Orthogonal Matrices and Mutually Orthogonal Vectors

Orthogonal Matrices and Mutually Orthogonal Vectors

Orthogonal matrices appear naturally throughout linear algebra, geometry, physics, and computer graphics. They preserve lengths, angles, and orientation, which makes them fundamental in describing rotations and rigid motions in three-dimensional space. This article provides a clear and carefully structured explanation of what orthogonal matrices are, why they matter, and how to verify that a given matrix is orthogonal.

1. Definition of an Orthogonal Matrix

Let M be an n × n square matrix. M is called orthogonal if it satisfies:

M MT = I

Here:

  • MT is the transpose of M.
  • I is the identity matrix of the same size.

Because of this property, every orthogonal matrix has a very useful consequence:

M-1 = MT

This means that the inverse of an orthogonal matrix is obtained simply by transposing it. This property is central to rigid-body transformations in 3D geometry and computer graphics.

2. Orthogonal Vectors

Two vectors x1 and x2 are said to be orthogonal if their scalar product (dot product) is zero:

x1 · x2 = 0

Orthogonality means that the two vectors meet at a right angle. In matrix form, this is the condition that ensures no component of one vector “spills into” the direction of the other.

3. Columns of an Orthogonal Matrix

Suppose the matrix M is orthogonal, and its columns are the vectors:

x1,   x2,   x3

If M is orthogonal, then:

  • Each column vector has length 1 (they are normalised).
  • Any two different columns are orthogonal to one another.

In particular:

  • x1 · x2 = 0
  • x2 · x3 = 0
  • x3 · x1 = 0

Therefore, an orthogonal 3 × 3 matrix is formed from three mutually orthogonal, normalised basis vectors in ℝ³. These vectors span the space and preserve geometric structure under transformation.

4. How to Verify That a Matrix Is Orthogonal

There are two equivalent and commonly used methods to show that a matrix M is orthogonal:

Method 1 — Check the Matrix Equation

Compute M MT. If the result is the identity matrix I, then M is orthogonal.

Method 2 — Check the Column Structure

A matrix M is orthogonal if and only if its columns:

  • each have length 1 (normalised)
  • are mutually orthogonal (their dot products are zero)

Both conditions guarantee that the transformation represented by M preserves distances and angles in space.

This completes the core theory behind orthogonal matrices. The topic is essential for understanding rotations, rigid transformations, eigenvector structure, and many areas of vector geometry.

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