follows the formula U^H = U^-1OR U^H.U = I.

What Are the Properties of Unitary Matrix?

The properties of a unitary matrix are as follows.

The unitary matrix is a non-singular matrix.
The unitary matrix is an invertible matrix
The product of two unitary matrices is a unitary matrix.

is a Unitary Matrix Also a Hermitian Matrix?

The unitary matrix is not a hermitian matrix but is made up of a hermitian matrix. By definition, a hermitian matrix is a matrix that is equal to its conjugate transpose and a unitray matrix refers to a matrix if the product of the matrix and its transpose conjugate matrix results in an identity matrix. If A is a hermitian matrix, then e^iA.

What is the Order of a Unitary Matrix?

The unitary matrix is a square matrix and has an order of n x n. i.e., a unitray matrix has an equal number of rows and columns.

A hermitian matrix is a square matrix, with equal number of rows and columns, and has an order n x n.

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