8.3 Characteristic polynomials and diagonalization tests

The null-space criterion

$$\det(A-\lambda I)=0$$

tells you how to test a proposed eigenvalue. The next step is to package that test into one polynomial whose roots are exactly the eigenvalues of the matrix.

That polynomial is strong enough to limit how many eigenvalues a matrix can have, describe repeated eigenvalues, and give practical tests for diagonalization.

Throughout this note, the base field is $F=\mathbb{R}$. Unless a complex extension is explicitly mentioned, matrices, vectors, scalars, and eigenvalues are over $F$; every determinant is taken for a square matrix with $n\ge1$. We use $t$ for the polynomial indeterminate, so the convention is $p_A(t)=\det(A-tI_n)$.

Characteristic polynomial

Definition 8.3.1

(Characteristic polynomial)

Let $A\in M_n(F)$ with $F=\mathbb{R}$ and $n\ge1$. The polynomial

$$p_A(t)=\det(A-tI_n)$$

is called the characteristic polynomial of $A$.

The variable $t$ is only an indeterminate. Once you substitute a scalar $\lambda$, the value $p_A(\lambda)$ becomes the determinant of $A-\lambda I$.

Theorem 8.3.2

(Eigenvalues are exactly the roots of the characteristic polynomial)

Let $A$ be an $n\times n$ matrix and $\lambda$ a scalar. Then the following are equivalent:

1. $\lambda$ is an eigenvalue of $A$;
2. $\lambda$ is a root of $p_A(t)$.

This theorem is just the determinant test from the previous note written in a more organized way:

$$\lambda\text{ is an eigenvalue} \iff \det(A-\lambda I)=0 \iff p_A(\lambda)=0.$$

A root finds a value; an eigenspace counts directions

The roots of $p_A(t)=\det(A-tI)$ identify the eigenvalues. The corresponding eigenspace dimensions determine whether they supply enough independent vectors for diagonalization.

Theorem 8.3.3

(Basic shape of the characteristic polynomial)

If $A$ is an $n\times n$ matrix, then $p_A(t)$ is a degree-$n$ polynomial whose leading coefficient is $(-1)^n$ and whose constant term is $\det(A)$.

So the characteristic polynomial is never an arbitrary expression. Its degree is fixed by the matrix size, and its constant term already remembers the determinant.

Proof

(Degree and leading sign from column multilinearity)

Write the columns of $A$ as $a_1,\ldots,a_n$. The columns of $A-tI_n$ are $a_j-t e_j$. Column multilinearity expands the determinant by choosing either $a_j$ or $-t e_j$ in each position. Choosing $-t e_j$ in every column gives

$$(-t)^n\det(I_n)=(-1)^n t^n.$$

Every other choice contains at least one $a_j$, so it has at most $n-1$ factors of $t$. Hence the degree is exactly $n$ and the leading coefficient is $(-1)^n$. The term with no $t$ is $\det(A)$, so $p_A(0)=\det(A)$. This proof keeps the convention $\det(A-tI_n)$; the alternative $\det(tI_n-A)$ is $(-1)^n p_A(t)$, with the same roots and multiplicities but different odd-degree coefficient signs.

First examples

Worked example 8.3.4

(A 2×2 characteristic polynomial)

Let

$$A= \begin{bmatrix} 3&2\\ 3&-2 \end{bmatrix}.$$

Then

$$p_A(t)= \det \begin{bmatrix} 3-t&2\\ 3&-2-t \end{bmatrix} =(3-t)(-2-t)-6.$$

Simplifying gives

$$p_A(t)=t^2-t-12=(t-4)(t+3).$$

The coefficient calculation is worth checking rather than treating the factorization as a pattern to memorize:

$$(3-t)(-2-t)-6=t^2-t-6-6=t^2-t-12.$$

The constant term is $-12$, which agrees with $\det(A)=-12$, and the leading coefficient is $+1$ because this is a 2 by 2 instance of the sign $(-1)^n$. The factors therefore give the roots $4$ and $-3$, and the root theorem turns those scalar roots into the two eigenvalues. Notice that the sign convention does not change this root set: replacing $\det(A-tI)$ by $\det(tI-A)$ multiplies the whole polynomial by $(-1)^n$, rather than changing its zeros. It does change the leading coefficient when $n$ is odd, which is why the convention must be stated before reading coefficient identities such as Cayley-Hamilton. Under our convention $\det(A-tI)$, the constant term remains $\det(A)$, as the shape proof predicted.

Worked example 8.3.5

(A repeated root)

Let

$$B= \begin{bmatrix} 2&3\\ 0&2 \end{bmatrix}.$$

Then

$$p_B(t)= \det \begin{bmatrix} 2-t&3\\ 0&2-t \end{bmatrix} =(2-t)^2=(t-2)^2.$$

So $2$ is the only eigenvalue, but it appears with multiplicity $2$. The factorization alone does not say that there are two independent eigenvectors: that question belongs to the null space of $B-2I$, which is why algebraic and geometric multiplicity must be kept separate.

Algebraic and geometric multiplicity

Repeated roots need extra language.

Definition 8.3.6

(Algebraic multiplicity and geometric multiplicity)

Let $\lambda\in F$ be a root of $p_A(t)$. Its algebraic multiplicity $m_a(\lambda)$ is the greatest positive integer $m$ for which $(t-\lambda)^m$ divides $p_A(t)$. Equivalently, $p_A(t)=(t-\lambda)^m q(t)$ for a polynomial $q$ with $q(\lambda)\ne0$. Its geometric multiplicity $m_g(\lambda)$ is $\dim_F E_A(\lambda)=\dim_F N(A-\lambda I_n)$.

If $p_A$ splits completely over $F$, its distinct roots can be listed as $\lambda_1,\dots,\lambda_s$ and

$$p_A(t)=(-1)^n(t-\lambda_1)^{m_1}\cdots(t-\lambda_s)^{m_s},\qquad\sum_i m_i=n.$$

The factorization and sum require complete splitting over the chosen field; the local multiplicity definition itself does not.

Algebraic multiplicity comes from the polynomial. Geometric multiplicity comes from the null space $N(A-\lambda I)$. They measure different things, and they do not always agree.

Theorem 8.3.7

(Multiplicity inequality)

If $\lambda$ is an eigenvalue of an $n\times n$ matrix $A$, then

$$1\le m_g(\lambda)\le m_a(\lambda)\le n.$$

So every eigenspace has dimension at least $1$, but it can never be larger than the algebraic multiplicity of its eigenvalue.

Proof

(Why geometric multiplicity cannot exceed algebraic multiplicity)

Fix an eigenvalue $\lambda$ and let $g=\dim E_A(\lambda)$. Choose a basis of that eigenspace and extend it to a basis of $F^n$. Let $P$ have this extended basis as columns. The first $g$ columns of $B=P^{-1}AP$ are $\lambda e_1,\dots,\lambda e_g$, so

$$B=\begin{bmatrix}\lambda I_g&C\\0&D\end{bmatrix}.$$

Similarity preserves characteristic polynomials. If $g<n$, the first $g$ columns of $B-tI$ have at most one potentially nonzero entry, namely $\lambda-t$; at $t=\lambda$ that entry is zero, and the repeated column expansion is a polynomial identity giving

$$p_A(t)=\det(B-tI)=(\lambda-t)^g\det(D-tI).$$

Thus $(t-\lambda)^g$ divides $p_A(t)$, so $g\le m_a(\lambda)$. If $g=n$, then $B=\lambda I_n$ and $p_A(t)=(\lambda-t)^n$. This composition uses basis extension and similarity already established earlier and does not assume splitting.

Worked example 8.3.8

(A repeated eigenvalue with too small an eigenspace)

For

$$B= \begin{bmatrix} 2&3\\ 0&2 \end{bmatrix},$$

the characteristic polynomial is $(t-2)^2$, so the eigenvalue $2$ has algebraic multiplicity $2$.

Now solve $(B-2I)x=0$:

$$B-2I= \begin{bmatrix} 0&3\\ 0&0 \end{bmatrix}.$$

So $x_2=0$ and $x_1$ is free. The eigenspace is

$$\operatorname{span}\left\{ \begin{bmatrix} 1\\0 \end{bmatrix} \right\},$$

which has dimension $1$. Therefore

$$m_a(2)=2,\qquad m_g(2)=1.$$

This mismatch is exactly why the matrix is not diagonalizable.

Distinct eigenvalues force independence

Theorem 8.3.9

(Eigenvectors of distinct eigenvalues are linearly independent)

If $v_1,\dots,v_k$ are eigenvectors of $A$ corresponding to pairwise distinct eigenvalues $\lambda_1,\dots,\lambda_k$, then

$$v_1,\dots,v_k$$

are linearly independent.

Proof

(Induction without assuming independence)

Suppose $\lambda_1,\dots,\lambda_s$ are pairwise distinct and $w_j\in E_A(\lambda_j)$. We prove by induction on $s$ that $\sum\limits_{j=1}^s w_j=0$ forces every $w_j=0$, allowing some $w_j$ to be zero. The case $s=1$ is immediate. For the induction step, apply $A-\lambda_s I$ to the relation. The last term vanishes and we obtain

$$\sum_{j=1}^{s-1}(\lambda_j-\lambda_s)w_j=0.$$

Each scaled vector remains in its own eigenspace. The induction hypothesis forces every $(\lambda_j-\lambda_s)w_j$ to be zero; distinctness makes each scalar nonzero, so $w_j=0$ for $j<s$, and the original relation gives $w_s=0$. Grouping a relation by eigenvalue then proves that the union of bases chosen inside the eigenspaces is independent: the grouped vector in each eigenspace is zero, and independence inside that eigenspace kills every coefficient.

This theorem has two major consequences.

Theorem 8.3.10

(Upper bound on the number of distinct eigenvalues)

An $n\times n$ matrix can have at most $n$ distinct eigenvalues.

Theorem 8.3.11

(Distinct eigenvalue test for diagonalizability)

If an $n\times n$ matrix has exactly $n$ distinct eigenvalues, then it is diagonalizable.

The preceding independence result gives both consequences: there can be at most $n$ distinct eigenvalues, and when there are $n$, one nonzero eigenvector from each one is an independent list of length $n$, hence a basis of $F^n$.

The converse is false. A diagonalizable matrix may still have repeated eigenvalues. The identity matrix is the simplest example: every nonzero vector is an eigenvector with eigenvalue $1$, yet the matrix is already diagonal.

A sharper diagonalization criterion

Distinct eigenvalues are a sufficient test, but not the only one.

Theorem 8.3.12

(Dimension-sum test for diagonalizability)

List all distinct eigenvalues of $A$ that lie in the chosen field $F$ as $\lambda_1,\dots,\lambda_s$; the list may be empty. Then $A$ is diagonalizable over $F$ if and only if

$$\dim_F E_A(\lambda_1)+\cdots+\dim_F E_A(\lambda_s)=n.$$

This criterion is a basis count and does not require the characteristic polynomial to split. Choose a basis in each nonzero eigenspace and concatenate these finite lists. The distinct-eigenspace proof shows that the union of eigenspace bases is independent. If the sum is $n$, that union is a basis and diagonalization follows. Conversely, a diagonalizing basis is partitioned by its diagonal eigenvalues; its vectors lie in the corresponding eigenspaces. The independent union of eigenspace bases has dimension at most $n$; its span contains every vector of the diagonalizing basis, so it spans $F^n$ and has dimension at least $n$. Thus the sum is exactly $n$. If there are no eigenvalues in $F$, the sum is $0$, and an $n\ge1$ space cannot be diagonalizable over $F$.

A sharper multiplicity version does require splitting:

Theorem 8.3.13

(Algebraic-geometric multiplicity test)

If $p_A$ splits completely over $F$, then $A$ is diagonalizable over $F$ if and only if $m_g(\lambda)=m_a(\lambda)$ for every eigenvalue $\lambda$ in $F$.

Indeed, complete splitting gives $\sum\limits_i m_a(\lambda_i)=n$, while the multiplicity inequality gives $m_g(\lambda_i)\le m_a(\lambda_i)$ term by term. Thus equality of every pair is equivalent to the dimension sum being $n$, and the preceding criterion applies. Without splitting, the condition is not a safe replacement. Over the reals,

$$R=\begin{bmatrix}1&-1\\1&1\end{bmatrix},\qquad p_R(t)=(t-1)^2+1$$

has no real eigenvalue. The statement “$m_g=m_a$ for every real eigenvalue” would be vacuous, but $R$ is not diagonalizable over $\mathbb R$. Over $\mathbb C$, its roots $1+i$ and $1-i$ are distinct, so the field-qualified criterion applies there.

Worked example 8.3.14

(Repeated eigenvalue but still diagonalizable)

Let

$$A= \begin{bmatrix} 2&1&1\\ 1&2&1\\ 1&1&2 \end{bmatrix}.$$

Put $d=2-t$. Expanding along the first row, the first minor is $\det\begin{bmatrix}d&1\\1&d\end{bmatrix}=d^2-1$ with positive sign. The second minor is $\det\begin{bmatrix}1&1\\1&d\end{bmatrix}=d-1$ and receives the cofactor sign $-$; the third is $\det\begin{bmatrix}1&d\\1&1\end{bmatrix}=1-d$ with sign $+$. Thus the three signed terms are $d(d^2-1)$, $-(d-1)$, and $+(1-d)$, so

$$d(d^2-1)-(d-1)+(1-d)=d^3-3d+2=(d-1)^2(d+2).$$

Substituting back gives

$$p_A(t)=(1-t)^2(4-t)=-(t-1)^2(t-4).$$

The factorization records root $1$ twice and root $4$ once, while the odd degree explains the negative leading coefficient. Since the polynomial has degree three, these are all of its roots over $\mathbb R$, so the later eigenspace count accounts for every real eigenvalue of this matrix, before we inspect eigenspace dimensions.

For $\lambda=1$, solving $(A-I)x=0$ gives $x+y+z=0$, so

$$E_A(1)=\operatorname{span}\left\{\begin{bmatrix}1\\-1\\0\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix}\right\}.$$

For $\lambda=4$, the equations give $x=y=z$, hence $E_A(4)=\operatorname{span}\{(1,1,1)^T\}$. The two vectors in $E_A(1)$ are independent because their second and third coordinates force both coefficients to vanish. The distinct-eigenspace lemma then makes their union with the $E_A(4)$ vector independent, and

$$\dim E_A(1)+\dim E_A(4)=2+1=3.$$

For an audit of the eigenspace calculations,

$$A-I=\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}, \qquad A-4I=\begin{bmatrix}-2&1&1\\1&-2&1\\1&1&-2\end{bmatrix}.$$

The first matrix gives exactly the single equation $x+y+z=0$, so its null space has dimension $2$; the second has row equations equivalent to $x=y=z$, so its null space has dimension $1$. The polynomial and these dimensions therefore agree with the factorization: the repeated root $1$ contributes two independent directions, while root $4$ contributes one. Their union is a basis of $F^3$, which is the concrete diagonalization test in this example.

Thus $A$ is diagonalizable even though $1$ is a repeated root.

Cayley-Hamilton gives a polynomial identity

One further theorem relates the characteristic polynomial back to the matrix itself.

Theorem 8.3.15

(Cayley-Hamilton theorem)

If

$$p_A(t)=c_0+c_1t+c_2t^2+\cdots+c_nt^n,$$

then

$$c_0I+c_1A+c_2A^2+\cdots+c_nA^n=0.$$

Equivalently, $p_A(A)=0$. The theorem is general; the proof supplied here handles the diagonalizable case. For every square matrix, including nondiagonalizable ones, follow the general coefficient-comparison proof in MATH1550. Its prerequisites are the determinant and adjugate identities over the commutative ring $F[t]$, established just before that proof. It uses $\chi_A(t)=\det(tI-A)=(-1)^n p_A(t)$, so $\chi_A(A)=0$ is equivalent to the convention here. Compare coefficients in $(tI-A)\operatorname{adj}(tI-A)=\chi_A(t)I$, then left-multiply by powers of $A$ and sum; do not directly substitute $t=A$ in a matrix-coefficient identity. Return here for the elementary diagonalizable proof and the directly checked Jordan example.

For diagonalizable matrices this is easy to believe: once $A$ is written as $SDS^{-1}$, the identity reduces to applying the scalar polynomial $p_A$ to each diagonal entry of $D$, and every diagonal entry is an eigenvalue, hence a root of $p_A$.

Proof

(Cayley-Hamilton in the diagonalizable case)

Assume $A=SDS^{-1}$ with $D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$. For a scalar polynomial $q(t)=c_0+\cdots+c_rt^r$, define $q(A)=c_0I+c_1A+\cdots+c_rA^r$; this is scalar substitution into matrix powers, not substitution inside a determinant. Since powers respect similarity, $q(A)=S q(D)S^{-1}$, and $q(D)$ is diagonal with entries $q(\lambda_1),\dots,q(\lambda_n)$. Taking $q=p_A$, every diagonal entry is zero by the eigenvalue-root theorem. The constant term uses $I=SIS^{-1}$, so $p_A(A)=0$.

The practical consequence is that high powers of $A$ can be rewritten in terms of lower powers $I,A,\dots,A^{n-1}$. If

$$p_A(t)=c_0+c_1t+\cdots+c_{n-1}t^{n-1}+c_nt^n,$$

then $c_n=(-1)^n$ is nonzero, so the Cayley-Hamilton identity solves for $A^n$. Multiplying that expression by $A$ and reducing again gives an algorithm for all later powers. When $c_0=\det(A)\ne0$, multiplying the identity by $A^{-1}$ and solving gives the explicit formula

$$A^{-1}=-\frac{c_1I+c_2A+\cdots+c_nA^{n-1}}{c_0}.$$

This use is valid only after invertibility has been established.

Worked example 8.3.16

(A small Cayley-Hamilton identity)

Let

$$J= \begin{bmatrix} 1&4\\ 0&1 \end{bmatrix}$$

Its characteristic polynomial is

$$p_J(t)=(1-t)^2=t^2-2t+1$$

Therefore Cayley-Hamilton says

$$J^2-2J+I=0$$

Directly,

$$J^2=\begin{bmatrix}1&8\\0&1\end{bmatrix}, \qquad J^2-2J+I=0$$

Multiplying this verified identity by $J$ gives

$$J^3=2J^2-J=2(2J-I)-J=3J-2I$$

More explicitly, multiplying $J^2-2J+I=0$ by $J^k$ gives the recurrence $J^{k+2}=2J^{k+1}-J^k$ for every $k\ge0$. Thus the next terms are determined without multiplying two large matrices from scratch; for example, $J^4=2J^3-J^2=4J-3I$. This is the concrete reduction mechanism behind the polynomial identity. In fact, for every positive integer $m$,

$$J^m=mJ-(m-1)I=\begin{bmatrix}1&4m\\0&1\end{bmatrix}.$$

The case $m=1$ is the definition of $J$. If the formula holds for $m$, then $J^{m+1}=mJ^2-(m-1)J=m(2J-I)-(m-1)J=(m+1)J-mI$, proving the next case. This makes the reduction concrete even though $J$ is not diagonalizable.

Earlier, the coefficient formula can now be applied concretely: here $c_0=1$, $c_1=-2$, and $c_2=1$, so

$$J^{-1}=-\frac{-2I+J}{1}=2I-J=\begin{bmatrix}1&-4\\0&1\end{bmatrix}.$$

Both cancellation orders can be checked directly:

$$J(2I-J)=2J-J^2=I, \qquad (2I-J)J=2J-J^2=I.$$

Common mistake

Counterexample mode

A repeated eigenvalue can still be diagonalizable

False claim: every repeated eigenvalue prevents diagonalization. Take $A=2I_2$. Then $p_A(t)=(2-t)^2$, so $2$ has algebraic multiplicity $2$, but $E_A(2)=\mathbb R^2$ has basis $e_1,e_2$; the matrix is already diagonal. The earlier matrix $B$ has the same repeated-root pattern but only a one-dimensional eigenspace, so it is not diagonalizable. The repair is to compare $m_g(\lambda)$ with $m_a(\lambda)$ under the stated splitting condition, or to use the unconditioned dimension-sum criterion over the chosen field.

Quick check

Quick check

(What is the characteristic polynomial of $\begin{bmatrix}2&0\\0&5\end{bmatrix}$?)

Use $p_A(t)=\det(A-tI)$.

Solution · Answer

$$p_A(t)= \begin{vmatrix} 2-t&0\\ 0&5-t \end{vmatrix} =(2-t)(5-t).$$

Quick check

(If an $n\times n$ matrix has n distinct eigenvalues, what can you conclude immediately?)

Use the distinct-eigenvalue theorem.

Solution · Answer

It is diagonalizable, because eigenvectors corresponding to distinct eigenvalues are linearly independent.

Quick check

(Can geometric multiplicity ever exceed algebraic multiplicity?)

Use the multiplicity inequality.

Solution · Answer

No. For every eigenvalue $\lambda$,

$$m_g(\lambda)\le m_a(\lambda).$$

Exercises

Quick check

(Find the characteristic polynomial of $\begin{bmatrix}0&1\\-2&3\end{bmatrix}$ and list its eigenvalues.)

Compute $\det(A-tI)$ and factor the result.

Solution · Guided solution

$$p_A(t)= \begin{vmatrix} -t&1\\ -2&3-t \end{vmatrix} =t^2-3t+2=(t-1)(t-2).$$

So the eigenvalues are $1$ and $2$.

Quick check

(A real 4×4 matrix has four distinct real eigenvalues. What is the dimension of the span of one chosen eigenvector from each eigenvalue?)

Use linear independence of eigenvectors with distinct eigenvalues.

Solution · Guided solution

The four chosen eigenvectors are linearly independent, so their span has dimension $4$. Therefore they already form a basis of $\mathbb{R}^4$.

Quick check

(Suppose $p_A(t)=-t^3+6t^2-11t+6$. What matrix identity does Cayley-Hamilton give?)

Replace the variable $t$ by the matrix $A$.

Solution · Guided solution

$$-A^3+6A^2-11A+6I=0,$$

or, after multiplying by $-1$,

$$A^3-6A^2+11A-6I=0.$$

That is the Cayley-Hamilton identity for this characteristic polynomial.

Related notes

Keep 8.1 Eigenvalues, eigenvectors, and eigenspaces open for the null-space interpretation of eigenvalues.

Keep 8.2 Diagonalization and similarity nearby, because this note supplies the polynomial tests that feed that chapter.

The determinant machinery inside $p_A(t)=\det(A-tI)$ depends on 7.1 Determinants and cofactor expansion.