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The Numerical Range of a Matrix The Numerical Range of a Matrix Kristin A. Camenga and Patrick X. Rault, joint work with Dan Rossi, Tsvetanka Sendova, and Ilya M. Spitkovsky Houghton College and SUNY Geneseo January 13, 2015 Rault & Camenga 1/13/15 January 13, 2015 1 / 23 Numerical Range Definition (Numerical Range) Let A be an n × n matrix with entries in C. Then the numerical range of A is given by n ∗ n W (A) = fhAx; xi : x 2 C ; kxk = 1g = fx Ax : x 2 C ; kxk = 1g: Rault & Camenga 1/13/15 January 13, 2015 2 / 23 Some Examples 2 1 0 1 3 2 −1 2 2 3 A = 4 0 i :5 5 B = 4 0 −1 2 5 0 0 −i 0 0 −1 Rault & Camenga 1/13/15 January 13, 2015 3 / 23 W (A) also contains the eigenvalues of A. Proof: If λ is an eigenvalue of A, then we pick a corresponding unit eigenvector, x. Then hAx; xi = hλx; xi = λhx; xi = λ. Basic Properties The numerical range of a matrix, W (A) is a compact and convex subset of C. Rault & Camenga 1/13/15 January 13, 2015 4 / 23 Basic Properties The numerical range of a matrix, W (A) is a compact and convex subset of C. W (A) also contains the eigenvalues of A. Proof: If λ is an eigenvalue of A, then we pick a corresponding unit eigenvector, x. Then hAx; xi = hλx; xi = λhx; xi = λ. Rault & Camenga 1/13/15 January 13, 2015 4 / 23 Unitary Matrices A matrix U is unitary if U∗U = I or U−1 = U∗. If U is a unitary n × n matrix, W (U∗AU) = W (A) for any n × n matrix A. We say W (A) is invariant under unitary similarities. If A is unitarily reducible, that is, unitarily similar to the direct sum A1 ⊕ A2, then W (A) is the convex hull of W (A1) [ W (A2). A1 0 A1 ⊕ A2 = 0 A2 Rault & Camenga 1/13/15 January 13, 2015 5 / 23 Unitarily Reducible Example 2 1 0 0 0 3 1 0 2 i 6 1 2 0 0 7 A1 = ; A2 = ; A1 ⊕ A2 = 6 7 1 2 1 3 4 0 0 2 i 5 0 0 1 3 Rault & Camenga 1/13/15 January 13, 2015 6 / 23 Normal Matrices A square matrix A is normal if A∗A = AA∗. A matrix is normal if and only if it is unitarily similar to a diagonal matrix. In this case, A is unitarily reducible to a direct sum of 1 × 1 matrices which are its eigenvalues, so W (A) is the convex hull of its eigenvalues. 2 2 0 0 0 3 6 0 1 − i 0 0 7 A = 6 7 4 0 0 1 + i 0 5 0 0 0 0 Rault & Camenga 1/13/15 January 13, 2015 7 / 23 Characterizing Shapes: 2 × 2 matrices If A is an irreducible 2 × 2 matrix, W (A) is an ellipse with foci at the eigenvalues. 0 3 A = −1 4 Rault & Camenga 1/13/15 January 13, 2015 8 / 23 Characterizing Shapes: 3 × 3 matrices Rault & Camenga 1/13/15 January 13, 2015 9 / 23 Characterizing Shapes: 4 × 4 Doubly Stochastic matrices A doubly stochastic matrix is one whose real number entries are non-negative and each row and column sums to 1. For example: 2 2 2 1 3 5 0 5 5 2 1 2 6 5 0 5 5 7 6 3 2 7 4 0 5 0 5 5 1 2 2 5 5 5 0 Key fact: Every doubly stochastic matrix is unitarily reducible to the direct sum of the matrix [1] with another matrix: ∗ U AU = [1] ⊕ A1; where U is a unitary matrix with real entries. Rault & Camenga 1/13/15 January 13, 2015 10 / 23 Characterizing Shapes: 4 × 4 Doubly Stochastic matrices Given the characterization of shapes of the numerical range for 3 × 3 T matrices and unitary reducibility U AU = [1] ⊕ A1;, there are three possibilities: W (A1) is the convex hull of a point and an ellipse (with the point lying either inside or outside the ellipse); the boundary of W (A1) contains a flat portion, with the rest of it lying on a 4th degree algebraic curve; W (A1) has an ovular shape, bounded by a 6th degree algebraic curve. All possibilities occur in these three categories and we can characterize which occurs. Rault & Camenga 1/13/15 January 13, 2015 11 / 23 4 × 4 D-S matrices with ellipse-based numerical range Theorem Let A be a 4 × 4 doubly stochastic matrix. Then the boundary of W (A) consists of elliptical arcs and line segments if and only if tr A3 − tr(AT A2) µ := tr A − 1 + tr(AT A) − tr A2 is an eigenvalue of A (multiple, if µ = 1). If, in addition, α = tr A−1−3µ > 0; β = (tr A−1−3µ)2−tr(AT A)+1+2(det A)/µ+µ2 > 0; then W (A) also has corner points at µ and 1, and thus four flat portions on the boundary. Otherwise, 1 is the only corner point of W (A); and @W (A) consists of two flat portions and one elliptical arc. Rault & Camenga 1/13/15 January 13, 2015 12 / 23 4 × 4 D-S examples: ellipses and line segments α = 7=24, β = −59=576 α = 43=96, β = −779=9216 α ≈ 0:65, β ≈ 0:31 α ≈ 0:77, β ≈ 0:21 Rault & Camenga 1/13/15 January 13, 2015 13 / 23 4 × 4 D-S Examples: 4th degree curves and flat portions 4th degree curves and 1 with flat 4th degree curve and 1 with flat portion on the left portion on the right Rault & Camenga 1/13/15 January 13, 2015 14 / 23 4 × 4 D-S Examples: Ovular Rault & Camenga 1/13/15 January 13, 2015 15 / 23 Graphing the boundary of the numerical range iφ Let Aφ = Ae . W (A) vs. W (Aφ) 0.4 0.2 0.2 0.1 f -0.4 -0.2 0.2 0.4 -0.4 -0.2 0.2 0.4 -0.1 -0.2 -0.2 -0.4 Rault & Camenga 1/13/15 January 13, 2015 16 / 23 Alternative (Kippenhahn, 1951): Let F (x : y : t) := det(H0x + K0y + It): Then W (A) is the convex hull of the dual curve to F (x : y : t) = 0. Graphing the boundary of the numerical range iφ Let Aφ = Ae . 0.4 W (A) vs. W (Aφ) ∗ ∗ A Aφ+Aφ Aφ−Aφ W f Hφ := 2 ; Kφ := 2i 0.2 W (Hφ) = Re(W (Aφ)) =[ λmin; λmax]. H L lmin W H f Hφv = λmaxv. f -0.3 -0.2 -0.1 0.1 0.2 0.3 λmax = hHφv; vi = RehAv; vi. lmax hAv; vi2 W (A). H L -0.2 Singularity: W (Aφ) has a vertical flat portion ) λmax has multiplicity 2 -0.4 Rault & Camenga 1/13/15 January 13, 2015 17 / 23 Graphing the boundary of the numerical range iφ Let Aφ = Ae . 0.4 W (A) vs. W (Aφ) ∗ ∗ A Aφ+Aφ Aφ−Aφ W f Hφ := 2 ; Kφ := 2i 0.2 W (Hφ) = Re(W (Aφ)) =[ λmin; λmax]. H L lmin W H f Hφv = λmaxv. f -0.3 -0.2 -0.1 0.1 0.2 0.3 λmax = hHφv; vi = RehAv; vi. lmax hAv; vi2 W (A). H L -0.2 Singularity: W (Aφ) has a vertical flat portion ) λmax has multiplicity 2 -0.4 Alternative (Kippenhahn, 1951): Let F (x : y : t) := det(H0x + K0y + It): Then W (A) is the convex hull of the dual curve to F (x : y : t) = 0. Rault & Camenga 1/13/15 January 13, 2015 17 / 23 The Gau-Wu number Definition (Gau-Wu number, 2013) k(A) = max #fx1;:::; xk g 8j;hAxj ;xj i2@W (A) fx1;:::;xk g orthonormal n fx1;:::;xk }⊂C Basic results: 1 ≤ k(A) ≤ n Points on parallel support lines of W (A) come from orthogonal vectors. Rault & Camenga 1/13/15 January 13, 2015 18 / 23 Vertical flat portion `1 ) pair of orthogonal eigenvectors u; v of H0, with hBu; ui; hBv; vi 2 `1 \ @W (B). Let `2 be a parallel support line, and let hAw; wi 2 `2 \ @W (B). Then w ? u; w ? v. Figure: B 2 M ( ) 3 C Thus k(B) = 3. Basic examples Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2 Figure: A 2 M2(C). k(A) = 2 Rault & Camenga 1/13/15 January 13, 2015 19 / 23 Let `2 be a parallel support line, and let hAw; wi 2 `2 \ @W (B). Then w ? u; w ? v. Thus k(B) = 3. Basic examples Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2 Figure: A 2 M2(C). k(A) = 2 Vertical flat portion `1 ) pair of orthogonal eigenvectors u; v of H0, with hBu; ui; hBv; vi 2 `1 \ @W (B). Figure: B 2 M3(C) Rault & Camenga 1/13/15 January 13, 2015 19 / 23 Thus k(B) = 3. Basic examples Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2 Figure: A 2 M2(C). k(A) = 2 Vertical flat portion `1 ) pair of orthogonal eigenvectors u; v of H0, with hBu; ui; hBv; vi 2 `1 \ @W (B). Let `2 be a parallel support line, and let hAw; wi 2 `2 \ @W (B). Then w ? u; w ? v. Figure: B 2 M3(C) Rault & Camenga 1/13/15 January 13, 2015 19 / 23 Basic examples Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2 Figure: A 2 M2(C).
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