Characteristic Equation: Polynomial & Eigenvalues

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:

Characteristic Equation

\det(A - \lambda I) = 0

Characteristic equation →

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?"

Eigenvalue Notation

Notation

Eigenvalue Notation

The Greek letter no one dares rename, the identity matrix that must not be dropped, and the subscript that names a space after a number.

\det

and its bars — determinant notation;

I

and matrix algebra — matrix notation;

\text{Null}

— the four subspaces.

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

A v equals lambda v

\lambda

is the subject's reserved letter — eigenvalues are

\lambda

in every textbook on earth, enumerated

\lambda_1, \lambda_2, \ldots

when there are several. The defining equation reads as a balance: matrix action on the left, plain scaling on the right, same vector both sides.

CasesThe full set of eigenvalues has its own mark in advanced texts: the spectrum

\sigma(A)

— a set-valued operator, sibling to

\text{Col}

and

\text{Null}

.

Also written

\mu

when

\lambda

is spoken for, in perturbation and two-matrix arguments; German texts historically used

\lambda

from the start — the letter is Hilbert-era and stuck.

Do not confuseA matrix.

\lambda

is a scalar — which is exactly why

A - \lambda

is illegal and the next entry's

I

exists.

\det(A - \lambda I) = 0

The determinant of A minus lambda I, set to zero

The

I

is load-bearing: a scalar cannot be subtracted from a matrix, so

\lambda

must first become

\lambda I

— a scaling of the identity. Writing

A - \lambda

is the classic species error; the notation

A - \lambda I

is its permanent correction. The equation itself is derived in From Eigenvectors to the Determinant Condition above.

CasesTwo sign conventions circulate:

\det(A - \lambda I)

— this site's, keeping

A

first — and

\det(\lambda I - A)

, which makes the polynomial monic. They differ by

(-1)^n

; the roots, being what matters, agree.

Also written

p(\lambda)

, or

p_A(\lambda)

with the matrix as subscript — the characteristic polynomial named as a function, per The Characteristic Polynomial below.

Do not confuseAn equation to solve for

A

. The unknown is

\lambda

— the matrix is data, the scalar is the variable, an inversion of the usual roles that the notation quietly performs.

E_{\lambda} = \text{Null}(A - \lambda I)

The eigenspace for lambda

A space named after a number: the subscript on

E_{\lambda}

is the eigenvalue whose eigenvectors (plus

\mathbf{0}

) fill it. One subscripted letter per eigenvalue —

E_3

is the eigenspace for

\lambda = 3

, not a third eigenspace.

CasesEach

\lambda

carries two counted quantities: algebraic multiplicity — its exponent as a root of

p(\lambda)

, the multiplicity notation of polynomials — and geometric multiplicity,

\dim E_{\lambda}

. The pair drives Algebraic Multiplicity below and the whole of diagonalization.

Also written

\ker(A - \lambda I)

— the map-dialect spelling, per kernel notation.

Do not confuseThe standard-basis letter.

\mathbf{e}_i

is a basis vector,

E_{ij}

a matrix unit,

E_{\lambda}

an eigenspace — one letter, three species, separated by bold, double subscripts, and a Greek subscript respectively.

All linear algebra symbols →↑ Eigenvalues and Eigenvectors

The Characteristic Polynomial

The characteristic polynomial of

A

is the determinant of

A - \lambda I

viewed as a function of

\lambda

:

Characteristic Polynomial

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

Characteristic polynomial →

It is a polynomial of degree

n

in the variable

\lambda

.

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}

is

(-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.

The determinant expanded into a polynomial

Taking the determinant of the shifted matrix has produced a polynomial in λ whose degree equals the size of the matrix. Its roots are the eigenvalues, so the question has been converted from a matrix problem into a root-finding problem. Expand it for a matrix of your own on the eigenvalues and eigenvectors visualizer.

Because the degree matches the size, an n×n matrix has exactly n eigenvalues once multiplicity and complex roots are counted.

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)

Rewritten in terms of trace and determinant:

Characteristic Polynomial 2x2

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

Characteristic polynomial 2x2 →

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

.

Discriminant Δ = tr(A)² − 4·det(A)	Eigenvalue type	Independent real eigenvectors
Δ > 0	two distinct real eigenvalues	2 — always diagonalizable
Δ = 0	one repeated real eigenvalue	1 or 2 — diagonalizable only when A = λ · I (otherwise defective)
Δ < 0	complex conjugate pair a ± bi	0 real (one conjugate pair of complex eigenvectors over ℂ)

Roots extracted from the characteristic polynomial

The polynomial has been solved and its roots listed — these are the eigenvalues, and a repeated root is recorded with its multiplicity rather than collapsed into one. That bookkeeping matters later, because algebraic multiplicity and the number of independent eigenvectors need not agree. See the repeated and complex cases on the eigenvalues and eigenvectors visualizer.

Beyond 4×4 no formula exists for these roots, which is why large eigenvalue problems are solved iteratively rather than algebraically.

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.

Matrix structure	How to find the eigenvalues
Diagonal	eigenvalues are the diagonal entries — read off by inspection
Triangular (upper or lower)	eigenvalues are the diagonal entries (det of a triangular matrix is the product of its diagonal)
Block triangular	multiply the characteristic polynomials of each diagonal block; eigenvalues are the union of block eigenvalues
General 2×2	quadratic formula on λ² − tr(A)·λ + det(A) = 0
General 3×3 or 4×4	expand det(A − λI) and factor (rational-root testing, cubic/quartic formulas, or inspection)
General n ≥ 5 (numerical)	iterative algorithms such as QR iteration — the polynomial is never formed explicitly

Algebraic Multiplicity

If

\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

.

Solving the shifted system for one eigenvalue

With a specific eigenvalue substituted, the shifted matrix is being reduced and its null space read off. Any non-zero vector in that space is an eigenvector, and the whole space is the eigenspace — which is why eigenvectors are never unique and are usually quoted normalised. Run the solve for each eigenvalue in turn on the eigenvalues and eigenvectors visualizer.

Finding the eigenvalues is therefore only half the work; each one still has its own null space to compute.

The Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic polynomial. If

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

is the characteristic polynomial, then

Cayley-Hamilton

p(A) = O

Cayley hamilton →

where

O

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.

Summary: The Characteristic Polynomial at a Glance

The characteristic polynomial p(λ) = det(A − λI) carries every spectral fact about A — its degree, individual coefficients, roots, and behavior under similarity all encode information about the matrix. The table below collects each structural property of the polynomial alongside its statement and the connection it makes to A.

The polynomial is usually formed in order to solve it, and the roots are indeed the point. But the coefficients are informative before any solving happens: the constant term is the determinant, the next coefficient is the trace, and both are readable off the matrix directly. The grouping below separates what the coefficients tell you from what the roots tell you, with the closed forms and their consequences last.

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Eigen · characteristic polynomial

What the polynomial encodes

The roots give the eigenvalues, which is the reason to form it. The coefficients give trace and determinant for free, which is the reason it is worth reading rather than only solving.

9readings

What the coefficients carry3

1

Degree and leading term§ 2

the sign alternates with

n

\deg p = n

, leading term

(-1)^n\lambda^n

One root per dimension, counted with multiplicity, over

\mathbb{C}

. Some texts negate the whole polynomial to make it monic — harmless, but it flips the sign of every coefficient below, so the convention has to be fixed before the formulas are quoted.

2

Coefficient of

\lambda^{n-1}

§ 2

with algebraic multiplicity

\sum \lambda_i = \operatorname{tr}(A)

The trace is available without solving anything — it is the sum of the diagonal, and the polynomial says that sum is also the sum of the roots. A cheap consistency check on any eigenvalue computation.

3

Constant term§ 2

always

p(0) = \det(A) = \prod \lambda_i

Setting

\lambda = 0

in

\det(A - \lambda I)

leaves

\det(A)

. So

A

is invertible exactly when

p(0) \neq 0

, which is the same as saying no eigenvalue is zero — three statements that turn out to be one.

What the roots carry3

4

Roots are the eigenvalues§ 1

over

\mathbb{C}

, with multiplicity

p(\lambda) = 0 \iff \lambda

is an eigenvalue

The determinant condition is what turns "

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

has a nontrivial solution" into an equation that can be solved. Over

\mathbb{R}

some roots may not exist — a rotation has none — which is why the spectrum is stated over

\mathbb{C}

.

5

Algebraic multiplicity§ 6

m_g \leq m_a

, never the reverse

multiplicity of

\lambda

as a root

How many times a root repeats, which is not the same as how many independent eigenvectors it has. When the geometric count falls short the matrix is defective and cannot be diagonalized — the polynomial cannot detect this on its own.

6

Similarity invariance§ 9

P

invertible

p_{P^{-1}AP}(\lambda) = p_A(\lambda)

Similar matrices share the polynomial, hence the eigenvalues, trace and determinant. Two matrices with different characteristic polynomials cannot represent the same transformation in different bases — which makes

p

a usable test for similarity in one direction.

Closed forms and consequences3

7

2×2 form§ 3

worth memorising

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

The discriminant

\operatorname{tr}^2 - 4\det

classifies the pair before any solving: positive gives two distinct reals, zero gives a repeated root, negative gives a complex conjugate pair.

8

3×3 form§ 4

M_2

= sum of principal

2 \times 2

minors

-\lambda^3 + \operatorname{tr}(A)\lambda^2 - M_2\lambda + \det(A)

The pattern continues: each coefficient is a sum of principal minors of one size. Beyond

3 \times 3

the expansion is not worth doing by hand, and numerical work does not compute the polynomial at all.

9

Cayley–Hamilton§ 8

every square matrix satisfies its own polynomial

p(A) = O

Substituting the matrix into its own characteristic polynomial gives the zero matrix. The practical consequence is that

A^n

can be rewritten in terms of lower powers, so any power or the inverse is a polynomial in

A

of degree below

n

.

Every entry is a property of the transformation rather than of the matrix expressing it: similar matrices have the same polynomial, so all of trace, determinant and spectrum survive a change of basis. That invariance is what makes the polynomial a description of the map and not merely of the array.

What the polynomial encodes·/linear-algebra/eigen/characteristic-equationLearn Math Class

Two of these are worth using as checks rather than as facts. Since the roots must sum to the trace and multiply to the determinant, both are available from the matrix in seconds and can be compared against any computed spectrum — a mismatch means an arithmetic error, found without redoing the work. For a

2 \times 2

matrix the discriminant

\operatorname{tr}^2 - 4\det

goes further and classifies the eigenvalues before they are computed at all.

The one thing the polynomial cannot report is how many independent eigenvectors an eigenvalue has. Algebraic multiplicity counts root repetition; geometric multiplicity counts eigenvectors, and it can be strictly smaller. That gap is invisible in

p(\lambda)

and only appears when the null space of

A - \lambda I

is actually computed — which is why diagonalizability is a separate question from finding the eigenvalues.

Characteristic Equation FAQ

Why is it $\det(A - \lambda I)$ rather than $\det(A - \lambda)$ ?

+

Because

\lambda

is a scalar and

A

is a matrix, so

A - \lambda

subtracts incompatible objects and is simply not defined. Multiplying by the identity turns the scalar into a matrix of the right shape,

\lambda I

, which subtracts

\lambda

from each diagonal entry and leaves everything else alone. The

I

is doing real work.Read more →

Is it $\det(A - \lambda I)$ or $\det(\lambda I - A)$ ?

+

Both conventions circulate and neither is wrong. They differ by a factor of

(-1)^n

, so the polynomials are not identical, but their roots are the same and the roots are what you are after. The second form has the advantage of coming out monic. Pick one and stay with it inside a single calculation.Read more →

What is the difference between $\mathbf{e}_i$ , $E_{ij}$ and $E_{\lambda}$ ?

+

Three different objects sharing one letter, separated only by their decoration. Bold

\mathbf{e}_i

is a standard basis vector,

E_{ij}

with two subscripts is a matrix unit, and

E_{\lambda}

with a Greek subscript is an eigenspace, a whole subspace rather than a single vector or array.Read more →

Characteristic Equation

The Polynomial Whose Roots Are the Eigenvalues

From Eigenvectors to the Determinant Condition

Eigenvalue Notation

Eigenvalue Notation

The Characteristic Polynomial

Computing the Characteristic Polynomial: 2×2

Worked Example

Computing the Characteristic Polynomial: 3×3

Worked Example

Larger Matrices

Algebraic Multiplicity

Finding Eigenvectors After Finding Eigenvalues

Worked Example

The Cayley-Hamilton Theorem

Characteristic Polynomial and Similarity

Summary: The Characteristic Polynomial at a Glance

What the polynomial encodes

Characteristic Equation FAQ

Why is it det⁡(A−λI)\det(A - \lambda I)det(A−λI) rather than det⁡(A−λ)\det(A - \lambda)det(A−λ)?

Is it det⁡(A−λI)\det(A - \lambda I)det(A−λI) or det⁡(λI−A)\det(\lambda I - A)det(λI−A)?

What is the difference between ei\mathbf{e}_iei​, EijE_{ij}Eij​ and EλE_{\lambda}Eλ​?

Why is it $\det(A - \lambda I)$ rather than $\det(A - \lambda)$ ?

Is it $\det(A - \lambda I)$ or $\det(\lambda I - A)$ ?

What is the difference between $\mathbf{e}_i$ , $E_{ij}$ and $E_{\lambda}$ ?