Special Matrices: Identity, Diagonal, Orthogonal

Some matrices have special properties that make them particularly important in machine learning and linear algebra. In this lesson, we'll cover identity matrices, diagonal matrices, and orthogonal matrices.

Identity Matrix

The identity matrix $I_n$ is a square matrix with ones on the diagonal and zeros elsewhere:

$$I_n=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

Property:

$$AI=IA=A$$

for any matrix $A$ of compatible dimensions.

Explanation of Identity

The identity matrix is like the number 1 for matrices. Multiplying by $I$ leaves the matrix unchanged.

• In ML: shows up in regularization (e.g., $X^{T}X+\lambda I$ in Ridge regression).

Diagonal Matrix

A diagonal matrix has nonzero entries only on its main diagonal:

$$D=\left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right]$$

Property: multiplying a vector scales each coordinate by the corresponding diagonal element.

Mini Example:

$$D=\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right],x=\left[\begin{array}{c}4\\ 5\end{array}\right]\Rightarrow Dx=\left[\begin{array}{c}8\\ 15\end{array}\right]$$

Explanation of Diagonal

Diagonal matrices represent scaling transformations.

• In ML: eigenvalues often appear in diagonal matrices after decomposition (PCA, SVD).

Orthogonal Matrix

A square matrix $Q$ is orthogonal if:

$$Q^{T}Q=Q{Q}^T=I$$

This means:

• Rows (and columns) of $Q$ are orthonormal vectors (unit length and mutually perpendicular).
• $Q^{-1}=Q^T$.

Mini Example:

$$Q=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$$

Here, $Q^{T}Q=I$.

Explanation of Orthogonal

Orthogonal matrices preserve lengths and angles. They represent rotations and reflections.

• In ML: orthogonal transformations are used in PCA, whitening, and optimization stability.

Hands-on with Python and Rust

Python

import numpy as np

# Identity matrix
I = np.eye(3)

# Diagonal matrix
D = np.diag([2, 3])

# Orthogonal matrix (rotation by 90 degrees)
Q = np.array([[0, 1], [-1, 0]])

print("Identity matrix:\n", I)
print("Diagonal matrix:\n", D)
print("Orthogonal matrix:\n", Q)
print("Check Q^T Q = I:\n", Q.T.dot(Q))

Rust

use ndarray::{array, Array2};
use ndarray_linalg::Norm;

fn main() {
    // Identity matrix
    let i: Array2<f64> = Array2::eye(3);

    // Diagonal matrix
    let d: Array2<f64> = array![[2.0, 0.0], [0.0, 3.0]];

    // Orthogonal matrix (rotation by 90 degrees)
    let q: Array2<f64> = array![[0.0, 1.0], [-1.0, 0.0]];
    let qtq = q.t().dot(&q);

    println!("Identity matrix:\n{:?}", i);
    println!("Diagonal matrix:\n{:?}", d);
    println!("Orthogonal matrix:\n{:?}", q);
    println!("Q^T Q =\n{:?}", qtq);
}

Connection to ML

• Identity → regularization, basis for “do nothing” transformations.
• Diagonal → scaling features, eigenvalues, covariance structure.
• Orthogonal → rotations, PCA, numerical stability in optimization.

Next Steps

Continue to Rank, Determinant, and Inverses.