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Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that the trace is the sum of the eigenvalues by comparing coefficients

Builds on Trace is a coefficient of the characteristic polynomial Characteristic polynomial Degree of a polynomial

Proof

We compute the coefficient of $t^{n-1}$ of $p_A$ in two ways.

Let $A$ be an $n \times n$ matrix over $F$ . The characteristic polynomial is defined for $n \ge 1$ , so $n \ge 1$ . Assume that its characteristic polynomial is $p_A(t) = (t - \lambda_1) \cdots (t - \lambda_n)$ with $\lambda_i \in F$ .
Claim: for $k \ge 1$ , $\prod_{i=1}^{k} (t - \lambda_i) = t^k - (\lambda_1 + \cdots + \lambda_k) t^{k-1} + r_k$ , where $r_k$ is $0$ or has degree at most $k - 2$ .
For $k = 1$ , the claim holds with $r_1 = 0$ .
Suppose the claim holds for $k$ , and let $s = \lambda_1 + \cdots + \lambda_k$ and $c = \lambda_{k+1}$ . Multiply both sides by $t - c$ .
This gives $t^{k+1} - (s + c) t^k + r_{k+1}$ , with $r_{k+1} = c s\, t^{k-1} + t\, r_k - c\, r_k$ .
Each of the three terms of $r_{k+1}$ is $0$ or has degree at most $k - 1$ . So the claim holds for $k + 1$ , and by induction for all $k$ .
By step 2 with $k = n$ , the coefficient of $t^{n-1}$ in $p_A$ is $-(\lambda_1 + \cdots + \lambda_n)$ .
By the rule for the trace as a coefficient, the same coefficient is $-\tr A$ .
By steps 7 and 8, $\tr A = \lambda_1 + \lambda_2 + \cdots + \lambda_n$ .

Idea

Multiplying out $(t - \lambda_1) \cdots (t - \lambda_n)$ gives $t^n - (\lambda_1 + \cdots + \lambda_n) t^{n-1} + \cdots$ . The same coefficient of $t^{n-1}$ is $-\tr A$ when $\det(t I - A)$ is expanded. Comparing the two gives the result.

Reading path

To understand this page from the ground up, read these 106 pages in order. The foundations come first. The path takes the proofs Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof that a matrix with two equal rows has determinant zero, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that the trace is a coefficient of the characteristic polynomial by the Leibniz formula, Proof that an intersection of subgroups is a subgroup, Proof that the determinant of the identity matrix is one, Proof of the decomposition of a permutation into disjoint cycles, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof that the determinant is multilinear in the rows, Proof of the subgroup criterion.

Binary operation definition
Cycle notation concept
Definition by recursion concept
Divisibility definition
Element definition
Group concept
Identity matrix object
Mathematical induction concept
Matrix definition
Multilinear map concept
Natural numbers object
Peano axioms concept
Permutation concept
Proposition concept
Transposition concept
Vector space definition
Identity element definition
Set definition
Ring definition
Subgroup definition
Symmetric group definition
Square matrix definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Linear map definition
Inverse element definition
Empty set object
Family of sets definition
Ordered pair definition
Polynomial ring definition
Transposition definition
Intersection definition
Proof that an intersection of subgroups is a subgroup proof
Union definition
Trace of a matrix definition
Set-builder notation concept
Addition of natural numbers definition
Biconditional definition
The successor function is injective statement
Zero is not a successor statement
Existential quantifier definition
Universal quantifier definition
Multilinear map definition
Proof of the uniqueness of inverses proof
Cartesian product definition
Disjoint sets definition
Degree of a polynomial definition
Subset definition
Proof of the commutativity of addition by induction proof
Axiom of extensionality statement
Union of a family of sets definition
Order on the natural numbers definition
Uniqueness quantifier definition
Alternating multilinear map definition
Uniqueness of inverses statement
Binary relation definition
Partition of a set definition
Determinant definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof of the laws of exponents in a group proof
Proof of the subgroup criterion proof
Proof that a matrix with two equal rows has determinant zero proof
Characteristic polynomial definition
Proof that the determinant of the identity matrix is one proof
Function definition
Proof that the determinant is multilinear in the rows proof
Strong induction statement
Equivalence relation definition
Subgroup criterion statement
Determinant of a matrix with two equal rows statement
Laws of exponents in a group statement
Determinant of the identity matrix statement
Determinant is multilinear in the rows statement
Proof of the well-ordering principle from strong induction proof
Injective function definition
Proof that disjoint cycles commute proof
Intersection of subgroups is a subgroup statement
Equivalence class definition
Well-ordering principle statement
Disjoint cycles commute statement
Subgroup generated by a subset definition
Proof that equivalence classes form a partition proof
Proof of the division algorithm proof
Equivalence classes form a partition statement
Division algorithm statement
Proof of the decomposition of a permutation into disjoint cycles proof
Decomposition of a permutation into disjoint cycles statement
Proof of the decomposition of a permutation into transpositions proof
Decomposition of a permutation into transpositions statement
Sign of a permutation definition
Proof that the symmetric group is generated by transpositions proof
The symmetric group is generated by transpositions statement
Proof that permuting rows multiplies the determinant by the sign proof
Permuting rows multiplies the determinant by the sign statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Leibniz formula for determinants statement
Proof that the trace is a coefficient of the characteristic polynomial by the Leibniz formula proof
Trace is a coefficient of the characteristic polynomial statement
Proof that the trace is the sum of the eigenvalues by comparing coefficients proof

Where this leads

1 page builds on this one.

Trace is the sum of the eigenvalues

Further reading

Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Trace as the sum of eigenvalues

About this page

Tag	`92AK0G8APE`
URI	`m2://92AK0G8APE`
Kind	proof
Writing	W0 stub
MSC	15A15, 15A18
Proves	Trace is the sum of the eigenvalues
Uses	Trace is a coefficient of the characteristic polynomial, Characteristic polynomial, Degree of a polynomial
Changed	4 November 2026

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