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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.
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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,
Chi-Kwong Li Eigenvalues of the sum of matrices.
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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,
or simply,A + V ∗BV V is unitary.
Chi-Kwong Li Eigenvalues of the sum of matrices.
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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,
or simply,A + V ∗BV V is unitary.
Why study?
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 5: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/5.jpg)
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,
or simply,A + V ∗BV V is unitary.
Why study?
It is natural to make predictions about U∗AU + V ∗BV based oninformation of A and B.
Chi-Kwong Li Eigenvalues of the sum of matrices.
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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,
or simply,A + V ∗BV V is unitary.
Why study?
It is natural to make predictions about U∗AU + V ∗BV based oninformation of A and B.
Knowing E(A,B) is helpful in the study of perturbations,approximations, stability, convergence, spectral variations, ....
Chi-Kwong Li Eigenvalues of the sum of matrices.
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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,
or simply,A + V ∗BV V is unitary.
Why study?
It is natural to make predictions about U∗AU + V ∗BV based oninformation of A and B.
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.
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Hermitian matrices
Example Let A =
(
1 00 2
)
and B =
(
3 00 4
)
.
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Hermitian matrices
Example Let A =
(
1 00 2
)
and B =
(
3 00 4
)
. Then E(A,B) = [4, 6].
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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].
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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].
Example Let A =
(
10 00 20
)
and B =
(
3 00 4
)
.
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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].
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],
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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].
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].
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Results on Hermitian matrices
Theorem
Let A = diag (a1, . . . , an) and B = diag (b1, . . . , bn) with
a1 ≥ · · · ≥ an and b1 ≥ · · · ≥ bn.
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Results on Hermitian matrices
Theorem
Let A = diag (a1, . . . , an) and B = diag (b1, . . . , bn) with
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.
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Results on Hermitian matrices
Theorem
Let A = diag (a1, . . . , an) and B = diag (b1, . . . , bn) with
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.
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)) .
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Results on Hermitian matrices
Theorem
Let A = diag (a1, . . . , an) and B = diag (b1, . . . , bn) with
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.
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).
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Theorem [Klychko, Fulton, A. Horn, Thompson, Kuntson, Tao, ... ]
Let a1 ≥ · · · ≥ an, b1 ≥ · · · ≥ bn and c1 ≥ · · · ≥ cn be given.
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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
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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
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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∑
r∈R
ar +∑
s∈S
bs ≥∑
t∈T
ct
for all subsequences R,S, T of (1, . . . , n) determined by theLittlewood-Richardson rules.
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Normal matrices
Example 1 Suppose σ(A) = {1,−1} and σ(B) = {i,−i}.
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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
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Example 2 Suppose σ(A) = {1,−1} and σ(B) = {0.8i,−0.8i}.
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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
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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}.
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Example 3 Suppose w = ei2π/3,σ(A) = {−iw,−iw2} and σ(B) = {−i,−wi,−w2i}.
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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
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Example 4 Suppose w = ei2π/3,σ(A) = {−0.95wi,−0.95w2i) and σ(B) = {−i,−wi,−w2i}.
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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
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Example 5 Suppose σ(A) = {0, 1 + i} and σ(B) = {0, 1, 4}.
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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
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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).
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One more example on normal matrices
Example 6 Suppose σ(A) = {0, 1, 4, 6} and σ(B) = {0, i, 2i).
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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
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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
)) .
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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.
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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}.
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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}.
Proposition
Let A ∈ Mn.
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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}.
Proposition
Let A ∈ Mn.
Then µ ∈ σ(A) if and only if (µ, |µ|2) ∈ DW (A).
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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}.
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.
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Theorem [LPS,2008]
Suppose A,B ∈ Mn. Then µ ∈ E(A,B) if and only if any one of thefollowing holds.
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Theorem [LPS,2008]
Suppose A,B ∈ Mn. Then µ ∈ E(A,B) if and only if any one of thefollowing holds.
DW (A) ∩ DW (µI − B) 6= ∅.
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Theorem [LPS,2008]
Suppose A,B ∈ Mn. Then µ ∈ E(A,B) if and only if any one of thefollowing holds.
DW (A) ∩ DW (µI − B) 6= ∅.
For any ξ ∈ C,
conv σ(|A + ξI|) ∩ conv σ(|B − ξI − µI|) 6= ∅.
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Theorem [LPS,2008]
Suppose A,B ∈ Mn. Then µ ∈ E(A,B) if and only if any one of thefollowing holds.
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.
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 46: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/46.jpg)
Further research
Develop computer programs to generate E(A,B) for generalA,B ∈ Mn.
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 47: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/47.jpg)
Further research
Develop computer programs to generate E(A,B) for generalA,B ∈ Mn.
Determine the entire set or a subset of eigenvalues of A + V ∗BVfor given (normal) matrices A,B ∈ Mn.
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 48: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/48.jpg)
Further research
Develop computer programs to generate E(A,B) for generalA,B ∈ Mn.
Determine the entire set or a subset of eigenvalues of A + V ∗BVfor given (normal) matrices A,B ∈ Mn.
Determine all possible eigenvalues for∑k
j=1U∗
j AjUj for givenA1, . . . , Ak ∈ Mn.
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 49: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/49.jpg)
Further research
Develop computer programs to generate E(A,B) for generalA,B ∈ Mn.
Determine the entire set or a subset of eigenvalues of A + V ∗BVfor given (normal) matrices A,B ∈ Mn.
Determine all possible eigenvalues for∑k
j=1U∗
j AjUj for givenA1, . . . , Ak ∈ Mn.
Study the spectrum of A + V ∗BV for infinite dimensional boundedlinear operators A,B.
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 50: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/50.jpg)
Further research
Develop computer programs to generate E(A,B) for generalA,B ∈ Mn.
Determine the entire set or a subset of eigenvalues of A + V ∗BVfor given (normal) matrices A,B ∈ Mn.
Determine all possible eigenvalues for∑k
j=1U∗
j AjUj for givenA1, . . . , Ak ∈ Mn.
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).
Chi-Kwong Li Eigenvalues of the sum of matrices.
![Page 51: Eigenvalues of the sum of matrices from unitary similarity](https://reader030.vdocuments.us/reader030/viewer/2022041108/625007965be8974671067655/html5/thumbnails/51.jpg)
Thank you for your attention!
Chi-Kwong Li Eigenvalues of the sum of matrices.