What is the characteristic equation?

The characteristic equation is det(A - λI) = 0. It arises from the eigenvector equation Av = λv rewritten as (A - λI)v = 0. Nontrivial solutions exist only when A - λI is singular, which happens when its determinant is zero.

What is the characteristic polynomial?

The characteristic polynomial is p(λ) = det(A - λI), a degree-n polynomial for an n×n matrix. Its roots are the eigenvalues. The constant term is det(A), and the coefficient of λⁿ⁻¹ involves the trace.

How do you find the characteristic polynomial of a 2×2 matrix?

For A = [[a,b],[c,d]]: p(λ) = λ² - tr(A)λ + det(A) = λ² - (a+d)λ + (ad-bc). Use the quadratic formula: λ = (tr(A) ± √(tr(A)² - 4det(A)))/2.

How do you find eigenvalues of a 3×3 matrix?

Expand det(A - λI) using cofactors to get a cubic polynomial. For triangular matrices, eigenvalues are the diagonal entries. Otherwise, factor the cubic by finding rational roots or using the cubic formula.

How are eigenvalues found for large matrices?

For n ≥ 5, no closed-form root formula exists (Abel-Ruffini). Large matrices use iterative algorithms like QR iteration, which converge to eigenvalues without forming the characteristic polynomial—direct polynomial root-finding is numerically unstable.

What is algebraic multiplicity?

Algebraic multiplicity is the power k such that (λ - λ₀)ᵏ divides the characteristic polynomial—the multiplicity of λ₀ as a root. It's an upper bound for geometric multiplicity. All algebraic multiplicities sum to n.

How do you find eigenvectors after finding eigenvalues?

For each eigenvalue λᵢ, solve the homogeneous system (A - λᵢI)v = 0. Row reduce A - λᵢI and express the solution in parametric form. Free variables give basis vectors for the eigenspace.

What is the Cayley-Hamilton theorem?

Every matrix satisfies its own characteristic polynomial: p(A) = 0 (zero matrix). This lets you express A⁻¹ as a polynomial in A and reduce high powers Aᵏ to polynomials of degree at most n-1.

Characteristic Equation

Q: Do similar matrices have the same characteristic polynomial?

Yes. det(P⁻¹AP - λI) = det(A - λI) because det(P⁻¹)det(P) = 1. The characteristic polynomial is an invariant of the linear transformation, not the specific matrix representation. Similar matrices share eigenvalues with same multiplicities.

From Eigenvectors to the Determinant Condition

The Characteristic Polynomial

Computing the Characteristic Polynomial: 2×2

Computing the Characteristic Polynomial: 3×3

Larger Matrices

Algebraic Multiplicity

Finding Eigenvectors After Finding Eigenvalues

The Cayley-Hamilton Theorem

Characteristic Polynomial and Similarity

The Polynomial Whose Roots Are the Eigenvalues

The eigenvalue problem Av = λv converts into a determinant condition: det(A − λI) = 0. This determinant is a polynomial in λ whose roots are the eigenvalues. Computing the characteristic polynomial and factoring it is the standard method for finding eigenvalues of small matrices — and the polynomial's coefficients encode the trace, determinant, and other invariants of the matrix.

From Eigenvectors to the Determinant Condition

The equation

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

rearranges to

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

. This is a homogeneous system, and eigenvectors are its nontrivial solutions. Nontrivial solutions exist if and only if the coefficient matrix

A - \lambda I

is singular:

\det(A - \lambda I) = 0

This is the characteristic equation. It holds for exactly those values of

\lambda

that are eigenvalues of

A

. Every other value of

\lambda

makes

A - \lambda I

invertible, the system has only the trivial solution, and no eigenvector exists for that

\lambda

.

The characteristic equation transforms the geometric question "which directions does

A

preserve?" into the algebraic question "for which

\lambda

is this determinant zero?"

The Characteristic Polynomial

The expression

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

is a polynomial of degree

n

in the variable

\lambda

. It is called the characteristic polynomial of

A

.

For an

n \times n

matrix,

p(\lambda)

has degree

n

with leading term

(-1)^n \lambda^n

. The constant term is

p(0) = \det(A)

— the determinant of the matrix itself. The coefficient of

\lambda^{n-1}

(-1)^{n-1}\text{tr}(A)

, connecting the next-to-leading term to the trace.

The eigenvalues are precisely the roots of

p(\lambda) = 0

. Every root is an eigenvalue, and every eigenvalue is a root. The characteristic polynomial packages the entire eigenvalue structure of the matrix into a single algebraic expression.

Computing the Characteristic Polynomial: 2×2

For

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

, the characteristic polynomial is

p(\lambda) = \det\begin{pmatrix} a - \lambda & b \\ c & d - \lambda \end{pmatrix} = (a - \lambda)(d - \lambda) - bc = \lambda^2 - (a + d)\lambda + (ad - bc)

This is

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

. The eigenvalues follow from the quadratic formula:

\lambda = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2}

The discriminant

\Delta = \text{tr}(A)^2 - 4\det(A)

classifies the eigenvalue type. When

\Delta > 0

, there are two distinct real eigenvalues. When

\Delta = 0

, there is one repeated real eigenvalue. When

\Delta < 0

, the eigenvalues are a complex conjugate pair.

Worked Example

For

A = \begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}

\text{tr}(A) = 9

\det(A) = 14

\Delta = 81 - 56 = 25

. The eigenvalues are

\lambda = \frac{9 \pm 5}{2}

, giving

\lambda_1 = 7

and

\lambda_2 = 2

Computing the Characteristic Polynomial: 3×3

For a

3 \times 3

matrix, expanding

\det(A - \lambda I)

using cofactor expansion produces a cubic polynomial:

p(\lambda) = -\lambda^3 + \text{tr}(A)\lambda^2 - (\text{sum of } 2 \times 2 \text{ principal minors})\lambda + \det(A)

The computation is lengthier but follows the same cofactor mechanics as any

3 \times 3

determinant.

Worked Example

For

A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{pmatrix}

, this is upper triangular, so

A - \lambda I

is also upper triangular with diagonal entries

2 - \lambda

3 - \lambda

1 - \lambda

p(\lambda) = (2 - \lambda)(3 - \lambda)(1 - \lambda)

The eigenvalues are

\lambda = 1, 2, 3

— readable directly from the diagonal. For triangular matrices, the characteristic polynomial always factors as the product of the diagonal terms, making the eigenvalues visible by inspection.

For non-triangular

3 \times 3

matrices, the cubic must be factored by finding rational roots (testing factors of the constant term), by inspection, or by the cubic formula.

Larger Matrices

For an

n \times n

matrix, the characteristic polynomial has degree

n

, and finding its roots becomes increasingly difficult as

n

grows. There is no general closed-form formula for roots of polynomials of degree

5

or higher (Abel-Ruffini theorem), so explicit factoring is limited to small matrices or matrices with special structure.

Diagonal and triangular matrices are immediate: the eigenvalues are the diagonal entries. Block triangular matrices factor block by block: the characteristic polynomial is the product of the characteristic polynomials of the diagonal blocks.

For general large matrices, eigenvalues are computed numerically by iterative algorithms — most importantly the QR algorithm, which repeatedly applies QR decompositions to converge on the eigenvalues without ever forming the characteristic polynomial explicitly. Computing the polynomial and then finding its roots is numerically unstable for large

n

and is never used in practice.

Algebraic Multiplicity

\lambda_0

is a root of the characteristic polynomial

p(\lambda)

, its algebraic multiplicity is the largest power

k

such that

(\lambda - \lambda_0)^k

divides

p(\lambda)

. Equivalently, it is the multiplicity of

\lambda_0

as a root.

If

p(\lambda) = (\lambda - 2)^3(\lambda + 1)

, then

\lambda = 2

has algebraic multiplicity

3

and

\lambda = -1

has algebraic multiplicity

1

. The algebraic multiplicities of all eigenvalues sum to

n

— the degree of the polynomial — when complex roots are included.

The algebraic multiplicity is an upper bound for the geometric multiplicity:

1 \leq m_g(\lambda) \leq m_a(\lambda)

. The geometric multiplicity is the dimension of the eigenspace, and it can be strictly smaller than the algebraic multiplicity. When this gap occurs for any eigenvalue, the matrix is not diagonalizable.

Finding Eigenvectors After Finding Eigenvalues

Once the eigenvalues are known, the eigenvectors for each

\lambda_i

are found by solving the homogeneous system

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

.

Row reduce

A - \lambda_i I

and express the general solution in parametric form. Each free variable contributes one basis vector for the eigenspace

E_{\lambda_i}

Worked Example

For

A = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}

, the characteristic polynomial is

\lambda^2 - 4\lambda - 5 = (\lambda - 5)(\lambda + 1)

. Eigenvalues:

\lambda_1 = 5

\lambda_2 = -1

.

For

\lambda_1 = 5

A - 5I = \begin{pmatrix} -4 & 2 \\ 4 & -2 \end{pmatrix}

. Row reducing:

\begin{pmatrix} 1 & -1/2 \\ 0 & 0 \end{pmatrix}

. Free variable

v_2 = t

, so

v_1 = t/2

. Eigenvector:

\mathbf{v}_1 = (1, 2)^T

.

For

\lambda_2 = -1

A + I = \begin{pmatrix} 2 & 2 \\ 4 & 4 \end{pmatrix}

. Row reducing:

\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}

. Free variable

v_2 = t

, so

v_1 = -t

. Eigenvector:

\mathbf{v}_2 = (-1, 1)^T

.

Verification:

A\mathbf{v}_1 = \begin{pmatrix} 5 \\ 10 \end{pmatrix} = 5\mathbf{v}_1

and

A\mathbf{v}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix} = -1 \cdot \mathbf{v}_2

The Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic polynomial. If

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

is the characteristic polynomial, then

p(A) = 0

where

0

is the zero matrix and

\lambda

is replaced by

A

(with constant terms multiplied by

I

).

For example, if

p(\lambda) = \lambda^2 - 5\lambda + 6

, then

A^2 - 5A + 6I = O

. This can be rearranged to express

A^{-1}

as a polynomial in

A

A^{-1} = \frac{1}{6}(5I - A)

(provided

\det(A) = 6 \neq 0

). More generally, the Cayley-Hamilton theorem guarantees that

A^{-1}

can always be written as a polynomial in

A

of degree at most

n - 1

.

The theorem also shows that any power

A^k

with

k \geq n

can be reduced to a polynomial in

A

of degree at most

n - 1

— the characteristic polynomial provides a recurrence that expresses higher powers in terms of lower ones.

Characteristic Polynomial and Similarity

Similar matrices have the same characteristic polynomial:

\det(P^{-1}AP - \lambda I) = \det(P^{-1}(A - \lambda I)P) = \det(A - \lambda I)

The second equality uses the multiplicative property of the determinant:

\det(P^{-1})\det(A - \lambda I)\det(P) = \det(A - \lambda I)

, since

\det(P^{-1})\det(P) = 1

.

This means the characteristic polynomial is a property of the linear transformation itself, not of any particular matrix representation. Changing the basis changes the matrix but not the polynomial. Since the eigenvalues are the roots of the polynomial, similar matrices have the same eigenvalues with the same algebraic multiplicities.

The trace and determinant are just two of the

n

coefficients of the characteristic polynomial. The polynomial carries more information than either one alone — it determines the complete multiset of eigenvalues, not just their sum and product.