1. What is an Orthogonal Matrix?

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (orthonormal vectors). The key property is that the transpose of an orthogonal matrix is equal to its inverse: \( Q^T = Q^{-1} \).

2. How Does the Calculator Work?

The calculator verifies orthogonality by checking if \( Q^T Q = I \), where:

Where:

• \( Q \) — The input matrix
• \( Q^T \) — The transpose of Q
• \( I \) — The identity matrix

Explanation: The calculator multiplies the matrix by its transpose and checks if the result is the identity matrix (within numerical tolerance).

3. Importance of Orthogonal Matrices

Details: Orthogonal matrices preserve lengths and angles, making them essential in many areas including computer graphics, signal processing, and numerical linear algebra. They provide numerical stability in algorithms.

4. Using the Calculator

Tips: Enter your matrix with comma-separated values within rows and semicolon-separated rows. For example, "1,0;0,1" for a 2×2 identity matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between orthogonal and orthonormal?
A: For matrices, they mean the same thing - columns/rows are orthonormal vectors (orthogonal and unit length).

Q2: Are rotation matrices orthogonal?
A: Yes, all rotation matrices are orthogonal with determinant +1.

Q3: What's the determinant of an orthogonal matrix?
A: The determinant is always either +1 or -1.

Q4: Are orthogonal matrices always square?
A: Yes, by definition orthogonal matrices must be square.

Q5: Why are orthogonal matrices important in QR factorization?
A: QR factorization decomposes a matrix into an orthogonal matrix Q and an upper triangular matrix R, which is numerically stable.