The Numerical Range of a Matrix

Home , Numerical range

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) = {hAx, xi : x ∈ C , kxk = 1} = {x Ax : x ∈ C , kxk = 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 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

 1 0 0 0  1 0 2 i  1 2 0 0  A1 = , A2 = , A1 ⊕ A2 =   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  5 0 5 5 2 1 2  5 0 5 5   3 2   0 5 0 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 

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  Hφ := 2 , Kφ := 2i 0.2 W (Hφ) = Re(W (Aφ)) =[ λmin, λmax]. H L min W H  Hφv = λmaxv.  0.3 0.2 0.1 0.1 0.2 0.3 λmax = hHφv, vi = RehAv, vi. max hAv, vi∈ 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 .

Rault & Camenga 1/13/15 January 13, 2015 17 / 23 The Gau-Wu number

Deﬁnition (Gau-Wu number, 2013)

k(A) = max #{x1,..., xk } ∀j,hAxj ,xj i∈∂W (A) {x1,...,xk } orthonormal n {x1,...,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 ∈ `1 ∩ ∂W (B).

Let `2 be a parallel support line, and let hAw, wi ∈ `2 ∩ ∂W (B). Then w ⊥ u, w ⊥ v. Figure: B ∈ M ( ) 3 C Thus k(B) = 3.

Basic examples

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

Figure: A ∈ 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 ∩ ∂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 ∈ M2(C). k(A) = 2

Vertical ﬂat portion `1 ⇒ pair of orthogonal eigenvectors u, v of H0, with hBu, ui, hBv, vi ∈ `1 ∩ ∂W (B).

Figure: B ∈ 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 ∈ M2(C). k(A) = 2

Vertical ﬂat portion `1 ⇒ pair of orthogonal eigenvectors u, v of H0, with hBu, ui, hBv, vi ∈ `1 ∩ ∂W (B).

Let `2 be a parallel support line, and let hAw, wi ∈ `2 ∩ ∂W (B). Then w ⊥ u, w ⊥ v. Figure: B ∈ 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 ∈ M2(C). k(A) = 2

Vertical ﬂat portion `1 ⇒ pair of orthogonal eigenvectors u, v of H0, with hBu, ui, hBv, vi ∈ `1 ∩ ∂W (B).

Let `2 be a parallel support line, and let hAw, wi ∈ `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 Thus k(A) = 4.

Example 1 of an irreducible 4 × 4 matrix

1.0

0.5 Let A ∈ M4(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 `1 ⇒ pair of orthogonal eigenvectors u, v of H0, with hAu, ui, hAv, vi ∈ `1 ∩ ∂W (A).

Let `2 be a parallel support line, and let hAw, wi ∈ `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 ∈ M4(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 `1 ⇒ pair of orthogonal eigenvectors u, v of H0, with hAu, ui, hAv, vi ∈ `1 ∩ ∂W (A).

Let `2 be a parallel support line, and let hAw, wi ∈ `2 ∩ ∂W (A). Then w ⊥ u, w ⊥ v. Thus k(A) = 4.

Rault & Camenga 1/13/15 January 13, 2015 20 / 23 Lemma ∗ (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + Aφ)/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 6= ∅ ⇒ k(A) > 2. Thus k(A) = 3.

Example 2 of an irreducible 4 × 4 matrix

2 Let A ∈ M4(C) irreducible, with

F (x : y : t) = 0 a curve having three nodes. 0 Dual curve:

2

4

2 0 2 4

Rault & Camenga 1/13/15 January 13, 2015 21 / 23 2 S 6= ∅ ⇒ k(A) > 2. Thus k(A) = 3.

Example 2 of an irreducible 4 × 4 matrix

2 Let A ∈ M4(C) irreducible, with

F (x : y : t) = 0 a curve having three nodes. 0 Dual curve:

2

4

2 0 2 4 Lemma ∗ (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + Aφ)/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.

Rault & Camenga 1/13/15 January 13, 2015 21 / 23 Thus k(A) = 3.

Example 2 of an irreducible 4 × 4 matrix

2 Let A ∈ M4(C) irreducible, with

F (x : y : t) = 0 a curve having three nodes. 0 Dual curve:

2

4

2 0 2 4 Lemma ∗ (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + Aφ)/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 6= ∅ ⇒ k(A) > 2.

Rault & Camenga 1/13/15 January 13, 2015 21 / 23 Example 2 of an irreducible 4 × 4 matrix

2 Let A ∈ M4(C) irreducible, with

F (x : y : t) = 0 a curve having three nodes. 0 Dual curve:

2

4

2 0 2 4 Lemma ∗ (C,R,S,S, 2015) Let A ∈ M4(C), Hφ := (Aφ + Aφ)/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 6= ∅ ⇒ k(A) > 2. Thus k(A) = 3.

Rault & Camenga 1/13/15 January 13, 2015 21 / 23 Example in M7(C)

 a c 0 ... 0   .. .   b a c . .     ......  A =  0 . . . 0     . ..   . . b a c  0 ... 0 b a

The numerical range of a 7 × 7 tri-diagonal Toeplitz matrix, with a = 5 + 4i, b = −1 + i, c = −3. Theorem n (C, R, S, S, 2014) k(A) = 2 .

Rault & Camenga 1/13/15 January 13, 2015 22 / 23 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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