The Numerical Range of a Matrix Kristin A. Camenga and Patrick X. - PowerPoint PPT Presentation

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 W ( A ) = {� Ax , x � : x ∈ C n , � x � = 1 } = { x ∗ Ax : x ∈ C n , � x � = 1 } . Rault & Camenga 1/13/15 January 13, 2015 2 / 23

Some Examples     1 0 1 − 1 2 2 A = 0 i . 5 B = 0 − 1 2     0 0 − i 0 0 − 1 Rault & Camenga 1/13/15 January 13, 2015 3 / 23

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 � Ax , x � = � λ x , x � = λ � x , x � = 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 A 1 ⊕ A 2 , then W ( A ) is the convex hull of W ( A 1 ) ∪ W ( A 2 ). � A 1 � 0 A 1 ⊕ A 2 = A 2 0 Rault & Camenga 1/13/15 January 13, 2015 5 / 23

Unitarily Reducible Example   1 0 0 0 � 1 � 2 � � 0 i 1 2 0 0   A 1 = , A 2 = , A 1 ⊕ A 2 =   1 2 1 3 0 0 2 i   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 0 0 0  0 1 − i 0 0   A =   0 0 1 + i 0   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 1   0 5 5 5 2 1 2 0   5 5 5  3 2  0 0   5 5 1 2 2 0 5 5 5 Key fact: Every doubly stochastic matrix is unitarily reducible to the direct sum of the matrix [1] with another matrix: U ∗ AU = [1] ⊕ A 1 , 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 matrices and unitary reducibility U T AU = [1] ⊕ A 1 , , there are three possibilities: W ( A 1 ) is the convex hull of a point and an ellipse (with the point lying either inside or outside the ellipse); the boundary of W ( A 1 ) contains a flat portion, with the rest of it lying on a 4th degree algebraic curve; W ( A 1 ) 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 µ := tr A − 1 + tr A 3 − tr( A T A 2 ) tr( A T A ) − tr A 2 is an eigenvalue of A (multiple, if µ = 1 ). If, in addition, α = tr A − 1 − 3 µ > 0 , β = (tr A − 1 − 3 µ ) 2 − tr( A T A )+1+2(det A ) /µ + µ 2 > 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 Let A φ = Ae i φ . W ( A ) vs. W ( A φ ) 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

Graphing the boundary of the numerical range Let A φ = Ae i φ . W ( A ) vs. W ( A φ ) 0.4 W H A f L A φ + A ∗ A φ − A ∗ H φ := φ , K φ := φ 2 2 i 0.2 W ( H φ ) = Re( W ( A φ )) = [ λ min , λ max ]. W H H f L l min f H φ v = λ max v . - 0.3 - 0.2 - 0.1 0.1 0.2 l max 0.3 λ max = � H φ v , v � = Re � Av , v � . � Av , v � ∈ W ( A ). - 0.2 Singularity: W ( A φ ) has a vertical flat portion ⇒ λ max has multiplicity 2 - 0.4 Rault & Camenga 1/13/15 January 13, 2015 17 / 23

Graphing the boundary of the numerical range Let A φ = Ae i φ . W ( A ) vs. W ( A φ ) 0.4 W H A f L A φ + A ∗ A φ − A ∗ H φ := φ , K φ := φ 2 2 i 0.2 W ( H φ ) = Re( W ( A φ )) = [ λ min , λ max ]. W H H f L l min f H φ v = λ max v . - 0.3 - 0.2 - 0.1 0.1 0.2 l max 0.3 λ max = � H φ v , v � = Re � Av , v � . � Av , v � ∈ W ( A ). - 0.2 Singularity: W ( A φ ) has a vertical flat portion ⇒ λ max has multiplicity 2 - 0.4 Alternative (Kippenhahn, 1951): Let F ( x : y : t ) := det( H 0 x + K 0 y + 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 # { x 1 , . . . , x k } ∀ j , � A x j , x j �∈ ∂ W ( A ) { x 1 ,..., x k } orthonormal { x 1 ,..., x k }⊂ C n 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

Basic examples Basic results: 1 ≤ k ( A ) ≤ n k ( A ) ≥ 2 if n ≥ 2 Figure: A ∈ M 2 ( C ). k ( A ) = 2 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 ∈ M 2 ( C ). k ( A ) = 2 Vertical flat portion ℓ 1 ⇒ pair of orthogonal eigenvectors u , v of H 0 , with � Bu , u � , � Bv , v � ∈ ℓ 1 ∩ ∂ W ( B ). Figure: B ∈ M 3 ( 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 ∈ M 2 ( C ). k ( A ) = 2 Vertical flat portion ℓ 1 ⇒ pair of orthogonal eigenvectors u , v of H 0 , with � Bu , u � , � Bv , v � ∈ ℓ 1 ∩ ∂ W ( B ). Let ℓ 2 be a parallel support line, and let � Aw , w � ∈ ℓ 2 ∩ ∂ W ( B ). Then w ⊥ u , w ⊥ v . Figure: B ∈ M 3 ( 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 ∈ M 2 ( C ). k ( A ) = 2 Vertical flat portion ℓ 1 ⇒ pair of orthogonal eigenvectors u , v of H 0 , with � Bu , u � , � Bv , v � ∈ ℓ 1 ∩ ∂ W ( B ). Let ℓ 2 be a parallel support line, and let � Aw , w � ∈ ℓ 2 ∩ ∂ W ( B ). Then w ⊥ u , w ⊥ v . Figure: B ∈ M 3 ( C ) Thus k ( B ) = 3. Rault & Camenga 1/13/15 January 13, 2015 19 / 23

Example 1 of an irreducible 4 × 4 matrix 1.0 0.5 Let A ∈ M 4 ( C ) irreducible, with F ( x : y : t ) = 0 a curve having two nodes. - 1.0 - 0.5 0.5 1.0 Dual curve: - 0.5 - 1.0 Vertical flat portion ℓ 1 ⇒ pair of orthogonal eigenvectors u , v of H 0 , with � Au , u � , � Av , v � ∈ ℓ 1 ∩ ∂ W ( A ). Let ℓ 2 be a parallel support line, and let � Aw , w � ∈ ℓ 2 ∩ ∂ W ( A ). Then w ⊥ u , w ⊥ v . Rault & Camenga 1/13/15 January 13, 2015 20 / 23

Example 1 of an irreducible 4 × 4 matrix 1.0 0.5 Let A ∈ M 4 ( C ) irreducible, with F ( x : y : t ) = 0 a curve having two nodes. - 1.0 - 0.5 0.5 1.0 Dual curve: - 0.5 - 1.0 Vertical flat portion ℓ 1 ⇒ pair of orthogonal eigenvectors u , v of H 0 , with � Au , u � , � Av , v � ∈ ℓ 1 ∩ ∂ W ( A ). Let ℓ 2 be a parallel support line, and let � Aw , w � ∈ ℓ 2 ∩ ∂ W ( A ). Then w ⊥ u , w ⊥ v . Thus k ( A ) = 4. Rault & Camenga 1/13/15 January 13, 2015 20 / 23