Orthogonal Matrix Formula Proof and Solved Examples
The concept of orthogonal matrix plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios—especially in linear algebra, engineering, and computer science. Understanding orthogonal matrices can help you ace your board exams and competitive tests. Let’s break it down with stepwise methods, solved examples, and fast tricks for exam success.
What Is Orthogonal Matrix?
An orthogonal matrix is a type of square matrix where every row and every column forms an orthonormal set of vectors. This means any two different rows (or columns) are perpendicular to each other (their dot product is 0), and every row or column has a length (magnitude) of 1. You’ll find this concept applied in linear algebra, physics (rotations and reflections), and computer graphics (image transformations).
Key Formula for Orthogonal Matrix
Here’s the standard formula: \( AA^T = I \ ) or \( A^T = A^{-1} \ )
Where:
AT = transpose of A
I = identity matrix of order n × n
Core Properties of Orthogonal Matrices
- If \( A \) is orthogonal, then \( AA^T = A^TA = I \).
- The inverse of an orthogonal matrix is its transpose: \( A^{-1} = A^T \).
- All rows and columns are orthonormal vectors.
- The determinant of an orthogonal matrix is always +1 or −1.
- The product of two orthogonal matrices is also orthogonal.
- If A is orthogonal, then AT is also orthogonal.
- Identity matrix is always orthogonal.
Step-by-Step Illustration: Orthogonal Matrix Check (2 × 2)
Let’s check if this matrix is orthogonal:
Given \( A = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \)
2. Multiply A by its transpose:
\( A A^T = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \)
3. Result is the 2 × 2 identity matrix, so A is orthogonal!
Step-by-Step Illustration: Orthogonal Matrix Check (3 × 3)
Let’s use a 3 × 3 matrix:
\( A = \dfrac{1}{2}\begin{bmatrix} 1 & 1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2 \end{bmatrix} \)
\( A^T = \dfrac{1}{2}\begin{bmatrix} 1 & -1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 2 \end{bmatrix} \)
2. Multiply \( AA^T \) and verify you get the 3 × 3 identity matrix.
3. If you do, A is orthogonal. Otherwise, it isn’t.
How to Check If a Matrix Is Orthogonal (Exam Checklist)
- Check the matrix is square (n × n).
- Find transpose (\( A^T \)).
- Multiply original and transpose (\( AA^T \)).
- If result is identity matrix (all diagonal 1, all other elements 0), the matrix is orthogonal.
Orthogonal Matrix vs Orthonormal Matrix vs Symmetric Matrix
| Type | Definition | Key Property |
|---|---|---|
| Orthogonal Matrix | Square matrix with orthonormal rows and columns | \( AA^T = I \) |
| Orthonormal Matrix | Usually means orthogonal; rows/columns are unit and mutually perpendicular | Same as orthogonal in real matrices |
| Symmetric Matrix | Square matrix equal to its transpose | \( A = A^T \) |
Cross-Disciplinary Usage
Orthogonal matrices are used not only in Maths but also frequently in Physics (representing rotations and reflections), Computer Science (3D graphics, image processing), and Statistics (PCA, QR decomposition). Students preparing for exams like JEE, NEET, or Olympiads often encounter orthogonal matrices in advanced problems.
Speed Trick or Vedic Shortcut
Here’s a quick tip: If a matrix’s transpose is its inverse (\( A^T = A^{-1} \)), it’s orthogonal! For 2 × 2 rotation matrices using cosine and sine, you can often quickly check the determinant to spot orthogonality (it should be ±1).
Relation to Other Concepts
The idea of orthogonal matrix connects to different types of matrices, matrix inverses, and symmetric matrices. Mastering orthogonality builds a strong base for topics like eigenvalues, transformations, and vectors.
Try These Yourself
- Check if \( \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \) is an orthogonal matrix.
- Find the determinant of \( \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix} \).
- Decide if all orthogonal matrices are symmetric.
- Is every diagonal matrix orthogonal?
Frequent Errors and Misunderstandings
- Confusing “orthogonal” (perpendicular vectors) with “orthonormal” (perpendicular and unit length)
- Forgetting that not every square matrix is orthogonal (must check the identity property)
- Assuming all diagonal/symmetric matrices are automatically orthogonal
- Missing the determinant being ±1 as a quick check
Classroom Tip
A good way to remember orthogonal matrices: “Transpose equals inverse.” Vedantu’s maths teachers often repeat this rule for board and entrance exam readiness!
We explored orthogonal matrix from its definition and formula, to properties, stepwise checks, and examples. Practicing these helps you solve questions much faster in exams. Continue learning with Vedantu for a clear understanding and expert support in all maths topics!
Internal Links for Further Study
FAQs on Orthogonal Matrix Explained with Definition and Properties
1. What is an orthogonal matrix?
An orthogonal matrix is a square matrix whose transpose is equal to its inverse, that is QTQ = QQT = I. This means:
- Q-1 = QT
- The rows and columns of Q are orthonormal vectors
- It preserves lengths and angles in linear transformations
2. What is the condition for a matrix to be orthogonal?
A matrix is orthogonal if and only if QTQ = I, where I is the identity matrix. This condition implies:
- The matrix must be square
- Each row and column has unit length
- Different rows (and columns) are mutually perpendicular
3. How do you check if a matrix is orthogonal?
To check if a matrix is orthogonal, compute QTQ and verify whether it equals the identity matrix I. Follow these steps:
- Step 1: Find the transpose QT
- Step 2: Multiply QT by Q
- Step 3: Check if the result equals I
4. What is the determinant of an orthogonal matrix?
The determinant of an orthogonal matrix is always +1 or -1. This follows from the property QTQ = I, which gives:
- det(QTQ) = det(I) = 1
- (det Q)2 = 1
5. What is the inverse of an orthogonal matrix?
The inverse of an orthogonal matrix is equal to its transpose, meaning Q-1 = QT. This simplifies matrix computations because:
- You do not need to compute a separate inverse
- Transposition is computationally easier
- It preserves numerical stability
6. Can you give an example of an orthogonal matrix?
Yes, a simple example of an orthogonal matrix is the 2×2 rotation matrix Q = [[cosθ, -sinθ], [sinθ, cosθ]]. For example, when θ = 90°,
- cos90° = 0
- sin90° = 1
7. What are the properties of an orthogonal matrix?
An orthogonal matrix has several important properties, the key one being QTQ = I. Other properties include:
- Q-1 = QT
- det(Q) = ±1
- Rows and columns form an orthonormal set
- It preserves vector length and inner products
8. What is the difference between an orthogonal matrix and an orthonormal matrix?
An orthogonal matrix is a square matrix with orthonormal rows and columns, while orthonormal typically describes a set of vectors. In other words:
- Orthogonal matrix: A square matrix satisfying QTQ = I
- Orthonormal vectors: Vectors that are mutually perpendicular and have unit length
9. Why are orthogonal matrices important in linear algebra?
Orthogonal matrices are important because they preserve lengths, angles, and inner products during transformations. Specifically:
- They represent rotations and reflections
- They simplify inverse computations using Q-1 = QT
- They are used in QR decomposition and eigenvalue problems
10. Do orthogonal matrices preserve vector length?
Yes, orthogonal matrices preserve vector length because they maintain the inner product. If Q is orthogonal and x is a vector, then:
- ‖Qx‖ = ‖x‖
- (Qx)·(Qy) = x·y
