Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof of the Cayley–Hamilton theorem for linear maps

Builds on Characteristic polynomial of a linear mapPolynomial of an endomorphismMatrix of a linear mapMatrix of a composition is the product of matricesIsomorphism with the coordinate spaceCayley–Hamilton theorem

Proof

We reduce the statement to the Cayley–Hamilton theorem for matrices.

  1. If V=0V = 0, then every endomorphism of VV is the zero map, so pT(T)=0p_T(T) = 0. From now on let n=dim⁡V≥1n = \dim V \ge 1.
  2. Choose a basis B=(b1,…,bn)B = (b_1, \dots, b_n) of VV. For an endomorphism SS of VV, let [S]B[S]_B be its matrix in BB. Let M=[T]BM = [T]_B.
  3. Column jj of [S]B[S]_B is the coordinate vector [S(bj)]B[S(b_j)]_B. The coordinate map is linear by the isomorphism with the coordinate space. So [S+S′]B=[S]B+[S′]B[S + S']_B = [S]_B + [S']_B and [cS]B=c[S]B[c S]_B = c [S]_B for c∈Fc \in F.
  4. By the matrix of a composition, [S∘S′]B=[S]B[S′]B[S \circ S']_B = [S]_B [S']_B. Also [id⁡V]B=In[\id_V]_B = I_n, since id⁡V(bj)=bj\id_V(b_j) = b_j.
  5. By step 4 and induction on kk, [Tk]B=Mk[T^k]_B = M^k for all k≥0k \ge 0.
  6. Let p(x)=∑kckxk∈F[x]p(x) = \sum_k c_k x^k \in F[x]. By the definition of the value at TT and steps 3 and 5, [p(T)]B=∑kckMk=p(M)[p(T)]_B = \sum_k c_k M^k = p(M).
  7. By the definition of pTp_T, pT=pMp_T = p_M, the characteristic polynomial of MM.
  8. By the Cayley–Hamilton theorem for matrices, applied over the field FF, pM(M)=0p_M(M) = 0.
  9. By steps 6, 7 and 8, [pT(T)]B=pM(M)=0[p_T(T)]_B = p_M(M) = 0.
  10. So every column of [pT(T)]B[p_T(T)]_B is zero, that is, pT(T)(bj)=0p_T(T)(b_j) = 0 for each jj.
  11. A linear map that vanishes on a basis vanishes on every linear combination of it. Hence pT(T)=0p_T(T) = 0.

Idea

Choosing a basis turns endomorphisms into matrices, and this translation respects sums, scalar multiples and composition. So it sends p(T)p(T) to p(M)p(M) for every polynomial pp. The matrix theorem says pM(M)=0p_M(M) = 0, and a map with zero matrix is zero.

Reading path

To understand this page from the ground up, read these 155 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 Laplace expansion along a column by transposition, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that the matrix of a composition is the product of the matrices, Proof that permuting rows multiplies the determinant by the sign, Proof that disjoint cycles commute, Proof that a function is bijective iff it has a two-sided inverse, Proof that a matrix with two equal rows has determinant zero, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof that an intersection of subgroups is a subgroup, Proof that the sign is a homomorphism, Proof that the determinant of the identity matrix is one, Proof of the decomposition of a permutation into disjoint cycles, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the sign of a cycle, 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, Proof of the Cayley–Hamilton theorem with the adjugate.

  1. Basis concept
  2. Binary operation definition
  3. Composition of linear maps concept
  4. Coordinate space object
  5. Coordinate vector concept
  6. Cycle notation concept
  7. Definition by recursion concept
  8. Divisibility definition
  9. Element definition
  10. Group concept
  11. Identity matrix object
  12. Linear isomorphism concept
  13. Linear map concept
  14. Mathematical induction concept
  15. Matrix definition
  16. Matrix multiplication concept
  17. Matrix of a linear map concept
  18. Multilinear map concept
  19. Natural numbers object
  20. Peano axioms concept
  21. Permutation concept
  22. Permutation definition
  23. Proposition concept
  24. Transposition concept
  25. Vector space definition
  26. Identity element definition
  27. Set definition
  28. Group homomorphism definition
  29. Proof that the matrix of a composition is the product of the matrices proof
  30. Subgroup definition
  31. Symmetric group definition
  32. Matrix multiplication definition
  33. Square matrix definition
  34. Transpose definition
  35. Successor function definition
  36. Conjunction definition
  37. Disjunction definition
  38. Implication definition
  39. Predicate definition
  40. Linear combination definition
  41. Linear map definition
  42. Linear subspace definition
  43. Inverse element definition
  44. Empty set object
  45. Family of sets definition
  46. Ordered pair definition
  47. Transposition definition
  48. Intersection definition
  49. Proof that an intersection of subgroups is a subgroup proof
  50. Union definition
  51. Set-builder notation concept
  52. Addition of natural numbers definition
  53. Biconditional definition
  54. Existential quantifier definition
  55. Universal quantifier definition
  56. Linear independence definition
  57. The successor function is injective statement
  58. Zero is not a successor statement
  59. Composition of linear maps definition
  60. Endomorphism definition
  61. Linear isomorphism definition
  62. Linear span definition
  63. Multilinear map definition
  64. Cartesian product definition
  65. Disjoint sets definition
  66. Proof of the uniqueness of inverses proof
  67. Union of a family of sets definition
  68. Axiom of extensionality statement
  69. Subset definition
  70. Proof of the commutativity of addition by induction proof
  71. Uniqueness quantifier definition
  72. Order on the natural numbers definition
  73. Polynomial of an endomorphism definition
  74. Spanning set definition
  75. Alternating multilinear map definition
  76. Binary relation definition
  77. Partition of a set definition
  78. Uniqueness of inverses statement
  79. Commutativity of addition statement
  80. Determinant definition
  81. Proof of strong induction from ordinary induction proof
  82. Basis definition
  83. Reflexive relation definition
  84. Symmetric relation definition
  85. Transitive relation definition
  86. Function definition
  87. Proof that a matrix with two equal rows has determinant zero proof
  88. Proof of the subgroup criterion proof
  89. Characteristic polynomial definition
  90. Proof of the laws of exponents in a group proof
  91. Coordinate vector definition
  92. Proof that the determinant of the identity matrix is one proof
  93. Strong induction statement
  94. Minor definition
  95. Proof that the determinant is multilinear in the rows proof
  96. Equivalence relation definition
  97. Codomain definition
  98. Function composition definition
  99. Identity function definition
  100. Injective function definition
  101. Determinant of a matrix with two equal rows statement
  102. Subgroup criterion statement
  103. Determinant of the identity matrix statement
  104. Determinant is multilinear in the rows statement
  105. Laws of exponents in a group statement
  106. Proof of the well-ordering principle from strong induction proof
  107. Matrix of a linear map definition
  108. Cofactor definition
  109. Surjective function definition
  110. Left inverse definition
  111. Right inverse definition
  112. Proof that disjoint cycles commute proof
  113. Equivalence class definition
  114. Intersection of subgroups is a subgroup statement
  115. Matrix of a composition is the product of matrices statement
  116. Adjugate matrix definition
  117. Well-ordering principle statement
  118. Characteristic polynomial of a linear map definition
  119. Bijective function definition
  120. Proof that equivalence classes form a partition proof
  121. Disjoint cycles commute statement
  122. Subgroup generated by a subset definition
  123. Proof of the division algorithm proof
  124. Equivalence classes form a partition statement
  125. Proof that a function is bijective iff it has a two-sided inverse proof
  126. Division algorithm statement
  127. Bijective iff it has a two-sided inverse statement
  128. Proof of the decomposition of a permutation into disjoint cycles proof
  129. Decomposition of a permutation into disjoint cycles statement
  130. Proof that the coordinate map is a linear isomorphism proof
  131. Proof of the decomposition of a permutation into transpositions proof
  132. Isomorphism with the coordinate space statement
  133. Decomposition of a permutation into transpositions statement
  134. Sign of a permutation definition
  135. Proof that the symmetric group is generated by transpositions proof
  136. The symmetric group is generated by transpositions statement
  137. Proof that the sign is a homomorphism proof
  138. Proof of the sign of a cycle proof
  139. Sign of a cycle statement
  140. Sign is a homomorphism statement
  141. Proof that permuting rows multiplies the determinant by the sign proof
  142. Permuting rows multiplies the determinant by the sign statement
  143. Proof of the Leibniz formula from the defining properties of the determinant proof
  144. Leibniz formula for determinants statement
  145. Proof of the Laplace expansion along a row by the Leibniz formula proof
  146. Proof that the determinant of a transpose equals the determinant proof
  147. Determinant of a transpose statement
  148. Laplace expansion along a row statement
  149. Proof of the Laplace expansion along a column by transposition proof
  150. Laplace expansion along a column statement
  151. Proof of the product of a matrix and its adjugate by Laplace expansion proof
  152. Product of a matrix and its adjugate statement
  153. Proof of the Cayley–Hamilton theorem with the adjugate proof
  154. Cayley–Hamilton theorem statement
  155. Proof of the Cayley–Hamilton theorem for linear maps proof

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cayley–Hamilton_theorem, lead section
About this page
TagRSNG3V23MA
URIm2://RSNG3V23MA
Kindproof
WritingW0 stub
MSC15A04, 15A18
ProvesCayley–Hamilton theorem for linear maps
UsesCharacteristic polynomial of a linear map, Polynomial of an endomorphism, Matrix of a linear map, Matrix of a composition is the product of matrices, Isomorphism with the coordinate space, Cayley–Hamilton theorem
Changed4 November 2026

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