Eigenvectors. Prop-Defn

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1 Eigenvectors Aim lecture: The simplest T -invariant subspaces are 1-dim & these give rise to the theory of eigenvectors. To compute these we introduce the similarity invariant, the characteristic polynomial. Prop-Defn Let T : V V be linear. The following are equivalent condns on v V. 1 F v is a T -invariant (automatically 1-dimensional) subspace. 2 T v = λv for some λ F. 3 v ker(t λ id ) for some λ F. If these hold & furthermore v 0 we say v is an eigenvector for T with eigenvalue λ. Proof. Clearly 2) 3) since T v = λv T v λ id v = 0 v ker(t λi ). 2) = 1) by lemma on testing invariance, lecture 20. For 1) = 2) note Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

2 Eigenvalues & eigenspaces of an endomorphism Defn Let T : V V be linear. An eigenvalue of T is a scalar λ F such that there is an e-vector v for T with eigenvalue λ. Given such an e-value, the λ-eigenspace of T is the subspace E λ = ker(t λ id ) V. The geometric multiplicity of λ is dim E λ. E.g. Any real number λ is an e-value for T = d dx : C (R) C (R) since Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

3 Characteristic polynomial of square matrices To find e-values of square matrices we need Prop-Defn Let A = (a ij ) M nn (F). The characteristic polynomial of A is the function cp A (λ) = det(a λi n ) where I n is the n n-identity matrix. This function is a polynomial function of degree n with co-efficients in F. Proof. Note that A λi n = (a ij ) where a ij = a ij if i j whilst a ii = a ii λ. Now det(a λi n ) = σ S n sgn(σ) which is clearly a polynomial function of λ. n i=1 a i σ(i) The summand corresponding to σ = id is (a 11 λ)(a 22 λ)... (a nn λ) which has degree n. Any other summand contains at least two non-diagonal entries so has degree n 2. Hence, deg cp A (λ) = n. Scholium The co-efficient of λ n 1 in cp A (λ) is ( 1) n 1 i a ii. Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

4 Relation with e-values. Algebraic multiplicity Prop-defn Let A M nn (F). Then λ is an e-value for A iff it is a root of the characteristic polynomial of A. In this case, the multiplicity of the root is called the algebraic multiplicity of the e-value. Proof. λ is an e-value iff A has an e-vector with e-value λ iff ker(a λi n ) 0 iff A λi n is not invertible iff det(a λi n ) = 0. E.g. Find the e-values of A = ( ) & their algebraic & geometric multiplicities. Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

5 Similar matrices Prop-Defn Two matrices A, B M nn (F) are similar if there exists C GL n (F) such that A = C 1 BC i.e. A is a matrix representing B wrt some co-ordinate system. Being similar is an equivalence relation. The set of all matrices similar to A is called the similarity class of A. Proof. Omitted. E.g. A = ( ) ( , B = 2 1 ) 1 1 are not similar for if A = C 1 BC then Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

6 Similarity invariants Defn A function of the form f : M nn (F) X is a similarity invariant if f (A) = f (B) whenever A, B are similar. E.g. 1 The characteristic polynomial cp : M nn (F) F[λ] : A cp A (λ) is a similarity invariant since if B = C 1 AC for some C GL n (F) then det(b λi n ) = det(c 1 AC λi n ) = det(c 1 [A λi n ]C) = E.g. 2 In particular, any of the co-efficients of the characteristic polynomial are similarity invariants. The important ones (up to sign) are the determinant det(a) = cp A (0) & the trace which is defined to be tr (A) = n i=1 a ii where A = (a ij ). E.g. 3 Similarly, the set of eigenvalues is a similarity invariant. Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

7 Characteristic polynomials of endomorphisms Similarity invariants can be extended to endomorphisms of finite dimensional vector spaces. For example Prop-Defn Let V = fin dim F-space & T : V V be linear. For any co-ordinate system C : F n V, we may define the characteristic polynomial of T, denoted cp T (λ), to be the characteristic polynomial of the representing matrix C 1 T C. This is well-defined since given any other co-ordinate system C 1 : F n V, the characteristic polynomials of C 1 T C & C 1 1 T C 1 are the same, so the definition is independent of the choice of co-ord system. Proof. We need only check equality of characteristic polynomials by showing C 1 T C & C 1 1 T C 1 are similar. Indeed C 1 1 T C 1 Rem We similarly can define det(t ), tr (T ) etc. E.g. We have seen that T = d dx : R cos x R sin x R cos x R sin x is represented by the matrix Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

8 Computing e-values & e-vectors of endomorphisms Using co-ordinates, we can calculate e-vectors, e-values & e-spaces using Prop Let V = fin dim F-space & T : V V be linear. Let C : F n V be a co-ord system & A = C 1 T C be the representing matrix. Then 1 x F n is an e-vector of A with e-value λ iff Cx is an e-vector of T with e-value λ. 2 The e-values of T & A are the same. They are the roots of cp A (λ) = cp T (λ). 3 If E λ is the λ-e-space of A then C(E λ ) is the λ-e-space of T. Proof. We just prove 1), as 2) & 3) readily then follow. Ax = λx C 1 T Cx = λx Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

9 Example E.g. We compute the e-vectors & e-values of T : R[x] 2 R[x] 2 defined by (Tp)(x) = xp (x) 2p (x) p(x). Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

10 Example continued Daniel Chan (UNSW) Lecture 21: Eigenvectors & similarity invariants Semester / 10

Jordan blocks. Defn. Let λ F, n Z +. The size n Jordan block with e-value λ is the n n upper triangular matrix. J n (λ) =

Jordan blocks. Defn. Let λ F, n Z +. The size n Jordan block with e-value λ is the n n upper triangular matrix. J n (λ) = Jordan blocks Aim lecture: Even over F = C, endomorphisms cannot always be represented by a diagonal matrix. We give Jordan s answer, to what is the best form of the representing matrix. Defn Let λ F,