What are the key properties of an orthogonal matrix?

Transpose = Inverse; Determinant is ±1' Orthonormal rows and columns; Product of two orthogonal matrices is orthogonal

Is every orthogonal matrix invertible?

Yes. Since A^(-1)=A^(T), all orthogonal matrices are non-singular (invertible).

Can a non-square matrix be orthogonal?

No. Only square matrices can be orthogonal, since A^(T)A must be an identity matrix.

What is the inverse of an orthogonal matrix?

The transpose of the matrix. That is, A^(-1)=A^T

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Orthogonal Matrix

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Orthogonal Matrix

Q: What is an orthogonal matrix?

A square matrix whose transpose equals its inverse, i.e.,A^T=A^(-1) , is called an 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:

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

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

Product of two orthogonal matrices is also orthogonal.
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:

$A = [cos θ - sin θ sin θ cos θ]$

Let’s compute $A^{T} A$ :

$A^{T} = [cos θ sin θ - sin θ cos θ]$

Now,

$A^{T} A = [cos θ sin θ - sin θ cos θ] [cos θ - sin θ sin θ cos θ] = [1001]$

Therefore, this is an orthogonal matrix.

5.0Orthogonal Matrix Example (3 × 3)

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

$A = \frac{1}{3} 111 1 - 1 1 10 - 2$

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 $A = [\frac{1}{2} - \frac{1}{2} \frac{1}{2} \frac{1}{2}]$ is orthogonal.

Solution:

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

$A^{T} = [\frac{1}{2} \frac{1}{2} - \frac{1}{2} \frac{1}{2}]$

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

$A^{T} A = [\frac{1}{2} \frac{1}{2} - \frac{1}{2} \frac{1}{2}] [\frac{1}{2} - \frac{1}{2} \frac{1}{2} \frac{1}{2}] = [1001]$

Hence, A is an orthogonal matrix.

Example 2: Show that $A = \frac{1}{3} 111 1 - 1 1 11 - 1$ 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 $A = [0 - 1 10]$

Solution:
Since AA is orthogonal, the inverse is: $A^{- 1} = A^{T} = [01 - 1 0]$

Inverse of orthogonal matrix is its transpose.

Example 4: Verify whether $A = [0 - 1 10]$ is orthogonal.

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

$A^{T} = [01 - 1 0] and A^{T} A = [01 - 1 0]^{T} [01 - 1 0] = [1001]$

Thus, A is orthogonal.

8.0Practice Questions on Orthogonal Matrix

Check whether the following matrix is orthogonal: $[cos 4 5^{\circ} - sin 4 5^{\circ} sin 4 5^{\circ} cos 4 5^{\circ}]$
Verify whether $A = 100 00 - 1 010$ is an orthogonal matrix.
Find the inverse of the orthogonal matrix: $A = [01 - 1 0]$
Give an example of an orthogonal matrix (3 × 3) with determinant -1.
True or False: Every orthogonal matrix is invertible.

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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:

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

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

Product of two orthogonal matrices is also orthogonal.
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:

$A = [cos θ - sin θ sin θ cos θ]$

Let’s compute $A^{T} A$ :

$A^{T} = [cos θ sin θ - sin θ cos θ]$

Now,

$A^{T} A = [cos θ sin θ - sin θ cos θ] [cos θ - sin θ sin θ cos θ] = [1001]$

Therefore, this is an orthogonal matrix.

5.0Orthogonal Matrix Example (3 × 3)

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

$A = \frac{1}{3} 111 1 - 1 1 10 - 2$

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 $A = [\frac{1}{2} - \frac{1}{2} \frac{1}{2} \frac{1}{2}]$ is orthogonal.

Solution:

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

$A^{T} = [\frac{1}{2} \frac{1}{2} - \frac{1}{2} \frac{1}{2}]$

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

$A^{T} A = [\frac{1}{2} \frac{1}{2} - \frac{1}{2} \frac{1}{2}] [\frac{1}{2} - \frac{1}{2} \frac{1}{2} \frac{1}{2}] = [1001]$

Hence, A is an orthogonal matrix.

Example 2: Show that $A = \frac{1}{3} 111 1 - 1 1 11 - 1$ 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 $A = [0 - 1 10]$

Solution:
Since AA is orthogonal, the inverse is: $A^{- 1} = A^{T} = [01 - 1 0]$

Inverse of orthogonal matrix is its transpose.

Example 4: Verify whether $A = [0 - 1 10]$ is orthogonal.

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

$A^{T} = [01 - 1 0] and A^{T} A = [01 - 1 0]^{T} [01 - 1 0] = [1001]$

Thus, A is orthogonal.

8.0Practice Questions on Orthogonal Matrix

Check whether the following matrix is orthogonal: $[cos 4 5^{\circ} - sin 4 5^{\circ} sin 4 5^{\circ} cos 4 5^{\circ}]$
Verify whether $A = 100 00 - 1 010$ is an orthogonal matrix.
Find the inverse of the orthogonal matrix: $A = [01 - 1 0]$
Give an example of an orthogonal matrix (3 × 3) with determinant -1.
True or False: Every orthogonal matrix is invertible.

Frequently Asked Questions

What is an orthogonal matrix?

What are the key properties of an orthogonal matrix?

Is every orthogonal matrix invertible?

Can a non-square matrix be orthogonal?

What is the inverse of an orthogonal matrix?

Join ALLEN!

Orthogonal Matrix

1.0What is an Orthogonal Matrix?

2.0 Properties of Orthogonal Matrix

3.0Inverse of Orthogonal Matrix

4.0Example of Orthogonal Matrix (2 × 2)

5.0Orthogonal Matrix Example (3 × 3)

6.0Real-World Applications of Orthogonal Matrices

7.0Solved Examples on Orthogonal Matrix

8.0Practice Questions on Orthogonal Matrix

Table of Contents

Frequently Asked Questions

What is an orthogonal matrix?

What are the key properties of an orthogonal matrix?

Is every orthogonal matrix invertible?

Can a non-square matrix be orthogonal?

What is the inverse of an orthogonal matrix?

Join ALLEN!

Orthogonal Matrix

1.0What is an Orthogonal Matrix?

2.0 Properties of Orthogonal Matrix

3.0Inverse of Orthogonal Matrix

4.0Example of Orthogonal Matrix (2 × 2)

5.0Orthogonal Matrix Example (3 × 3)

6.0Real-World Applications of Orthogonal Matrices

7.0Solved Examples on Orthogonal Matrix

8.0Practice Questions on Orthogonal Matrix

Table of Contents