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The Characteristic Polynomial

When you expand $\det(A - \lambda I)$ for an $n \times n$ matrix, you get a polynomial of degree $n$ in $\lambda$ . This is called the characteristic polynomial of $A$ . The eigenvalues are exactly its roots. For a $2 \times 2$ matrix $A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ , the characteristic polynomial simplifies to a familiar quadratic: $\lambda^2 - \text{tr}(A)\,\lambda + \det(A) = 0$ , where $\text{tr}(A) = a + d$ is the trace (sum of diagonal entries).

The quadratic formula gives $\lambda = \frac{\text{tr}(A) \pm \sqrt{\Delta}}{2}$ where the discriminant is $\Delta = \text{tr}(A)^2 - 4\det(A)$ . Three cases: $\Delta > 0$ means two distinct real eigenvalues; $\Delta = 0$ means one repeated real eigenvalue; $\Delta < 0$ means two complex conjugate eigenvalues (no real eigenvectors — the transformation rotates every direction).

Use the interactive diagram to build intuition. Set the matrix entries and watch how the characteristic polynomial and its roots change. Try to find a matrix with one positive and one negative eigenvalue, or push the discriminant negative to create a rotation. Notice how the trace equals $\lambda_1 + \lambda_2$ and the determinant equals $\lambda_1 \cdot \lambda_2$ .

Formal View

Definition 7.5 — Characteristic Polynomial

The characteristic polynomial of an

n \times n

matrix

A

p(\lambda) = \det(A - \lambda I)

. The characteristic equation is

p(\lambda) = 0

. The eigenvalues of

A

are exactly the roots of

p(\lambda)

Theorem 7.6 — Characteristic Polynomial of a 2×2 Matrix

For

A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}

, the characteristic polynomial is

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

where

\text{tr}(A) = a + d

. The eigenvalues satisfy

\lambda_1 + \lambda_2 = \text{tr}(A)

and

\lambda_1 \cdot \lambda_2 = \det(A)

These trace-determinant relations hold for any $n \times n$ matrix: the trace equals the sum of all eigenvalues and the determinant equals their product.

Interactive Visualization

Characteristic Polynomial

Why This Matters

The characteristic polynomial encodes all eigenvalue information in a single algebraic object.

The Cayley–Hamilton theorem says every matrix satisfies its own characteristic equation: $p(A) = 0$
The trace and determinant identities let engineers quickly estimate eigenvalue behavior without full computation
Checking whether a matrix is positive definite (all eigenvalues positive) is often done via its characteristic polynomial
Spectral analysis in signal processing uses characteristic polynomials of filter coefficient matrices

Learning Resources

Characteristic polynomial and eigenvalues

Professor Leonard

Detailed walkthrough of computing characteristic polynomials and finding eigenvalues.

22 min

A quick trick for computing eigenvalues

3Blue1Brown

An elegant shortcut using trace and determinant to find 2×2 eigenvalues mentally.

15 min

Quiz

Question 1

What is the characteristic polynomial of $A = \begin{pmatrix}4 & 1 \\ 2 & 3\end{pmatrix}$ ?

Question 2

The eigenvalues of $A = \begin{pmatrix}4 & 1 \\ 2 & 3\end{pmatrix}$ are:

Question 3

Every $2 \times 2$ real matrix has two real eigenvalues.

Common Mistakes

Computing $\det(A - \lambda)$ instead of $\det(A - \lambda I)$ — you must subtract a matrix, not a scalar.
Forgetting the minus sign: the characteristic polynomial is $\lambda^2 - \text{tr}(A)\lambda + \det(A)$ , not $+\text{tr}(A)\lambda$ .
Assuming the characteristic polynomial factors into real linear factors — complex roots are possible and common for rotation-like matrices.