Ann. Funct. Anal. 5 (2014), no. 1, 94–100
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL:www.emis.de/journals/AFA/
SINGULAR VALUES AND EIGENVALUES OF MATRICES IN
son (C) AND spn (C)
DURMUŞ BOZKURT1 , TIN-YAU TAM2∗ AND WEN YAN3
This paper is dedicated to Professor Tsuyoshi Ando for his significant contributions to linear
algebra, operator theory and functional analysis
Communicated by Q.-W. Wang
Abstract. We give a complete relation between the singular values and eigenvalues of a complex skew symmetric matrix in terms of multiplicative majorization and double occurrences of singular values and eigenvalues. Similar studies
are given for matrices in the algebras spn (C) and spn (R).
1. Introduction
Let A ∈ Cn×n with eigenvalues λ1 , . . . , λn arranged in descending order |λ1 | ≥
· · · ≥ |λn | according to their moduli. The singular values of A are the nonnegative
square roots of the eigenvalues of the positive semi-definite matrix A∗ A and are
denoted by s1 ≥ · · · ≥ sn . Weyl [9] established the multiplicative majorization
relation between the eigenvalues and singular values of A and Horn [3] established
the converse (see Ando’s paper [1] for some majorization results).
Theorem 1.1. (Weyl-Horn) Let A ∈ Cn×n with singular values s1 ≥ · · · ≥ sn
and eigenvalues λ1 , . . . , λn ordered as |λ1 | ≥ · · · ≥ |λn |. Then
k
Y
j=1
n
Y
j=1
|λj | ≤
|λj | =
k
Y
j=1
n
Y
sj ,
sj .
k = 1, . . . , n − 1,
(1.1)
(1.2)
j=1
Date: Received: 13 August 2013; Accepted: 6 September 2013.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 15A45; Secondary 15A18, 47BXX..
Key words and phrases. Eigenvalues, singular values, complex skew symmetric matrix.
94
SINGULAR VALUES AND EIGENVALUES
95
Conversely, if |λ1 | ≥ · · · ≥ |λn | and s1 ≥ · · · ≥ sn satisfy (1.1) and (1.2), then
there exists A ∈ Cn×n such that λ1 , . . . , λn are the eigenvalues and s1 , . . . , sn are
the singular values of A, respectively.
See [7] for a simple proof of Horn’s theorem. Thompson [8] studied the real
counterpart, i.e., A ∈ Rn×n . In this case the eigenvalues of A must occur in
complex conjugate pairs; and such is the only additional condition.
Our goal in Section 2 is to study the analogy of Theorem 1.1 for complex skew
symmetric matrix A ∈ Cm×m , i.e., A> = −A. This skew symmetry yields
det(A − tI) = det(A − tI)> = det(−A − tI),
i.e., the eigenvalues of A occur in pairs but opposite in sign, counting multiplicities. Moreover the singular values s1 , s1 , . . . , s[m/2] , s[m/2] , (0) of A also occur in
pairs. Here (0) refers to a zero singular value when m is odd [4, p.217]. Indeed,
according to Hua decomposition [5, Theorem 7, p.481], there exists U ∈ U(m)
such that
(
s1 J ⊕ s2 J ⊕ · · · ⊕ sn J
if m is even
U AU > =
s1 J ⊕ s2 J ⊕ · · · ⊕ sn J ⊕ (0) if m is odd,
where n := [m/2] and
J :=
0 1
.
−1 0
Here U(m) denotes the unitary group. Unlike unitary equivalence A 7→ U AV
(U, V ∈ U(m)) is used for the origin Weyl-Horn setting, unitary congruence
A 7→ U AU > (U ∈ U(m)) is only allowed to handle the skew symmetric case.
In Section 3 we have analogous study for the complex symplectic Lie algebra
spn (C). The doubly occurrence of the eigenvalues remains but the singular values
are arbitrary. The real case spn (R) is also studied.
2. Skew symmetric matrices
Denote by som (C) the set of m × m complex skew symmetric matrices. The
following theorem asserts that for the even case Weyl-Horn’s multiplicative majorization together with double occurrence of the eigenvalues and singular values
are the necessary and sufficient conditions. For the odd case, multiplicative weak
majoriation plays the role of multiplicative majorization.
Theorem 2.1.
(1) Let A ∈ so2n (C) with singular values s1 ≥ s1 ≥ · · · ≥
sn ≥ sn and eigenvalues ±λ1 , . . . , ±λn ordered as |λ1 | ≥ · · · ≥ |λn |. Then
(1.1) and (1.2) hold. Conversely, if |λ1 | ≥ · · · ≥ |λn | and s1 ≥ · · · ≥ sn
satisfy (1.1) and (1.2), then there exists A ∈ so2n (C) with eigenvalues
±λ1 , . . . , ±λn and singular values s1 , s1 , . . . , sn , sn .
(2) Let A ∈ so2n+1 (C) with singular values s1 ≥ s1 ≥ · · · ≥ sn ≥ sn ≥ 0 and
eigenvalues ±λ1 , . . . , ±λn , 0. Then
k
Y
j=1
|λj | ≤
k
Y
j=1
sj ,
k = 1, . . . , n.
(2.1)
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D. BOZKURT, T.Y. TAM AND W. YAN
Conversely, if |λ1 | ≥ · · · ≥ |λn | and s1 ≥ · · · ≥ sn satisfy (2.1), then there
exists A ∈ so2n+1 (C) with eigenvalues ±λ1 , . . . , ±λn , 0 and singular values
s1 , s1 , . . . , sn , sn , 0.
Proof. The necessity parts of both cases follow from Theorem 1.1. We now prove
the sufficiency.
Even case: Let
S
B :=
,
−S
where S := diag (s1 , . . . , sn ). Since s1 , s2 , . . . , sn and λ1 , λ2 , . . . , λn satisfy (1.1)
and (1.2), by Theorem 1.1, we can find U1 , V1 ∈ U(n) such that


λ1 ∗ ∗
..
A1 := U1 SV1 = 
. ∗ .
λn
Let U := U1 ⊕ V1> and V := V1 ⊕ U1> . Then U, V ∈ U(2n). Then
U1 SV1
0
A1
0
A2 := U BV =
=
.
0
−V1> SU1>
0 −A>
1
Clearly A2 has eigenvalues ±λ1 , . . . , ±λn and singular values s1 , s1 , . . . , sn , sn .
However A2 is not skew symmetric in general. We will provethat A2 is unitarily
iIn In
similar to a skew symmetric matrix. Let W := √12
∈ U(2n). Then
In iIn
1
A1 − A >
i(A1 + A>
∗
1
1)
A := W A2 W =
A 1 − A>
2 −i(A1 + A>
1)
1
is skew symmetric and has eigenvalues ±λ1 , ±λ2 , . . . , ±λn and singular values
s1 , s1 , . . . , sn , sn .
Odd case: Let m = 2n + 1. It is trivial if n = 0. For n ≥ 1, if sn = 0, then
s1 , . . . , sn and λ1 , . . . , λn satisfy (1.1) and (1.2). So it is reduced to the even case.
Suppose sn 6= 0. Let


Ŝ
0
u
A1 :=  0 −Ŝ 0  ,
0 −u> 0
where
Qn
p
j=1 λj
Ŝ := diag (s1 , . . . , sn−1 , ŝn ), ŝn := Qn−1 , u := (0, . . . , 0, s2n − ŝ2n )> .
j=1 sj
Direct computation shows that A1 has singular values s1 , s1 , . . . , sn , sn , 0. Clearly
s1 , . . . , sn−1 , ŝn and λ1 , . . . , λn satisfy (1.1) and (1.2). Then by Theorem 1.1,
there are U, V ∈ U(n) such that A2 := U ŜV is upper triangular with eigenvalues
λ1 , . . . , λn . Let
A3 := (U ⊕ V > ⊕ (1))A1 (V ⊕ U > ⊕ (1))


A2
0
Uu
−A>
0 .
= 0
2
>
0 −(U u)
0
SINGULAR VALUES AND EIGENVALUES
97
Note that A3 has eigenvalues ±λ1 , . . . , ±λn , 0 and singular values s1 , s1 , . . . , sn , sn , 0.
We will prove that A3 is unitarily similar to a skew symmetric matrix. Define
e−iπ/4 iIn In
W := √
⊕ (1).
In iIn
2
Then

A 2 − A>
i(A2 + A>
)
eiπ/4 U u
2
2
1
A := W A3 W ∗ = −i(A2 + A>
A2 − A >
e−iπ/4 U u
2)
2
2
iπ/4
>
−iπ/4
>
−e (U u)
−e
(U u)
0

is clearly skew symmetric and has the same eigenvalues and singular values as
A3 .
We remark that the real counterpart of Theorem 2.1 is trivial since by the
spectral decomposition of a real skew symmetric A ∈ so(m), the eigenvalues of
A are simply ±is1 , . . . , ±isn , (0).
3. Matrices in the symplectic algebras spn (C) and spn (R)
Consider the complex symplectic Lie algebra [6, p.128-129] which is simple for
n ≥ 1:
spn (C) := sp(n) ⊕ isp(n)
A1 A2
>
>
=
: A1 , A2 , A3 ∈ Cn×n , A2 = A2 , A3 = A3 .
A3 −A>
1
The compact group K = Sp(n, C) ∩ U(2n) [6] consists of the matrices
U −V
∈ U(2n).
V U
It is known [2, Proposition 3.1] that for any B ∈ spn (C), there is U ∈ K such
that U BU ∗ ∈ b ⊂ spn (C), where
A1 A2
>
b :=
, A1 ∈ Cn×n is upper triangular, A2 = A2
(3.1)
0 −A>
1
is a Borel subalgebra of spn (C). The eigenvalues of A ∈ spn (C) occur in pairs but
opposite in sign as we can see it from (3.1). However, unlike the complex skew
symmetric case in the previous section, the singular values of A do not generally
occur in pairs, e.g.,
1 2
A=
∈ sp1 (C)
0 −1
has distinct singular values.
We first have the following simple lemma.
Lemma 3.1. Let
s1 0
cos α sin α
sin α cos α
S=
, U=
, V =
,
0 s2
− sin α cos α
cos α − sin α
98
D. BOZKURT, T.Y. TAM AND W. YAN
where s1 , s2 ≥ 0. Then
(s1 + s2 ) sin α cos α
s1 cos2 α − s2 sin2 α
U SV =
.
−s1 sin2 α + s2 cos2 α −(s1 + s2 ) sin α cos α
√
s2
s1 +s2
√
s
1
and cos α = √s1 +s
, then
2
√
s1 s2 s1 − s2
√
U SV =
.
0
− s1 s2
In particular, if we choose sin α =
√
Theorem 3.2. Let A ∈ spn (C) with singular values s1 ≥ s2 ≥ · · · ≥ s2n−1 ≥ s2n
and eigenvalues ±λ1 , . . . , ±λn ordered as |λ1 | ≥ · · · ≥ |λn |. Then (1.1) and (1.2)
hold for them. Conversely, if |λ1 | ≥ |λ1 | ≥ · · · ≥ |λn | ≥ |λn | and s1 ≥ s2 ≥ · · · ≥
s2n−1 ≥ s2n satisfy (1.1) and (1.2), then there exists A ∈ spn (C) with eigenvalues
±λ1 , . . . , ±λn and singular values s1 ≥ s2 ≥ · · · ≥ s2n−1 ≥ s2n .
Proof. The necessity of (1.1) for s1 ≥ s2 ≥ · · · ≥ s2n−1 ≥ s2n and eigenvalues
±λ1 , . . . , ±λn follows from Theorem 1.1.
For the sufficiency part, suppose that λ1 , λ1 , . . . , λn , λn and s1 , . . . s2n satisfy
(1.1) and (1.2), i.e.,
√
√
s1 s2 ≥ · · · ≥ s2n−1 s2n , |λ1 | ≥ · · · ≥ |λn |
satisfy (1.1) and (1.2). Let S := diag (s1 , s3 , . . . , s2n−1 , s2 , s4 , . . . , s2n ),


cos α1
sin α1


...
...




cos αn
sin αn 

U := 

cos α1
− sin α1



..
..


.
.
− sin αn
cos αn


sin α1
cos α1

..
..

.
.




sin αn
cos αn 

V := 
.
− sin α1
cos α1



.
.
..
..


cos αn
− sin αn
By Lemma 3.1, we can choose appropriate α1 , . . . , αn such that
√

s1 s2
s1 − s2


..
..
.
.


√


s2n−1 s2n
s2n−1 − s2n 

√
U SV = 
.
− s1 s2




.
..


√
− s2n−1 s2n
√
√
√
√
Let B1 := diag ( s1 s2 , . . . , s2n−1 s2n ). Since s1 s2 ≥ · · · ≥ s2n−1 s2n ≥ 0
and |λ1 | ≥ · · · ≥ |λn | satisfy (1.1) and (1.2), by Theorem 1.1 we can find U1 , V1 ∈
SINGULAR VALUES AND EIGENVALUES
99
U(n) such that

λ1
A1 := U1 B1 V1 = 
∗
..
.
∗

(3.2)
∗
λn
√
√
has singular values s1 s2 , . . . , s2n−1 sn and eigenvalues λ1 , . . . , λn . Set Σ :=
diag (s1 − s2 , . . . , s2n−1 − sn ). Then
U1 B1 V1
U1 ΣU1>
>
>
A := (U1 ⊕ V1 )U SV (V1 ⊕ U1 ) =
.
(3.3)
0
−(U1 B1 V1 )>
Since U1 ΣU1> is symmetric, we conclude that A ∈ spn (C) and has eigenvalues
±λ1 , . . . , ±λn . Moreover A has singular values s1 , . . . , s2n because of the above
unitary equivalence. So A is the required matrix.
The split real form of spn (C) [6] is
spn (R)
A1 A 2
>
>
=
: A2 = A2 , A3 = A3 , A1 , A2 , A3 ∈ Rn×n
A3 −A>
1
= spn (C) ∩ R2n×2n .
The nonreal eigenvalues of each A ∈ spn (R) appear in quadruples (λ, −λ, λ̄, −λ̄).
The proof of the following result is similar to that of Theorem 3.2. The role of
Theorem 1.1 in the proof is played by Thompson’s result [8] and U1 , V1 in (3.2)
are orthogonal and A1 is a real “upper triangular” matrix with 2 × 2 diagonal
blocks for nonreal λ’s and 1 × 1 block for real λ’s.
Theorem 3.3. Let A ∈ spn (R) with singular values s1 ≥ s2 ≥ · · · ≥ s2n−1 ≥ s2n
and eigenvalues ±λ1 , . . . , ±λn (the nonreal λ’s appear in quadruples) ordered as
|λ1 | ≥ · · · ≥ |λn |. Then (1.1) and (1.2) hold for them. Conversely, if |λ1 | ≥ |λ1 | ≥
· · · ≥ |λn | ≥ |λn | and s1 ≥ s2 ≥ · · · ≥ s2n−1 ≥ s2n satisfy (1.1) and (1.2) and if
the nonreal λ’s appear in quadruples (λ, −λ, λ̄, −λ̄), then there exists A ∈ spn (R)
such that ±λ1 , . . . , ±λn are the eigenvalues and s1 ≥ s2 ≥ · · · ≥ s2n−1 ≥ s2n are
the singular values of A, respectively.
Acknowledgement. Part of the research was done during Tin-Yau Tam’s
visit at Selçuk University in Konya, Turkey (April 4-28, 2012) under the support
of the Scientific and Technological Research Council of Turkey (TUBITAK) and
Selçuk University Research Projects Coordinator (BAP).
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3. A. Horn, On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math.
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D. BOZKURT, T.Y. TAM AND W. YAN
5. L.K. Hua, On the theory of automorphic functions of a matrix level, I. Geometrical basis,
Amer. J. Math. 66 (1944), 470–488.
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1
Department of Mathematics, Selçuk University, 42075, Konya, Turkey.
E-mail address: dbozkurt@selcuk.edu.tr
2
Department of Mathematics and Statistics, 221 Parker Hall, Auburn University, AL 36849, USA.
E-mail address: tamtiny@auburn.edu
3
Department of Mathematics, Tuskegee University, AL 36088, USA.
E-mail address: wy0615900@mytu.tuskegee.edu