Unitary matrix

A matrix (plural: matrices) is a rectangular arrangement of symbols along rows and columns that might be real or complex numbers. Thus, a system of m x n symbols arranged in a rectangular formation along m rows and n columns is called an m by n matrix (which is written as m x n matrix). There are special types of matrices like Orthogonal matrices, Unitary matrices, and Idempotent matrices. In real life, we use unitary matrices in quantum mechanics.

This Story also Contains

1. Square matrix
2. Unitary matrix
3. Properties of Unitary Matrices
4. Solved Examples Based on Unitary Matrix

In this article, we will cover the concept of unitary matrices. This category falls under the broader category of Matrices, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of twelve questions have been asked on this topic in JEE MAINS(2013 - 2023) including one in 2021 and one in 2023.

Square matrix

The square matrix is the matrix in which the number of rows = number of columns. So a matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be a square matrix when m = n.

E.g.

$\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]_{3 \times 3}$ or, $\left[\begin{array}{cc}2 & -4 \\ 7 & 3\end{array}\right]_{2 \times 2}$

Unitary matrix

A unitary matrix is a square matrix of complex numbers, whose inverse is equal to its conjugate transpose.

The product of a unitary matrix and the conjugate transpose of a unitary matrix is equal to the identity matrix.

Let A be a square matrix, and if AA^H = I, where I is the identity matrix, then A is said to be a unitary matrix, and A^H is its conjugate transpose.

Properties of Unitary Matrices

If A is unitary matrices, I is identity matrices, A^-1 is the Inverse of matrix A, and A^H is the conjugate transpose of matrix A.

1) If A A^H = I, then A^-1 = A^H

2) If A and B are unitary, Then AB is also unitary.

3) If A is unitary, then A^-1 and A^H are also unitary.

4) A A^H = A^H A= I

5) A unitary matrix is a non-singular matrix.

6) The determinant of the unitary matrix is not equal to zero.

7) The inverse of a unitary matrix is another unitary matrix.

8) A matrix is unitary, if and only if its transpose is unitary

9) The unitary matrices can also be non-square matrices but have orthonormal columns and rows.

10) The sum or difference of two unitary matrices does 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.

Recommended Video Based on Unitary Matrix

Solved Examples Based on Unitary Matrix

Example 1: Which of the following properties is true for a unitary matrix \( U \)?

1) \( U^{-1} = U^T \)
2) The columns of \( U \) are orthonormal vectors.
3) The determinant of \( U \) is always zero.
4) \( U U^T = I \)

Solution:
1) False. For a unitary matrix \( U \), \( U^{-1} = U^* \), where \( U^* \) is the conjugate transpose of \( U \). \( U^T \) is used for orthogonal matrices.
2) True. The columns (and rows) of a unitary matrix are orthonormal vectors.
3) False. The determinant of a unitary matrix has an absolute value of 1, not zero.
4) False. For a unitary matrix, \( U U^* = I \), not \( U U^T \).

Hence, the answer is option 2.

Example 2: If the matrix $A=\left[\begin{array}{cc}a+b i & -c+i d \\ c+i d & a-i b\end{array}\right]$ is a Unitary matrix then
1) $a^2+b^2+c^2+d^2=1$
2) $a^2+b^2=c^2+d^2$
4) $a^2+b^2+c^2+d^2=0$
3) $a^2+c^2=b^2+d^2$

Solution

$
\begin{aligned}
& A=\left[\begin{array}{cc}
a+b i & -c+i d \\
c+i d & a-i b
\end{array}\right] \\
& A \cdot A^\theta=\left[\begin{array}{cc}
a+b i & -c+i d \\
c+i d & a-i b
\end{array}\right] \cdot\left[\begin{array}{cc}
a-b i & c-i d \\
-c-i d & a+i b
\end{array}\right]=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
& {\left[\begin{array}{cc}
a^2+b^2+c^2+d^2 & 0 \\
0 & a^2+b^2+c^2+d^2
\end{array}\right]=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]} \\
& a^2+b^2+c^2+d^2=1
\end{aligned}
$
Hence, the answer is option 1.

Example 3: If \( V \) is a 3x3 unitary matrix, what is the possible value of \( \det(V) \)?

A) 0
B) 1
C) -1
D) 1 or -1

Solution:
For a unitary matrix \( V \), the determinant must lie on the unit circle in the complex plane, which means it has an absolute value of 1. Therefore, the possible values are \( e^{i\theta} \) where \( \theta \) is any real number. Specifically, the determinant can be \( \pm 1 \) or any complex number of absolute value 1.

Hence, the answer is option 4.

Example 4: $\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & 1+i \\ 1-i & -1\end{array}\right]$ is a
1)Nilpotent Matrix
2)Unitary Matrix
3)Symmetric Matrix
4)Skew-Symmetric Matrix

Solution
Let $\mathrm{A}=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & 1+i \\ 1-i & -1\end{array}\right]$
Then $A^{\Theta}=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & 1-i \\ 1+i & -1\end{array}\right]$
Multiplying both the matrices, we get:

$
A A^{\Theta}=I
$
Thus, it is a unitary matrix.

Hence, the answer is the option (2).

Example 5: If A and B are unitary matrix. Then which of the following option is True.

1) A+B is unitary matrix.(always)

2) A-B is unitary matrix. (always)

3) AB is a unitary matrix.(always)

4) (1) and (3) both

Solution:

As we have learnt

Unitary matrix -

$
A A^{\Theta}=I
$
$A^{\Theta}$ is complex conjugate transpose matrix of matrix $A$ and $I$ is identity matrix
$A \cdot A^\theta=I$ and $B B^\theta=I$
(a) $
(A+B)^\theta(A+B)=\left(A^\theta+B^\theta\right)(A+B)=A^\theta A+A^\theta B+B^\theta A+B^\theta B
$

(b) $
(A-B)^\theta(A-B)=\left(A^\theta-B^\theta\right)(A-B)=A^\theta A-B^\theta A-A^\theta B+B^\theta B
$

(c) $(A B)^\theta(A B)=B^\theta A^\theta A B=B^\theta(I) B=B^\theta B=I$

Hence, the answer is option 4.