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Unitary Matrix

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Unitary Matrices are defined as square matrices of complex numbers such that the product of the conjugate transpose of a unitary matrix, with unitary matrix itself, gives an identity matrix. Matrices are defined as rectangular arrays in which numbers are arranged in rows and columns. Number of rows and columns determined the size of the matrix. We have different types of matrices, such as rectangular matrices, square matrices, triangular matrices, symmetric matrices, singular matrices, etc. In this article, we learn about a unitary matrix in detail.

What is Unitary Matrix?

A square matrix of complex numbers is said to be a unitary matrix if its inverse is equal to the conjugate transpose. In other words, the product of a unitary matrix and its conjugate transpose is equal to the identity matrix. As the determinant of a unitary matrix is not equal to zero, it is a non-singular matrix. The rows and columns of a unitary matrix are orthonormal. If “U” is a unitary matrix and “UH” is its conjugate transpose, then the following condition must be satisfied:

  • UUH = UHU = I
  • UH = U-1

Examples of a Unitary Matrix

  • Matrix given below is a unitary matrix of order “2 × 2.”

A_{2\times2} = \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1\\ i & -i \end{array}\right]

  • Matrix given below is a unitary matrix of order “3 × 3.”

B_{3\times3} = \left[\begin{array}{ccc} i & 0 & 0\\ 0 & i & 0\\ 0 & 0 & i \end{array}\right]

Properties of a Unitary Matrix

Following are some important properties of a unitary matrix:

  • Every unitary matrix is a square matrix.
  • A unitary matrix is a non-singular matrix.
  • Every unitary matrix is an invertible matrix.
  • Every unitary matrix is diagonalizable.
  • When two unitary matrices of the same order are multiplied, the resultant matrix is also unitary.
  • When two unitary matrices of the same order are added or subtracted, the resultant matrix is also unitary.
  • Absolute value of the determinant of a unitary matrix is one, i.e., |det(U)| = 1.
  • An identity matrix is also a unitary matrix.
  • In a unitary matrix, the modulus of every eigenvalue is always one, i.e., |λ| = 1.

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Solved Examples on Unitary Matrix

Example 1: Prove that the matrix given below is unitary.

U = \frac{1}{2}\left[\begin{array}{cc} 1+i & -1+i\\ 1+i & 1-i \end{array}\right]

Solution:

To prove that the given matrix is unitary, we need to prove that UUH = I

Conjugate matrix of U =\frac{1}{2} \left[\begin{array}{cc} 1-i & -1-i\\ 1-i & 1+i \end{array}\right]

U^{H} = \frac{1}{2}\left[\begin{array}{cc} 1-i & 1-i\\ -1-i & 1+i \end{array}\right]

UU^{H} = \frac{1}{2}\left[\begin{array}{cc} 1+i & -1+i\\ 1+i & 1-i \end{array}\right]\times\frac{1}{2}\left[\begin{array}{cc} 1-i & 1-i\\ -1-i & 1+i \end{array}\right]

UU^{H} = \frac{1}{4} \left[\begin{array}{cc} [(1+i)(1-i)+(-1+i)(-1-i)] & [(1+i)(1-i)+(-1+i)(1+i)]\\{} [(1+i)(1-i)+(1-i)(-1-i)] & [(1+i)(1-i)+(i-i)(1+i)] \end{array}\right]

UU^{H} = \frac{1}{4}\left[\begin{array}{cc} [1+1+1+1] & [1+1-1-1]\\{} [1+1-1-1] & [1+1+1+1] \end{array}\right]

UU^{H} = \frac{1}{4}\left[\begin{array}{cc} 4 & 0\\ 0 & 4 \end{array}\right]

UU^{H} = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right] = I

Hence the given matrix is unitary.

Example 2: Is the matrix given below a unitary matrix?

A = \left[\begin{array}{cc} i & 0\\ 0 & 1 \end{array}\right]

Solution:

To prove that the given matrix is unitary, we need to prove that AAH = I

The conjugate matrix of U = \left[\begin{array}{cc} -i & 0\\ 0 & 1 \end{array}\right]

A^{H} = \left[\begin{array}{cc} -i & 0\\ 0 & 1 \end{array}\right]

AA^{H} = \left[\begin{array}{cc} i & 0\\ 0 & 1 \end{array}\right]\times\left[\begin{array}{cc} -i & 0\\ 0 & 1 \end{array}\right]

AA^{H} = \left[\begin{array}{cc} [(i\times-i)+0] & [0+0]\\{} [0+0] & [0+1] \end{array}\right]

AA^{H} = \left[\begin{array}{cc} -i^{2} & 0\\ 0 & 1 \end{array}\right]

AA^{H} = \left[\begin{array}{cc} -(-1) & 0\\ 0 & 1 \end{array}\right] = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right] = I

Hence the given matrix is unitary.

Example 3: Prove that the absolute value of the determinant of a unitary matrix is one.

P =\frac{1}{3} \left[\begin{array}{cc} 2-i & 2\\ 2 & 2+i \end{array}\right]

Solution:

P = \left[\begin{array}{cc} \frac{2-i}{3} & \frac{-2}{3}\\ \frac{2}{3} & \frac{2+i}{3} \end{array}\right]

|P| = \left|\begin{array}{cc} \frac{2-i}{3} & \frac{-2}{3}\\ \frac{2}{3} & \frac{2+i}{3} \end{array}\right|

|P| = (\frac{2-i)}{3})(\frac{2+i}{3})-(\frac{-2}{3})(\frac{2}{3})

|P| = (\frac{4-i^{2}}{9})+(\frac{4}{9})

|P| = (\frac{4-(-1)}{9})+(\frac{4}{9})

|P| = \frac{5}{9}+\frac{4}{9}= \frac{(5+4)}{9}= \frac{9}{9}= 1

Hence proved.

FAQs on Unitary Matrix

Question 1: Define a matrix.

Answer:

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns.

Question 2: What is a unitary matrix?

Answer:

A square matrix of complex numbers is said to be a unitary matrix if its inverse is equal to the conjugate transpose. In other words, the product of a unitary matrix and its conjugate transpose is equal to the identity matrix.

Question 3: How to determine whether a matrix is unitary or not?

Answer:

A matrix can be determined to be unitary if the product of the given matrix and its conjugate transpose is equal to the identity matrix. If “U” is a unitary matrix and “UH” is its conjugate transpose, then the following condition must be satisfied:

  • UUH = UHU = I
  • UH = U-1

Question 4: How to find the complex conjugate transpose matrix?

Answer:

Follow the steps given below to find a complex conjugate transpose matrix:

  1. First, replace every element in the matrix with its complex conjugate.
  2. Now, find the transpose of the resultant matrix.

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Last Updated : 26 Mar, 2023
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