Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof that similar matrices have the same characteristic polynomial

Builds on Characteristic polynomialInvertible matrixMatrix multiplicationAssociativity of matrix multiplicationDistributivity of matrix multiplicationDeterminant is multiplicativeDeterminant of the identity matrixSimilar matricesPolynomial ring

Proof

We write tI−P−1APt I - P^{-1} A P as a conjugate of tI−At I - A and use the product rule.

  1. Let AA and PP be n×nn \times n matrices over KK, with PP invertible. Let Q=P−1Q = P^{-1}, so QP=PQ=InQ P = P Q = I_n.
  2. The ring KK is a subring of the polynomial ring K[t]K[t], which is commutative. So we view AA, PP, QQ as matrices over K[t]K[t]. The products QPQP and PQPQ do not change.
  3. The scalar tt commutes with every matrix in products. So Q(tIn)P=t(QP)=tInQ (t I_n) P = t (Q P) = t I_n, by the definition of the matrix product.
  4. By distributivity and associativity, Q(tIn−A)P=Q(tIn)P−QAP=tIn−QAPQ (t I_n - A) P = Q (t I_n) P - Q A P = t I_n - Q A P.
  5. By the definition of the characteristic polynomial and step 4, pQAP(t)=det⁡(Q(tIn−A)P)p_{QAP}(t) = \det\big(Q (t I_n - A) P\big), a determinant over K[t]K[t].
  6. By the product rule over K[t]K[t], this equals det⁡Q⋅det⁡(tIn−A)⋅det⁡P\det Q \cdot \det(t I_n - A) \cdot \det P.
  7. By commutativity of K[t]K[t] and the product rule, this equals det⁡(QP)det⁡(tIn−A)=det⁡In⋅pA(t)\det(Q P) \det(t I_n - A) = \det I_n \cdot p_A(t).
  8. By the determinant of the identity, det⁡In=1\det I_n = 1. So pP−1AP(t)=pA(t)p_{P^{-1} A P}(t) = p_A(t).
  9. If BB is similar to AA, then B=P−1APB = P^{-1} A P for an invertible PP. By step 8, pB=pAp_B = p_A.

Idea

The matrix tIt I commutes with everything, so tI−P−1AP=P−1(tI−A)Pt I - P^{-1} A P = P^{-1} (t I - A) P. Taking determinants, the factors det⁡P−1\det P^{-1} and det⁡P\det P cancel.

Reading path

To understand this page from the ground up, read these 112 pages in order. The foundations come first. The path takes the proofs Proof of the associativity of matrix multiplication, 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 of the distributivity of matrix multiplication, Proof that the determinant is multiplicative by expanding the rows, 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 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.

  1. Binary operation definition
  2. Cycle notation concept
  3. Definition by recursion concept
  4. Divisibility definition
  5. Element definition
  6. Group concept
  7. Identity matrix object
  8. Mathematical induction concept
  9. Matrix definition
  10. Multilinear map concept
  11. Natural numbers object
  12. Peano axioms concept
  13. Permutation concept
  14. Proposition concept
  15. Transposition concept
  16. Vector space definition
  17. Identity element definition
  18. Set definition
  19. Ring definition
  20. Subgroup definition
  21. Symmetric group definition
  22. Matrix addition definition
  23. Matrix multiplication definition
  24. Square matrix definition
  25. Successor function definition
  26. Conjunction definition
  27. Disjunction definition
  28. Implication definition
  29. Predicate definition
  30. Linear map definition
  31. Inverse element definition
  32. Empty set object
  33. Family of sets definition
  34. Ordered pair definition
  35. Polynomial ring definition
  36. Transposition definition
  37. Proof of the associativity of matrix multiplication proof
  38. Proof that an intersection of subgroups is a subgroup proof
  39. Proof of the distributivity of matrix multiplication proof
  40. Intersection definition
  41. Union definition
  42. Invertible matrix definition
  43. Set-builder notation concept
  44. Addition of natural numbers definition
  45. The successor function is injective statement
  46. Zero is not a successor statement
  47. Biconditional definition
  48. Existential quantifier definition
  49. Universal quantifier definition
  50. Multilinear map definition
  51. Proof of the uniqueness of inverses proof
  52. Cartesian product definition
  53. Disjoint sets definition
  54. Associativity of matrix multiplication statement
  55. Distributivity of matrix multiplication statement
  56. Subset definition
  57. Proof of the commutativity of addition by induction proof
  58. Union of a family of sets definition
  59. Axiom of extensionality statement
  60. Similar matrices definition
  61. Order on the natural numbers definition
  62. Uniqueness quantifier definition
  63. Alternating multilinear map definition
  64. Uniqueness of inverses statement
  65. Binary relation definition
  66. Partition of a set definition
  67. Determinant definition
  68. Commutativity of addition statement
  69. Proof of strong induction from ordinary induction proof
  70. Reflexive relation definition
  71. Symmetric relation definition
  72. Transitive relation definition
  73. Proof of the laws of exponents in a group proof
  74. Proof of the subgroup criterion proof
  75. Proof that a matrix with two equal rows has determinant zero proof
  76. Proof that the determinant of the identity matrix is one proof
  77. Characteristic polynomial definition
  78. Function definition
  79. Proof that the determinant is multilinear in the rows proof
  80. Strong induction statement
  81. Equivalence relation definition
  82. Subgroup criterion statement
  83. Determinant of a matrix with two equal rows statement
  84. Laws of exponents in a group statement
  85. Determinant of the identity matrix statement
  86. Determinant is multilinear in the rows statement
  87. Injective function definition
  88. Proof of the well-ordering principle from strong induction proof
  89. Proof that disjoint cycles commute proof
  90. Intersection of subgroups is a subgroup statement
  91. Equivalence class definition
  92. Well-ordering principle statement
  93. Disjoint cycles commute statement
  94. Subgroup generated by a subset definition
  95. Proof that equivalence classes form a partition proof
  96. Proof of the division algorithm proof
  97. Equivalence classes form a partition statement
  98. Division algorithm statement
  99. Proof of the decomposition of a permutation into disjoint cycles proof
  100. Decomposition of a permutation into disjoint cycles statement
  101. Proof of the decomposition of a permutation into transpositions proof
  102. Decomposition of a permutation into transpositions statement
  103. Sign of a permutation definition
  104. Proof that the symmetric group is generated by transpositions proof
  105. The symmetric group is generated by transpositions statement
  106. Proof that permuting rows multiplies the determinant by the sign proof
  107. Permuting rows multiplies the determinant by the sign statement
  108. Proof of the Leibniz formula from the defining properties of the determinant proof
  109. Leibniz formula for determinants statement
  110. Proof that the determinant is multiplicative by expanding the rows proof
  111. Determinant is multiplicative statement
  112. Proof that similar matrices have the same characteristic polynomial proof

Where this leads

90 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_polynomial, Properties
About this page
TagZWPWDMZKD2
URIm2://ZWPWDMZKD2
Kindproof
WritingW0 stub
MSC15A15, 15A18
ProvesCharacteristic polynomial does not depend on the basis
UsesCharacteristic polynomial, Invertible matrix, Matrix multiplication, Associativity of matrix multiplication, Distributivity of matrix multiplication, Determinant is multiplicative, Determinant of the identity matrix, Similar matrices, Polynomial ring
Changed4 November 2026

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