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REU 2013: Apprentice Program Summer 2013 Lecture 6: July 5, 2013 Madhur Tulsiani 1 Scribe: David Kim Eigenvalues & Eigenvectors Definition 1.1 (Eigenvalue & Eigenvector) Let A ∈ Cn×n . Then λ ∈ C is said to be an eigenvalue of A if ∃v 6= 0 such that Av = λv ⇐⇒ (A − λI)v = 0v . Such a v is an called eigenvector of A with eigenvalue λ. Exercise 1.2 λ is an eigenvalue of A if and only if det(λI − A) = 0 Exercise 1.3 Let Uλ = {v : Av = λv}. Prove: Uλ is a subspace of Cn . Definition 1.4 (Characteristic Polynomial) fA (t) = det(tI − A) is called the characteristic polynomial of A. From the above, we know that λ is an eigenvalue of A iff lambda is a root of the characteristic polynomial fA (t). 1 0 . The characteristic polynomial is fA (t) = det(tI − A) = Example 1.5 Consider A = 0 1 (t − 1)2 . The only eigenvalue is λ = 1, and Uλ = C2 , since Av = Iv = v for all v ∈ C2 . 1 1 Example 1.6 Consider A = . The characteristic polynomial is still fA (t) = det(tI − A) = 0 1 (t − 1)2 and the only eigenvalue is λ = 1. However, Uλ is now only the one-dimensional space 1 Uλ = α· | α∈C . 0 Exercise 1.7 Calculate the eigenvalues of A = 0 1 . −1 0 Note that the eigenvalue of a real matrix may be complex valued. Also, any polynomial p(t) of degree n over C can be factored as c(t − λ1 )...(t − λn ) for c, λ1 , ..., λn ∈ C. Thus, the characteristic polynomial of a matrix always has n (not necessarily distinct) complex roots. 1 Definition 1.8 (Algebraic Multiplicity) The algebraic multiplicity of an eigenvalue λ is the number of times t − λ appears as a factor in the characteristic polynomial fA (t). Definition 1.9 (Geometric Multiplicity) The geometric multiplicity of an eigenvalue λ is dim(Uλ ). Exercise 1.10 Algebraic multiplicity ≥ geometric multiplicity. 1 1 gives an example where the algebraic multiplicity is strictly greater 0 1 than the geometric multiplicity of an eigenvalue. Note that the matrix Exercise 1.11 Let Av = λv for v ∈ Cn , A ∈ Rn×n , λ ∈ R. Then Re(v), Im(v) are also eigenvectors with eigenvalue λ. Thus, if a real matrix A has an eigenvector with a real eigenvalue λ ∈ R, then it also has a real eigenvector with the same eigenvalue. Example 1.12 Calculate eigenvectors, and the characteristic polynomial for the the eigenvalues, cosθ −sinθ . rotation matrix, Rθ = sinθ cosθ ±iθ fRθ (t) = (t − cosθ)2 + (sinθ)2 = 0 gives t = cosθ ± isinθ = e . Let λ1 = cosθ + isinθ, 1 i : α ∈ C}. Note that the λ2 = cosθ − isinθ. Then Uλ1 = {α : α ∈ C}, Uλ2 = {α i 1 eigenvectors do not depend on θ. Exercise 1.13 If v1 , ..., vn are eigenvectors with distinct eigenvalues λ1 , ..., λn , then v1 , ..., vn are linearly independent. Definition 1.14 (Similar Matrices) A and B are similar, A ∼ B, if ∃S ∈ Cn×n such that A = S −1 BS. Definition 1.15 (Diagonalizable Matrices) A is diagonalizable if A = S −1 DS where D is a diagonal matrix. (A is similar to a diagonal matrix). Exercise 1.16 Prove that similarity between matrices is an equivalence relation. Exercise 1.17 Prove: If A ∼ B, then fA (t) = fB (t). Thus, if A ∼ B, then they have the same eigenvalues and each eigenvalue has the same algebraic multiplicity for both A and B. Exercise 1.18 Let A ∼ B. Then for each λ which is an eigenvalue of A (and hence also of (A) (B) B), show that Uλ is isomorphic to Uλ . Thus, each eigenvalue also has the same geometric multiplicity for both A and B. This follows by noting that S −1 BSv = λv ⇒ BSv = λSv. Hence, v 7→ Sv is a bijective linear map (A) (B) from Uλ to Uλ . 2 2 Inner Products and Unitary Matrices Definition 2.1 (Inner Product) Let u, v ∈ Cn . Then the Hermitian inner product of u and v is defined as n X hu, vi = ui vi , i=1 where ui denotes the conjugade of ui . Note that this the same as the usual dot-product if u, v ∈ Rn . P Note that thepquantity hu, ui = i |ui |2 is always non-negative and is zero only when u = 0. We define kuk = hu, ui, which extends the notion of length of a vector, to vectors in Cn . Definition 2.2 (Orthogonal Vectors) Two vectors u, v are said to be orthogonal if hu, vi = 0. Definition 2.3 (Orthonormal Basis) {v1 , ..., vk } is an orthonormal basis for V if it is a basis such that 0 if i = 6 j . ∀i, j hvi , vj i = 1 if i = j 1 1 are orthogonal. After and Example 2.4 For the rotation matrix Rθ , the eigenvectors i i 1 1 scaling, the vectors to √12 · and √12 · , we obtain an orthonormal basis of C2 consisting of i i eigenvectors of Rθ . Definition 2.5 (Unitary Matrix) A matrix U is called a unitary matrix if the columns of U form an orthonormal basis of Cn . Definition 2.6 (Adjoint of a matrix) For a matrix A ∈ Mn (C), its adjoint, denoted as A∗ is T the matrix defined as (A∗ )ij = aji i.e., A∗ = A . Note that it follows from the fact that (AB)T = B T AT that we have (AB)∗ = B ∗ A∗ . Exercise 2.7 U is unitary if and only if U ∗ U = U U ∗ = I. Note that (U ∗ U )ij = U (i) , U (j) , where U (i) denotes the ith column of U . Hence U ∗ U = I if and only if the columns form an orthonormal basis. Also, we have that U ∗U = I ⇔ U U ∗ = I ∗ = I . Exercise 2.8 Show that If U1 , U2 are unitary, then so is U1 U2 . 3