Orthogonal Matrix 

An orthogonal matrix is a special type of square matrix whose rows and columns are orthonormal vectors. In simple terms, a matrix A is orthogonal if the product of the matrix and its transpose equals the identity matrix.

An orthogonal matrix is a special type of square matrix whose transpose is equal to its inverse, i.e., $A^T=A^{-1}$. This means the rows and columns of the matrix are orthonormal vectors — they are perpendicular and of unit length. Orthogonal matrices preserve the length and angle of vectors under transformation, making them essential in geometry, computer graphics, and numerical analysis. Their determinant is always either +1 or –1, indicating volume-preserving transformations.

1.0What is an Orthogonal Matrix?

A matrix A is called orthogonal if:

$A^{T}A=A{A}^T=I$

Where:

• $A^T$ is the transpose of matrix A
• I is the identity matrix of the same order

2.0 Properties of Orthogonal Matrix

Here are some important orthogonal matrix properties:

1. The inverse of an orthogonal matrix is equal to its transpose, i.e., $A^T=A^{-1}$
2. The determinant of an orthogonal matrix is always +1 or –1 i.e., $|A|=\pm 1$
3. The rows and columns form orthonormal sets:

• Each row (or column) is a unit vector
• Dot product of distinct rows (or columns) is 0

4. Product of two orthogonal matrices is also orthogonal.
5. The transpose of an orthogonal matrix is also orthogonal.

3.0Inverse of Orthogonal Matrix

One of the most useful properties of an orthogonal matrix is:$A^{-1}=A^T$

This means that instead of performing complex inverse operations, we can simply transpose the matrix to get its inverse — saving both time and computation.

4.0Example of Orthogonal Matrix (2 × 2)

Let’s check whether the following matrix is orthogonal:

$A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$

Let’s compute $A^{T}A$:

$A^T=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$

Now,

$A^{T}A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Therefore, this is an orthogonal matrix.

5.0Orthogonal Matrix Example (3 × 3)

Let’s verify the orthogonality of this 3x3 matrix:

$A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 0\\ 1& 1& -2\end{array}\right]$

To verify it's orthogonal:

• Check if the rows are mutually orthogonal (dot product = 0)
• Each row has a magnitude of 1 (unit vectors)
• If both hold true, then $A^{T}A=I$

This is a valid orthogonal matrix example (3 × 3) after simplification.

6.0Real-World Applications of Orthogonal Matrices

• Computer graphics & animation: For rotating objects
• Quantum mechanics: Orthogonal matrices preserve length and angle
• Signal processing: Orthogonal transformations reduce computation
• Data science: PCA (Principal Component Analysis) uses orthogonal matrices

7.0Solved Examples on Orthogonal Matrix

Example 1: Check whether the matrix $A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]$ is orthogonal.

Solution:

Step 1: Find $A^T$ (transpose of A):

 $A^T=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]$

Step 2: Multiply $A^{T}A$:

$A^{T}A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Hence, A is an orthogonal matrix.

Example 2: Show that $A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right]$ is orthogonal.

Solution:

Step 1: Compute dot product of each pair of rows:

• Each row has magnitude = 1
• Dot product of different rows = 0

Step 2: Check $A^{T}A=I$(you can verify by matrix multiplication).

Hence, it is an orthogonal matrix example 3 x 3.

Example 3: Find the inverse of the matrix $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

Solution:
Since AA is orthogonal, the inverse is:$A^{-1}=A^T=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Inverse of orthogonal matrix is its transpose.

Example 4: Verify whether $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ is orthogonal.

Solution:
Find $A^{T}A$ :

$A^T=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\text{and}{A}^{T}A={\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]}^T\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Thus, A is orthogonal.

8.0Practice Questions on Orthogonal Matrix

1. Check whether the following matrix is orthogonal: $\left[\begin{array}{cc}\cos 45^{\circ }& \sin 45^{\circ }\\ -\sin 45^{\circ }& \cos 45^{\circ }\end{array}\right]$
2. Verify whether $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right]$is an orthogonal matrix.
3. Find the inverse of the orthogonal matrix: $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
4. Give an example of an orthogonal matrix (3 × 3) with determinant -1.
5. True or False: Every orthogonal matrix is invertible.