Orthogonal Matrix: Definition, Properties, Formula and Examples

Orthogonal matrix

Edited By Komal Miglani | Updated on Sep 19, 2024 11:25 AM IST

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 orthogonal matrices in Euclidean space, Multivariate time series analysis, and multichannel signal processing.

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This Story also Contains

Square matrix
Orthogonal matrix
Properties of Orthogonal matrix
Solved Examples Based on Orthogonal Matrices
Summary

In this article, we will cover the concept of Orthogonal 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 $\mathrm{A=[a_{ij}]_{m\times n}\;}$ is said to be a square matrix when m = n.

E.g.

$\\\mathrm{\begin{bmatrix} a_{11} &a_{12} &a_{13} \\ a_{21}& a_{22} &a_{23} \\ a_{31}& a_{32} &a_{33} \end{bmatrix}_{3\times3}} \mathrm{or,\;\;\;\begin{bmatrix} 2 &-4 \\ 7 & 3 \end{bmatrix}_{2\times2}}$

Orthogonal matrix

A matrix is said to be an orthogonal matrix if the product of a matrix and its transpose gives an identity matrix. A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

A square matrix is said to be an orthogonal matrix if AA’ = I, where I is the identity matrix.

Properties of Orthogonal matrix

1) . AA’ = I ⇒ A^-1 = A

2) The product of two orthogonal matrices is also an orthogonal matrix. If A and B are orthogonal then AB is also orthogonal.

3) The inverse of the orthogonal matrix is also orthogonal. If A is orthogonal the A^-1is also orthogonal.

4) All the orthogonal matrices are invertible.

5) The determinant of an orthogonal matrix is always equal to the -1 or +1. If A is orthogonal then | A| =1 or -1

6) All orthogonal matrices are square matrices but not all square matrices are orthogonal.

7) All identity matrices are orthogonal matrices.

8) The transpose of the orthogonal matrix is also orthogonal. If A is orthogonal then A’ is also orthogonal.

Solved Examples Based on Orthogonal Matrices

Example 1: A is a orthogonal matrix where $A=\begin{bmatrix} 5 & 5\alpha \\ 0& \alpha \end{bmatrix}$ . Then find the value of $\alpha$ .

1) 1

2) $\frac{1}{5}$

3) $\frac{1}{25}$

4) None of these

Solution:

Orthogonal matrix -

$AA{}'=I$

- wherein

$A{}'$ is transpose matrix of matrix $A$ and $I$ is identity matrix

Orthogonal matrix

$AA^{T}= I,A^{T}= \begin{bmatrix} 5 &0 \\ 5\alpha & \alpha \end{bmatrix}$ $AA^{T}= \begin{bmatrix} 5 &5\alpha \\ 0 & \alpha \end{bmatrix}\begin{bmatrix} 5 &0 \\ 5\alpha & \alpha \end{bmatrix}$

$=\begin{bmatrix} 25(1+\alpha ^{2}) &5\alpha ^{2} \\ 5\alpha ^{2} & \alpha ^{2}\end{bmatrix}$

$=\begin{bmatrix} 1 &0 \\ 0 & 1 \end{bmatrix}$

No value of 2 exist

Example 2: If A = $\begin{bmatrix} \frac{1}{3} & \frac{2}{3} &a \\ \frac{2}{3}&\frac{1}{3} &b \\ \frac{2}{3}& -\frac{2}{3} & c \end{bmatrix}$ is orthogonal, then find a,b,c

1) $\left(\pm \frac{1}{3}, \pm \frac{2}{3},\pm \frac{2}{3}\right )$

2) $\left(\pm \frac{2}{3}, \pm \frac{1}{3},\pm \frac{1}{3}\right )$

3) $\left(\pm \frac{2}{3}, \pm \frac{2}{3},\pm \frac{1}{3}\right )$

4) $\left(\pm \frac{2}{3}, \pm \frac{1}{3},\pm \frac{2}{3}\right )$

Solution: We know that Orthogonal matrix - $AA{}'=I$
where, $A{}'$ is a transpose matrix of matrix $A$ and $I$ is the identity matrix

For Orthogonal Matrices, AA' = I

$\begin{bmatrix} \frac{1}{3} & \frac{2}{3} &a \\ \frac{2}{3}&\frac{1}{3} &b \\ \frac{2}{3}& -\frac{2}{3} & c \end{bmatrix} \begin{bmatrix} \frac{1}{3} & \frac{2}{3} &\frac{2}{3} \\ \frac{2}{3} & \frac{1}{3} &-\frac{2}{3} \\ a& b & c \end{bmatrix} = \begin{bmatrix} 1 & 0 &0 \\ 0 & 1& 0\\ 0 & 0 & 1 \end{bmatrix}$

On comparing

$\frac{1}{}9 + \frac{4}{}9 + a^2 = 1 ; \frac{2}{}9 +\frac{2}{}9 + ab = 0 \\*\Rightarrow a^2 = \frac{4}{9} \Rightarrow a = \pm \frac{2}{3} \;and \;b = \mp \frac{2}{}3 \; and \; ||ly \; c =\pm \frac{1}{3}$

Hence, the answer is option 3.

Example 3: Which of the following statements about an orthogonal matrix $ A $ is **not** true?

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

Solution:
1) True. For an orthogonal matrix $ A $, $ A^{-1} = A^T $.
2) True. The columns (and rows) of an orthogonal matrix are orthonormal vectors.
3) False. The determinant of an orthogonal matrix is $ \pm 1 $, not zero.
4) True. For an orthogonal matrix, $ A A^T = I $.

Hence, the answer is option 3.

Example 4: Suppose that a, b, and c are real numbers such that a + b + c = 1. If the matrix $A=\begin{vmatrix} a &b &c \\ b & c& a\\ c& a& b \end{vmatrix}$ is orthogonal, then:

1) At least one of a, b, and c is negative

2) |A| is negative

3) a³ + b³ + c³ – 3abc = 1

4) All of these

Solution: we know that Circulant determinant - The elements of the rows (or columns) are in a cyclic arrangement

eg:- $\begin{vmatrix} a & b & c\\ b&c &a \\ c & a & b \end{vmatrix} = -\left ( a^{3} +b^{3}+c^{3}-3abc\right )$

AA^T = A^TA = I. Also A^T = A, so A² = I $\Rightarrow$ A is an involuntary matrix.

$\Rightarrow$ |A²| = |A|² = 1 or |A| = ±1

But |A| = $\begin{vmatrix} a &b & c\\ b & c& a\\ c& a&b \end{vmatrix}=(a+b+c)\begin{vmatrix} 1 & b&c \\ 1 & c & a\\ 1&a & b \end{vmatrix}=(a+b+c)(ab+bc+ca-a^2-b^2-c^2)$

|A| = ab + bc + ca – a² – b² – c² $(\because a+b+c=1)$

$\; \; \; \therefore \;$ a² + b² + c² – ab – bc – ca $\; \geq$ 0

So |A| = -1. Hence a³ + b³ + c³ – 3abc = 1.

Again a² + b² + c² – ab – bc – ca = 1 $\Rightarrow$ 1 – 3(ab + bc + c(A) = 1, so ab + bc + ca = 0

$\Rightarrow$ At least one of a, b, and c is negative.

Hence, the answer is the option (4).

Example 5: Let $A=\begin{bmatrix} x & y & z\\ y& z&x \\ z& x&y \end{bmatrix}$ , where x,y, and z are real numbers such that $x+y+z>0\; and\; xyz=2.$

If $A^{2}=I_{3},$ then the value of $x^{3}+y^{3}+z^{3}$ is______ [JEE MAINS 2021]

1) 7

2) 2

3) 5

4) 9

Solution

$\mathrm{A}^{2}=\mathrm{I} \\ \\\Rightarrow \mathrm{AA}^{\prime}=\mathrm{I}\left( \text{ as }\mathrm{A}^{\prime}=\mathrm{A}\right) \\\\\Rightarrow \mathrm{A}\text{ is orthogonal}$

$\begin{aligned} &\text { So, } x^{2}+y^{2}+z^{2}=1 \text { and } x y+y z+z x=0\\ &\Rightarrow(x+y+z)^{2}=1+2 \times 0\\ &\Rightarrow x+y+z=1 \end{aligned}$

$a^{3}+b^{3}+c^{3}=(a+b+c)\left[\left(a^{2}+b^{2}+c^{2}\right)-(a b+b c+c a)\right]+3 a b c$

Thus,

$\begin{aligned} \mathrm{x}^{3}+\mathrm{y}^{3}+\mathrm{z}^{3} &=3 \times 2+1 \times(1-0) =7 \end{aligned}$

Hence, the answer is the option 1.

Summary

Orthogonal matrices are a special type of matrices. Orthogonal matrices have the ability to preserve lengths and angles during transformations like rotations and reflections. They are fundamental in fields such as geometry, signal processing, and quantum mechanics, where their properties play a key role in both theoretical understanding and practical applications.

Frequently Asked Questions (FAQs)

1. What is an orthogonal matrix?

2. If A and B are orthogonal then AB is orthogonal or not?

The product of two orthogonal matrices is also an orthogonal matrix. If A and B are orthogonal then AB is also orthogonal.

3. What is the determinant of orthogonal matrices?

The determinant of an orthogonal matrix is always equal to the -1 or +1. If A is orthogonal then | A| =1 or -1

4. Are all square matrices are orthogonal matrices?

No, All orthogonal matrices are square matrices but not all square matrices are orthogonal.

5. What are square matrices?

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