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

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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

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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 ∈ Cn, x = 1} = {x∗Ax : x ∈ Cn, x = 1}.

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Some Examples

A =   1 1 i .5 −i   B =   −1 2 2 −1 2 −1  

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Basic Properties

The numerical range of a matrix, W (A) is a compact and convex subset of C.

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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 = λx, x = λx, x = λ.

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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 ⊕ A2 = A1 A2

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Unitarily Reducible Example

A1 = 1 1 2

, A2 =

2 i 1 3

, A1 ⊕ A2 =

    1 1 2 2 i 1 3    

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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. A =     2 1 − i 1 + i    

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Characterizing Shapes: 2 × 2 matrices

If A is an irreducible 2 × 2 matrix, W (A) is an ellipse with foci at the eigenvalues. A =

3

−1 4

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Characterizing Shapes: 3 × 3 matrices

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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 5 2 5 1 5 2 5 1 5 2 5 3 5 2 5 1 5 2 5 2 5

    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.

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Characterizing Shapes: 4 × 4 Doubly Stochastic matrices

Given the characterization of shapes of the numerical range for 3 × 3 matrices and unitary reducibility UTAU = [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.

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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 A3 − tr(ATA2) tr(ATA) − 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(ATA)+1+2(det A)/µ+µ2 > 0, then W (A) also has corner points at µ and 1, and thus four flat portions

n the boundary. Otherwise, 1 is the only corner point of W (A); and

∂W (A) consists of two flat portions and one elliptical arc.

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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

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4 × 4 D-S Examples: 4th degree curves and flat portions

4th degree curves and 1 with flat portion on the left 4th degree curve and 1 with flat portion on the right

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4 × 4 D-S Examples: Ovular

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Graphing the boundary of the numerical range

Let Aφ = Aeiφ. W (A) vs. W (Aφ)

0.4
0.2

0.2 0.4

0.2
0.1

0.1 0.2

f

0.4
0.2

0.2 0.4

0.4
0.2

0.2 0.4

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Graphing the boundary of the numerical range

Let Aφ = Aeiφ. W (A) vs. W (Aφ) Hφ :=

Aφ+A∗

φ

2

, Kφ :=

Aφ−A∗

φ

2i

W (Hφ) = Re(W (Aφ)) = [λmin, λmax]. Hφv = λmaxv. λmax = Hφv, v = ReAv, v. Av, v ∈ W (A). Singularity: W (Aφ) has a vertical flat portion ⇒ λmax has multiplicity 2

f lmin lmax WHAfL WHHfL

0.3
0.2
0.1

0.1 0.2 0.3

0.4
0.2

0.2 0.4

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Graphing the boundary of the numerical range

Let Aφ = Aeiφ. W (A) vs. W (Aφ) Hφ :=

Aφ+A∗

φ

2

, Kφ :=

Aφ−A∗

φ

2i

W (Hφ) = Re(W (Aφ)) = [λmin, λmax]. Hφv = λmaxv. λmax = Hφv, v = ReAv, v. Av, v ∈ W (A). Singularity: W (Aφ) has a vertical flat portion ⇒ λmax has multiplicity 2

f lmin lmax WHAfL WHHfL

0.3
0.2
0.1

0.1 0.2 0.3

0.4
0.2

0.2 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.

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The Gau-Wu number

Definition (Gau-Wu number, 2013) k(A) = max

∀j,Axj,xj∈∂W (A) {x1,...,xk} orthonormal {x1,...,xk}⊂Cn

#{x1, . . . , xk} Basic results: 1 ≤ k(A) ≤ n Points on parallel support lines of W (A) come from orthogonal vectors.

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Basic examples

Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2

Figure: A ∈ M2(C). k(A) = 2

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Basic examples

Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2

Figure: A ∈ M2(C). k(A) = 2 Figure: B ∈ M3(C)

Vertical flat portion ℓ1 ⇒ pair of

rthogonal eigenvectors u, v of H0, with

Bu, u, Bv, v ∈ ℓ1 ∩ ∂W (B).

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Basic examples

Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2

Figure: A ∈ M2(C). k(A) = 2 Figure: B ∈ M3(C)

Vertical flat portion ℓ1 ⇒ pair of

rthogonal eigenvectors u, v of H0, with

Bu, u, Bv, v ∈ ℓ1 ∩ ∂W (B). Let ℓ2 be a parallel support line, and let Aw, w ∈ ℓ2 ∩ ∂W (B). Then w ⊥ u, w ⊥ v.

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Basic examples

Basic results: 1 ≤ k(A) ≤ n k(A) ≥ 2 if n ≥ 2

Figure: A ∈ M2(C). k(A) = 2 Figure: B ∈ M3(C)

Vertical flat portion ℓ1 ⇒ pair of

rthogonal eigenvectors u, v of H0, with

Bu, u, Bv, v ∈ ℓ1 ∩ ∂W (B). Let ℓ2 be a parallel support line, and let Aw, w ∈ ℓ2 ∩ ∂W (B). Then w ⊥ u, w ⊥ v. Thus k(B) = 3.

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Example 1 of an irreducible 4 × 4 matrix

Let A ∈ M4(C) irreducible, with F(x : y : t) = 0 a curve having two nodes. Dual curve:

1.0
0.5

0.5 1.0

1.0
0.5

0.5 1.0

Vertical flat portion ℓ1 ⇒ pair of orthogonal eigenvectors u, v of H0, with Au, u, Av, v ∈ ℓ1 ∩ ∂W (A). Let ℓ2 be a parallel support line, and let Aw, w ∈ ℓ2 ∩ ∂W (A). Then w ⊥ u, w ⊥ v.

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Example 1 of an irreducible 4 × 4 matrix

Let A ∈ M4(C) irreducible, with F(x : y : t) = 0 a curve having two nodes. Dual curve:

1.0
0.5

0.5 1.0

1.0
0.5

0.5 1.0

Vertical flat portion ℓ1 ⇒ pair of orthogonal eigenvectors u, v of H0, with Au, u, Av, v ∈ ℓ1 ∩ ∂W (A). Let ℓ2 be a parallel support line, and let Aw, w ∈ ℓ2 ∩ ∂W (A). Then w ⊥ u, w ⊥ v. Thus k(A) = 4.

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Example 2 of an irreducible 4 × 4 matrix

Let A ∈ M4(C) irreducible, with F(x : y : t) = 0 a curve having three nodes. Dual curve:

2

2 4

4
2

2 4

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Example 2 of an irreducible 4 × 4 matrix

Let A ∈ M4(C) irreducible, with F(x : y : t) = 0 a curve having three nodes. Dual curve:

2

2 4

4
2

2 4

Lemma (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + A∗

φ)/2.

S = {φ : Hφ has a maximum e-value of multiplicity ≥ 2. Then:

1 ∀φ ∈ S, Hφ has exactly three distinct eigenvalues ⇒ k(A) < 4.

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Example 2 of an irreducible 4 × 4 matrix

Let A ∈ M4(C) irreducible, with F(x : y : t) = 0 a curve having three nodes. Dual curve:

2

2 4

4
2

2 4

Lemma (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + A∗

φ)/2.

S = {φ : Hφ has a maximum e-value of multiplicity ≥ 2. Then:

1 ∀φ ∈ S, Hφ has exactly three distinct eigenvalues ⇒ k(A) < 4. 2 S = ∅ ⇒ k(A) > 2.

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Example 2 of an irreducible 4 × 4 matrix

Let A ∈ M4(C) irreducible, with F(x : y : t) = 0 a curve having three nodes. Dual curve:

2

2 4

4
2

2 4

Lemma (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + A∗

φ)/2.

S = {φ : Hφ has a maximum e-value of multiplicity ≥ 2. Then:

1 ∀φ ∈ S, Hφ has exactly three distinct eigenvalues ⇒ k(A) < 4. 2 S = ∅ ⇒ k(A) > 2.

Thus k(A) = 3.

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Example in M7(C)

A =          a c . . . b a c ... . . . ... ... ... . . . ... b a c . . . b a          The numerical range of a 7 × 7 tri-diagonal Toeplitz matrix, with a = 5 + 4i, b = −1 + i, c = −3. Theorem (C, R, S, S, 2014) k(A) = n

2

.

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References

1 K.A. Camenga, P.X. Rault, D.J. Rossi, T. Sendova, I.M. Spitkovsky,

Numerical range of some doubly stochastic matrices. Applied Mathematics and Computation 221 (2013) 40-47.

2 K.A. Camenga, P.X. Rault, T. Sendova, and I.M. Spitkovsky, On the

Gau-Wu number for some classes of matrices, Linear Algebra and its Applications 444 (2014) 254-262.

3 H.L. Gau and P.Y. Wu, Numerical ranges and compressions of

Sn-matrices, Operators and Matrices 7 (2013) 465–476.

4 D.S. Keeler, L. Rodman, I.M. Spitkovsky, The numerical range of

3 × 3 matrices, Linear Algebra and its Applications 252 (1997) 115-139.

5 H. Lee, Diagonals and numerical ranges of direct sums of matrices,

Linear Algebra and its Applications, 439 (2013), 2584-2597.

6 P.X. Rault, T. Sendova, I.M. Spitkovsky 3-by-3 matrices with

elliptical numerical range revisited, Electronic Journal of Linear Algebra 26 (2013) 158-167.

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