A Quick Introduction to Orthonormal Matrices

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Setting the stage

Before dwelling into “orthonormal”, we must first understand what “orthogonal” means.

So how do we find if two vectors are orthogonal?

Well, that’s really simple! The vectors x and y are orthogonal if their dot product is equal to 0.

Now, what about “orthonormal”?

The vectors x and y are now orthonormal if they both are of unit length as well as orthogonal. Mathematically,

Moving from vector to matrix

An orthogonal matrix Q is a square matrix whose columns are all orthonormal i.e., orthogonal unit vectors.

Examples of orthonormal matrices

→ An identity matrix is the simplest orthonormal matrix

Visual example of rotation matrices:

In three dimensions, rotation can be applied in three directions i.e., X, Y and Z directions.

  1. Basis vectors rotated along X direction by 45 degrees
  2. Basis vectors rotated along Y direction by -15 degrees
  3. Basis vectors rotated along Z direction by 30 degrees

Few properties of orthonormal matrices:

Some of the properties of orthonormal matrices are listed below, along with the proofs. It is important to know that this list is non-exhaustive, and the goal here is to give

The inverse of an orthonormal matrix is its transpose

The transpose of the orthonormal matrix is also orthonormal

Given that Q is orthonormal,

The columns of the matrix are orthonormal

This directly follows the definition.

The rows of the matrix are orthonormal

Since the transpose of the matrix Q is orthonormal, the columns of Q^T are orthonormal. However, these columns form the rows in Q, thereby proving that the rows of Q are orthonormal.

An orthonormal matrix is non-singular

It is already established that for any orthonormal matrix, the inverse exists and is the transpose of the matrix itself.

  1. Determinant of an orthonormal matrix is NOT equal to 0
  2. 0 is NOT an eigen values of the orthonormal matrix

Transforming a vector by an orthonormal matrix does not change the magnitude of the vector

Mathematically,

The product of two orthonormal matrix is also orthonormal

Given two orthonormal matrices A and B

The determinant of an orthonormal matrix is +1 or -1

The eigenvalues of an orthonormal matrix are always +1 or -1

If Q is an orthonormal matrix,

Extending from real to complex matrices

So far so good. But what happens when Q is a complex matrix consisting of one or more complex numbers?

Conclusion:

For a novice reader, any rotation matrix is the most obvious example or orthonormal matrix. However, orthonormal and unitary matrices find applications in various aspects of linear algebra such as eigenvalue decomposition, spectral decomposition, Principal Component Analysis (PCA) etc. which form the basis for several real-world applications.

References and Further Reading:

  1. David W Lewis, “Matrix Theory
  2. R A Horn & C R Johnson, “Matrix Analysis
  3. Wikipedia article on orthonormal matrix: https://en.wikipedia.org/wiki/Orthogonal_matrix
  4. Wikipedia article on rotation matrix: https://en.wikipedia.org/wiki/Rotation_matrix

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