determin.ant — Linear Algebra from Scratch

Where you'll see this: stability analysis in control systems checks whether eigenvalues have negative real parts (system calms down) or positive ones (system blows up). Engineers solve the characteristic polynomial of a system matrix to find the stability threshold.

We know eigenvectors satisfy $A\mathbf{v} = \lambda\mathbf{v}$ . But how do we find λ? That's what the characteristic polynomial does.

Setting up the equation

Rewrite the eigenvector equation:

A\mathbf{v} = \lambda\mathbf{v}

A\mathbf{v} - \lambda\mathbf{v} = \mathbf{0}

(A - \lambda I)\mathbf{v} = \mathbf{0}

For this to have a non-zero solution v, the matrix $(A - \lambda I)$ must be singular — i.e. its determinant must be zero:

\det(A - \lambda I) = 0

The characteristic polynomial

For a 2×2 matrix, expanding this determinant gives a quadratic:

\lambda^2 - \text{tr}(A)\,\lambda + \det(A) = 0

The two solutions are the eigenvalues. The discriminant $\Delta = \text{tr}(A)^2 - 4\det(A)$ tells you whether they're real (Δ ≥ 0) or complex (Δ < 0).

Notice: λ₁ + λ₂ = tr(A) and λ₁ · λ₂ = det(A). These identities are always true — they're a quick sanity check.

Special cases

Symmetric matrices (A = Aᵀ) always have real eigenvalues.
Rotation matrices have complex eigenvalues — they rotate every direction so there's no real fixed direction.
Repeated eigenvalue: Δ = 0 means the matrix has one eigenvalue with multiplicity 2.

Try it

Edit the matrix on the right. The characteristic polynomial, discriminant, and eigenvalues all update live. Try to make the discriminant negative (complex eigenvalues) and positive (real eigenvalues).

In NumPy — the polynomial route:

import numpy as np
A = np.array([[3, 1], [1, 3]])
t = np.trace(A)       # 6
d = np.linalg.det(A)  # 8
# characteristic polynomial: λ² - 6λ + 8 = 0
roots = np.roots([1, -t, d])
print(roots)  # [4. 2.]

The Characteristic Polynomial

Setting up the equation

The characteristic polynomial

Special cases

Try it