Introduction

An orthogonal matrix is a type of square matrix whose columns and rows are orthogonal unit vectors. In simpler terms, an \(n \times n\) matrix \(A\) is orthogonal if \(A^T A = AA^T = I\), where \(I\) is the identity matrix of the same order and \(A^T\) denotes the transpose of \(A\).

Applications

Determining if a Matrix is Orthogonal

Determining if a matrix is orthogonal can be very important in various fields including computer graphics, quantum computing, and machine learning. The process is straightforward due to the property that when multiplied by its transpose, an orthogonal matrix results in the identity matrix.

In practice, due to numerical precision issues, one often checks if \(||A^T A - I|| < \varepsilon\) where \(\varepsilon\) is a small tolerance value.

References

• Orthogonal Matrix - Wikipedia
• Orthogonal Matrix - MathWorld