Eigenvalues of the sum of matrices from unitary similarity

Eigenvalues of the sum of matrices

from unitary similarity orbits

Chi-Kwong LiDepartment of Mathematics

The College of William and Mary

Based on some joint work with:Yiu-Tung Poon (Iowa State University),

Nung-Sing Sze (University of Connecticut),

Chi-Kwong Li Eigenvalues of the sum of matrices.

Introduction

Basic problem

Let A,B ∈ Mn. Determine the set E(A,B) of eigenvalues of matrices ofthe form

U∗AU + V ∗BV, U, V are unitary,


Introduction

Basic problem



or simply,A + V ∗BV V is unitary.


Introduction

Basic problem




Why study?


Introduction

Basic problem




Why study?

It is natural to make predictions about U∗AU + V ∗BV based oninformation of A and B.


Introduction

Basic problem




Why study?


Knowing E(A,B) is helpful in the study of perturbations,approximations, stability, convergence, spectral variations, ....


Introduction

Basic problem




Why study?


Knowing E(A,B) is helpful in the study of perturbations,approximations, stability, convergence, spectral variations, ....

Especially, in the study of quantum computing and quantuminformation theory, all measurements, control, perturbations, etc.are related to unitary similarity transforms.


Hermitian matrices

Example Let A =

(

1 00 2

)

and B =

(

3 00 4

)

.


Hermitian matrices

Example Let A =

(

1 00 2

)

and B =

(

3 00 4

)

. Then E(A,B) = [4, 6].


Hermitian matrices

Example Let A =

(

1 00 2

)

and B =

(

3 00 4

)

. Then E(A,B) = [4, 6].

Just consider the eigenvalues of

(

1 00 2

)

+

(

cos t − sin tsin t cos t

)(

3 00 4

)(

cos t sin t− sin t cos t

)

with t ∈ [0, π].


Hermitian matrices

Example Let A =

(

1 00 2

)

and B =

(

3 00 4

)

. Then E(A,B) = [4, 6].


(

1 00 2

)

+

(


)(

3 00 4

)(


)

with t ∈ [0, π].

Example Let A =

(

10 00 20

)

and B =

(

3 00 4

)

.


Hermitian matrices

Example Let A =

(

1 00 2

)

and B =

(

3 00 4

)

. Then E(A,B) = [4, 6].


(

1 00 2

)

+

(


)(

3 00 4

)(


)

with t ∈ [0, π].

Example Let A =

(

10 00 20

)

and B =

(

3 00 4

)

.

If A + V ∗BV has eigenvalues c1 ≥ c2, then c1 ∈ [23, 24], c2 ∈ [13, 14],


Hermitian matrices

Example Let A =

(

1 00 2

)

and B =

(

3 00 4

)

. Then E(A,B) = [4, 6].


(

1 00 2

)

+

(


)(

3 00 4

)(


)

with t ∈ [0, π].

Example Let A =

(

10 00 20

)

and B =

(

3 00 4

)

.

If A + V ∗BV has eigenvalues c1 ≥ c2, then c1 ∈ [23, 24], c2 ∈ [13, 14],

and E(A,B) = [13, 14] ∪ [23, 24].


Results on Hermitian matrices

Theorem

Let A = diag (a1, . . . , an) and B = diag (b1, . . . , bn) with

a1 ≥ · · · ≥ an and b1 ≥ · · · ≥ bn.



Theorem


a1 ≥ · · · ≥ an and b1 ≥ · · · ≥ bn.

If V is unitary and A + V ∗BV has eigenvalues c1 ≥ · · · ≥ cn, then

cj = [bj + an, bj + a1] ∩ [aj + bn, aj + b1] for j = 1, . . . , n.



Theorem


a1 ≥ · · · ≥ an and b1 ≥ · · · ≥ bn.



It follows that E(A,B) equals

[an + bn, a1 + b1] \n−1⋃

j=1

((aj+1 + b1, aj + bn) ∪ (bj+1 + a1, bj + an)) .



Theorem


a1 ≥ · · · ≥ an and b1 ≥ · · · ≥ bn.



It follows that E(A,B) equals

[an + bn, a1 + b1] \n−1⋃

j=1

((aj+1 + b1, aj + bn) ∪ (bj+1 + a1, bj + an)) .

Consequently, E(A,B) = [an + bn, a1 + b1] if

b1 − bn ≥ max1≤j≤n−1

(aj − aj+1) and a1 − an ≥ max1≤j≤n−1

(bj − bj+1).


Theorem [Klychko, Fulton, A. Horn, Thompson, Kuntson, Tao, ... ]

Let a1 ≥ · · · ≥ an, b1 ≥ · · · ≥ bn and c1 ≥ · · · ≥ cn be given.




There exist Hermitian matrices A, B and C = A + B with eigenvaluesa1 ≥ · · · ≥ an, b1 ≥ · · · ≥ bn, and c1 ≥ · · · ≥ cn if and only if





n∑

j=1

(aj + bj) =

n∑

j=1

cj ,

and





n∑

j=1

(aj + bj) =

n∑

j=1

cj ,

and∑

r∈R

ar +∑

s∈S

bs ≥∑

t∈T

ct

for all subsequences R,S, T of (1, . . . , n) determined by theLittlewood-Richardson rules.


Normal matrices

Example 1 Suppose σ(A) = {1,−1} and σ(B) = {i,−i}.


Normal matrices

Example 1 Suppose σ(A) = {1,−1} and σ(B) = {i,−i}.

Then E(A,B) equals

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Example 2 Suppose σ(A) = {1,−1} and σ(B) = {0.8i,−0.8i}.


Example 2 Suppose σ(A) = {1,−1} and σ(B) = {0.8i,−0.8i}.

Then E(A,B) equals

−1 −0.5 0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8


Proposition [LPS,2008]

Suppose A,B ∈ Mn are normal with

σ(A) = {a1, a2} and σ(B) = {b1, b2}.

Then E(A,B) are two (finite) segments of the hyperbola with end pointsin {a1 + b1, a1 + b2, a2 + b1, a2 + b2}.


Example 3 Suppose w = ei2π/3,σ(A) = {−iw,−iw2} and σ(B) = {−i,−wi,−w2i}.


Example 3 Suppose w = ei2π/3,σ(A) = {−iw,−iw2} and σ(B) = {−i,−wi,−w2i}.

Then E(A,B) equals

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2


Example 4 Suppose w = ei2π/3,σ(A) = {−0.95wi,−0.95w2i) and σ(B) = {−i,−wi,−w2i}.


Example 4 Suppose w = ei2π/3,σ(A) = {−0.95wi,−0.95w2i) and σ(B) = {−i,−wi,−w2i}.

Then E(A,B) equals

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Example 5 Suppose σ(A) = {0, 1 + i} and σ(B) = {0, 1, 4}.


Example 5 Suppose σ(A) = {0, 1 + i} and σ(B) = {0, 1, 4}.

Then E(A,B) equals

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Proposition [LPS,2008]

Suppose σ(A) = {a1, a2} and σ(B) = {b1, b2, b3}. ThenE(A,B) = E(a1, a2; b1, b2, b3) consists of connected components enclosedby the three pairs of hyperbola segments

E(a1, a2; b1, b2), E(a1, a2; b1, b3), E(a1, a2; b2, b3).


One more example on normal matrices

Example 6 Suppose σ(A) = {0, 1, 4, 6} and σ(B) = {0, i, 2i).


One more example on normal matrices

Example 6 Suppose σ(A) = {0, 1, 4, 6} and σ(B) = {0, i, 2i).

Then E(A,B) equals

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Results on normal matrices

Theorem [LPS,2008]

Suppose A,B ∈ Mn are normal with σ(A) = {a1, . . . , ap} andσ(B) = {b1, . . . , bq}. Then

E(A,B) = (∪E(ai1, ai2

, ai3; bj1

, bj2)) ∪ (∪E(ai1

, ai2; bj1

, bj2, bj3

)) .


Results on normal matrices

Theorem [LPS,2008]

Suppose A,B ∈ Mn are normal with σ(A) = {a1, . . . , ap} andσ(B) = {b1, . . . , bq}. Then

E(A,B) = (∪E(ai1, ai2

, ai3; bj1

, bj2)) ∪ (∪E(ai1

, ai2; bj1

, bj2, bj3

)) .

Theorem [Wielandt,1955], [LPS,2008]

Suppose A,B ∈ Mn are normal. Then µ /∈ E(A,B) if and only if there isa circular disk containing the eigenvalues of A or µI − B, and excludingthe eigenvalues of the other matrices.


General matrices

The Davis-Wielandt Shell of A ∈ Mn is the set

DW (A) = {(x∗Ax, ‖Ax‖2) : x ∈ Cn, x∗x = 1}

⊆ {(z, r) ∈ C × R : |z|2 ≤ r}.


General matrices


DW (A) = {(x∗Ax, ‖Ax‖2) : x ∈ Cn, x∗x = 1}

⊆ {(z, r) ∈ C × R : |z|2 ≤ r}.

Proposition

Let A ∈ Mn.


General matrices


DW (A) = {(x∗Ax, ‖Ax‖2) : x ∈ Cn, x∗x = 1}

⊆ {(z, r) ∈ C × R : |z|2 ≤ r}.

Proposition

Let A ∈ Mn.

Then µ ∈ σ(A) if and only if (µ, |µ|2) ∈ DW (A).


General matrices


DW (A) = {(x∗Ax, ‖Ax‖2) : x ∈ Cn, x∗x = 1}

⊆ {(z, r) ∈ C × R : |z|2 ≤ r}.

Proposition

Let A ∈ Mn.

Then µ ∈ σ(A) if and only if (µ, |µ|2) ∈ DW (A).

Then A is normal if and only if DW (A) is a polyhedron.


Theorem [LPS,2008]

Suppose A,B ∈ Mn. Then µ ∈ E(A,B) if and only if any one of thefollowing holds.


Theorem [LPS,2008]


DW (A) ∩ DW (µI − B) 6= ∅.


Theorem [LPS,2008]


DW (A) ∩ DW (µI − B) 6= ∅.

For any ξ ∈ C,

conv σ(|A + ξI|) ∩ conv σ(|B − ξI − µI|) 6= ∅.


Theorem [LPS,2008]


DW (A) ∩ DW (µI − B) 6= ∅.

For any ξ ∈ C,

conv σ(|A + ξI|) ∩ conv σ(|B − ξI − µI|) 6= ∅.

Equivalently, singular values of A + ξI and the singular values ofB − ξI − µI do not lie in two separate closed intervals.


Further research

Develop computer programs to generate E(A,B) for generalA,B ∈ Mn.


Further research


Determine the entire set or a subset of eigenvalues of A + V ∗BVfor given (normal) matrices A,B ∈ Mn.


Further research



Determine all possible eigenvalues for∑k

j=1U∗

j AjUj for givenA1, . . . , Ak ∈ Mn.


Further research




j=1U∗


Study the spectrum of A + V ∗BV for infinite dimensional boundedlinear operators A,B.


Further research




j=1U∗


Study the spectrum of A + V ∗BV for infinite dimensional boundedlinear operators A,B.

Study the above problems for unitary matrices chosen from acertain subgroups such as SU(2) ⊗ · · · ⊗ SU(2) (m copies).


Thank you for your attention!


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Eigenvalues of the sum of matrices from unitary similarity