Eigenvalues of the sum of matrices. from unitary similarity orbits

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Eigenvalues of the sum of matrices from unitary similarity orbits Department 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),

Introduction Basic problem Let A,B M n. Determine the set E(A,B) of eigenvalues of matrices of the form U AU + V BV, U,V are unitary,

Introduction Basic problem Let A,B M n. Determine the set E(A,B) of eigenvalues of matrices of the form U AU + V BV, U,V are unitary, or simply, A + V BV V is unitary.

Introduction Basic problem Let A,B M n. Determine the set E(A,B) of eigenvalues of matrices of the form U AU + V BV, U,V are unitary, or simply, A + V BV V is unitary. Why study?

Introduction Basic problem Let A,B M n. Determine the set E(A,B) of eigenvalues of matrices of the 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 on information of A and B.

Introduction Basic problem Let A,B M n. Determine the set E(A,B) of eigenvalues of matrices of the 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 on information of A and B. Knowing E(A, B) is helpful in the study of perturbations, approximations, stability, convergence, spectral variations,...

Introduction Basic problem Let A,B M n. Determine the set E(A,B) of eigenvalues of matrices of the 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 on information 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 quantum information theory, all measurements, control, perturbations, etc. are related to unitary similarity transforms.

Hermitian matrices Example Let A = ( ) 1 0 and B = 0 2 ( ) 3 0. 0 4

Hermitian matrices Example Let A = ( ) 1 0 and B = 0 2 ( ) 3 0. Then E(A,B) = [4,6]. 0 4

Hermitian matrices Example Let A = ( ) 1 0 and B = 0 2 ( ) 3 0. Then E(A,B) = [4,6]. 0 4 Just consider the eigenvalues of ( ) ( )( )( ) 1 0 cos t sin t 3 0 cos t sin t + 0 2 sin t cos t 0 4 sin t cos t with t [0,π].

Hermitian matrices Example Let A = ( ) 1 0 and B = 0 2 ( ) 3 0. Then E(A,B) = [4,6]. 0 4 Just consider the eigenvalues of ( ) ( )( )( ) 1 0 cos t sin t 3 0 cos t sin t + 0 2 sin t cos t 0 4 sin t cos t with t [0,π]. Example Let A = ( ) 10 0 and B = 0 20 ( ) 3 0. 0 4

Hermitian matrices Example Let A = ( ) 1 0 and B = 0 2 ( ) 3 0. Then E(A,B) = [4,6]. 0 4 Just consider the eigenvalues of ( ) ( )( )( ) 1 0 cos t sin t 3 0 cos t sin t + 0 2 sin t cos t 0 4 sin t cos t with t [0,π]. Example Let A = ( ) 10 0 and B = 0 20 ( ) 3 0. 0 4 If A + V BV has eigenvalues c 1 c 2, then c 1 [23,24], c 2 [13,14],

Hermitian matrices Example Let A = ( ) 1 0 and B = 0 2 ( ) 3 0. Then E(A,B) = [4,6]. 0 4 Just consider the eigenvalues of ( ) ( )( )( ) 1 0 cos t sin t 3 0 cos t sin t + 0 2 sin t cos t 0 4 sin t cos t with t [0,π]. Example Let A = ( ) 10 0 and B = 0 20 ( ) 3 0. 0 4 If A + V BV has eigenvalues c 1 c 2, then c 1 [23,24], c 2 [13,14], and E(A,B) = [13,14] [23,24].

Results on Hermitian matrices Theorem Let A = diag (a 1,...,a n ) and B = diag (b 1,...,b n ) with a 1 a n and b 1 b n.

Results on Hermitian matrices Theorem Let A = diag (a 1,...,a n ) and B = diag (b 1,...,b n ) with a 1 a n and b 1 b n. If V is unitary and A + V BV has eigenvalues c 1 c n, then c j = [b j + a n,b j + a 1 ] [a j + b n,a j + b 1 ] for j = 1,...,n.

Results on Hermitian matrices Theorem Let A = diag (a 1,...,a n ) and B = diag (b 1,...,b n ) with a 1 a n and b 1 b n. If V is unitary and A + V BV has eigenvalues c 1 c n, then c j = [b j + a n,b j + a 1 ] [a j + b n,a j + b 1 ] for j = 1,...,n. It follows that E(A, B) equals n 1 [a n + b n,a 1 + b 1 ] \ ((a j+1 + b 1,a j + b n ) (b j+1 + a 1,b j + a n )). j=1

Results on Hermitian matrices Theorem Let A = diag (a 1,...,a n ) and B = diag (b 1,...,b n ) with a 1 a n and b 1 b n. If V is unitary and A + V BV has eigenvalues c 1 c n, then c j = [b j + a n,b j + a 1 ] [a j + b n,a j + b 1 ] for j = 1,...,n. It follows that E(A, B) equals n 1 [a n + b n,a 1 + b 1 ] \ ((a j+1 + b 1,a j + b n ) (b j+1 + a 1,b j + a n )). j=1 Consequently, E(A,B) = [a n + b n,a 1 + b 1 ] if b 1 b n max (a j a j+1 ) and a 1 a n max (b j b j+1 ). 1 j n 1 1 j n 1

Theorem [Klychko, Fulton, A. Horn, Thompson, Kuntson, Tao,... ] Let a 1 a n, b 1 b n and c 1 c n be given.

Theorem [Klychko, Fulton, A. Horn, Thompson, Kuntson, Tao,... ] Let a 1 a n, b 1 b n and c 1 c n be given. There exist Hermitian matrices A, B and C = A + B with eigenvalues a 1 a n, b 1 b n, and c 1 c n if and only if

Theorem [Klychko, Fulton, A. Horn, Thompson, Kuntson, Tao,... ] Let a 1 a n, b 1 b n and c 1 c n be given. There exist Hermitian matrices A, B and C = A + B with eigenvalues a 1 a n, b 1 b n, and c 1 c n if and only if and n (a j + b j ) = j=1 n c j, j=1

Theorem [Klychko, Fulton, A. Horn, Thompson, Kuntson, Tao,... ] Let a 1 a n, b 1 b n and c 1 c n be given. There exist Hermitian matrices A, B and C = A + B with eigenvalues a 1 a n, b 1 b n, and c 1 c n if and only if and n (a j + b j ) = j=1 n c j, j=1 r R a r + s S b s t T for all subsequences R,S,T of (1,...,n) determined by the Littlewood-Richardson rules. c t

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 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1 0.5 0 0.5 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 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0.5 0 0.5 1

Proposition [LPS,2008] Suppose A,B M n are normal with σ(a) = {a 1,a 2 } and σ(b) = {b 1,b 2 }. Then E(A, B) are two (finite) segments of the hyperbola with end points in {a 1 + b 1,a 1 + b 2,a 2 + b 1,a 2 + b 2 }.

Example 3 Suppose w = e i2π/3, σ(a) = { iw, iw 2 } and σ(b) = { i, wi, w 2 i}.

Example 3 Suppose w = e i2π/3, σ(a) = { iw, iw 2 } and σ(b) = { i, wi, w 2 i}. Then E(A, B) equals 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 2 1.5 1 0.5 0 0.5 1 1.5 2

Example 4 Suppose w = e i2π/3, σ(a) = { 0.95wi, 0.95w 2 i) and σ(b) = { i, wi, w 2 i}.

Example 4 Suppose w = e i2π/3, σ(a) = { 0.95wi, 0.95w 2 i) and σ(b) = { i, wi, w 2 i}. Then E(A, B) equals 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 2 1.5 1 0.5 0 0.5 1 1.5 2

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 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Proposition [LPS,2008] Suppose σ(a) = {a 1,a 2 } and σ(b) = {b 1,b 2,b 3 }. Then E(A,B) = E(a 1,a 2 ;b 1,b 2,b 3 ) consists of connected components enclosed by the three pairs of hyperbola segments E(a 1,a 2 ;b 1,b 2 ), E(a 1,a 2 ;b 1,b 3 ), E(a 1,a 2 ;b 2,b 3 ).

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 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6

Results on normal matrices Theorem [LPS,2008] Suppose A,B M n are normal with σ(a) = {a 1,...,a p } and σ(b) = {b 1,...,b q }. Then E(A,B) = ( E(a i1,a i2,a i3 ;b j1,b j2 )) ( E(a i1,a i2 ;b j1,b j2,b j3 )).

Results on normal matrices Theorem [LPS,2008] Suppose A,B M n are normal with σ(a) = {a 1,...,a p } and σ(b) = {b 1,...,b q }. Then E(A,B) = ( E(a i1,a i2,a i3 ;b j1,b j2 )) ( E(a i1,a i2 ;b j1,b j2,b j3 )). Theorem [Wielandt,1955], [LPS,2008] Suppose A,B M n are normal. Then µ / E(A,B) if and only if there is a circular disk containing the eigenvalues of A or µi B, and excluding the eigenvalues of the other matrices.

General matrices The Davis-Wielandt Shell of A M n is the set DW(A) = {(x Ax, Ax 2 ) : x C n, x x = 1} {(z,r) C R : z 2 r}.

General matrices The Davis-Wielandt Shell of A M n is the set DW(A) = {(x Ax, Ax 2 ) : x C n, x x = 1} {(z,r) C R : z 2 r}. Proposition Let A M n.

General matrices The Davis-Wielandt Shell of A M n is the set DW(A) = {(x Ax, Ax 2 ) : x C n, x x = 1} {(z,r) C R : z 2 r}. Proposition Let A M n. Then µ σ(a) if and only if (µ, µ 2 ) DW(A).

General matrices The Davis-Wielandt Shell of A M n is the set DW(A) = {(x Ax, Ax 2 ) : x C n, x x = 1} {(z,r) C R : z 2 r}. Proposition Let A M n. 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 M n. Then µ E(A,B) if and only if any one of the following holds.

Theorem [LPS,2008] Suppose A,B M n. Then µ E(A,B) if and only if any one of the following holds. DW(A) DW(µI B).

Theorem [LPS,2008] Suppose A,B M n. Then µ E(A,B) if and only if any one of the following holds. DW(A) DW(µI B). For any ξ C, conv σ( A + ξi ) conv σ( B ξi µi ).

Theorem [LPS,2008] Suppose A,B M n. Then µ E(A,B) if and only if any one of the following holds. DW(A) DW(µI B). For any ξ C, conv σ( A + ξi ) conv σ( B ξi µi ). Equivalently, singular values of A + ξi and the singular values of B ξi µi do not lie in two separate closed intervals.

Further research Develop computer programs to generate E(A, B) for general A,B M n.

Further research Develop computer programs to generate E(A, B) for general A,B M n. Determine the entire set or a subset of eigenvalues of A + V BV for given (normal) matrices A,B M n.

Further research Develop computer programs to generate E(A, B) for general A,B M n. Determine the entire set or a subset of eigenvalues of A + V BV for given (normal) matrices A,B M n. Determine all possible eigenvalues for k j=1 U j A ju j for given A 1,...,A k M n.

Further research Develop computer programs to generate E(A, B) for general A,B M n. Determine the entire set or a subset of eigenvalues of A + V BV for given (normal) matrices A,B M n. Determine all possible eigenvalues for k j=1 U j A ju j for given A 1,...,A k M n. Study the spectrum of A + V BV for infinite dimensional bounded linear operators A, B.

Further research Develop computer programs to generate E(A, B) for general A,B M n. Determine the entire set or a subset of eigenvalues of A + V BV for given (normal) matrices A,B M n. Determine all possible eigenvalues for k j=1 U j A ju j for given A 1,...,A k M n. Study the spectrum of A + V BV for infinite dimensional bounded linear operators A, B. Study the above problems for unitary matrices chosen from a certain subgroups such as SU(2) SU(2) (m copies).

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