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 non singular matrix, and is invertible.
Let us learn more about the properties, examples of 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. A unitary matrix is a matrix, whose inverse is equal to its conjugate transpose. Alternatively, the product of the inverse of a unitary matrix, and the conjugate transpose of a unitary matrix is equal to the identity 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 nonsingular matrix.
 The unitary matrix is an invertible matrix
 The product of two unitary matrices is a unitary matrix.
 The sum or difference of two unitary matrices is also 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 nonsquare matrices but have orthonormal columns.
Terms Related to Unitary Matrix
The following terms related to matrices are helpful for a better understanding of this concept of unitary matrix.
 NonSingular 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 isA =ad  bc ≠ 0.
 Invertible Matrix: The matrix whoos 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 its conjugate complex numbers.
 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 nondiagonal 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 nonunitary 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
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 \(C_2\) = \(\begin{bmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1i}{\sqrt 2} &\frac{1i}{\sqrt 2}\end{bmatrix}\).
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.
Hence the two columns of the unitary matrix are orthonormal.
FAQs on Unitary Matrix
What Is 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. Alternatively, the product of the inverse of a unitary matrix, and the conjugate transpose of a unitary matrix is equal to the identity matrix. U^{H} = U^{1}.
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^{1 }OR 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 nonsingular matrix.
 The unitary matrix is an invertible matrix
 The product of two unitary matrices is a unitary matrix.
 The sum or difference of two unitary matrices is also 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 refer to a matrix, if the product of the matrix and its transpose conjugate matrix results in an identity matrix.
What Is the Order of a Unitary Matrix?
The unitary matrix is a square matrix and has an order of n x n. Further, 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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