Journal of Inequalities in Pure and
Applied Mathematics
THE QUATERNION MATRIX-VALUED YOUNG’S INEQUALITY
RENYING ZENG
Department of Mathematics
Saskatchewan Institute of Applied Science and Technology
Moose Jaw, Saskatchewan
CANADA S6H 4R4
volume 6, issue 3, article 89,
2005.
Received 09 December, 2002;
accepted 04 June, 2005.
Communicated by: G.P. Styan
EMail: zeng@siast.sk.ca
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper, we prove Young’s inequality in quaternion matrices: for any n × n
quaternion matrices A and B, any p, q ∈ (1, ∞) with p1 + 1q = 1, there exists
n × n unitary quaternion matrix Usuch that U|AB ∗ |U ∗ ≤ p1 |A|p + 1q |B|q .
Furthermore, there exists unitary quaternion matrix U such that the equality
holds if and only if |B| = |A|p−1 .
The Quaternion Matrix-Valued
Young’s Inequality
2000 Mathematics Subject Classification: 15A45, 15A42.
Key words: Quaternion, Matrix, Young’s inequality, Real representation.
Renying Zeng
Contents
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Matrix-valued Young’s inequality: the Quaternion Version . . 5
3
The Case of Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
The two most important classical inequalities probably are the triangle inequality and the arithmetic-geometric mean inequality.
The triangle inequality states that |α + β| ≤ |α| + |β| for any complex
numbers α, β.
Thompson [7] extended the classical triangle inequality to n × n complex
matrices: for any n × n complex matrices A and B, there are n × n unitary
complex matrices U and V such that
(1.1)
|A + B| ≤ U |A|U ∗ + V |B|V ∗ .
Thompson [6] proved that, the equality in the matrix-valued triangle inequality
(1.1) holds if and only if A and B have polar decompositions with a common
unitary factor.
Furthermore, Thompson [5] extended the complex matrix-valued triangle
inequality (1.1) to the quaternion matrices: for any n × n quaternion matrices
A and B, there are n × n unitary quaternion matrices U and V such that
|A + B| ≤ U |A|U ∗ + V |B|V ∗ .
The arithmetic-geometric mean inequality is as follows: for any complex numbers α, β,
p
1
|αβ| ≤ (|α| + |β|);
2
or,
1
|αβ| ≤ (|α|2 + |β|2 ),
2
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Young’s Inequality
Renying Zeng
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which is a special case of the classical Young’s inequality: for any complex
numbers α, β, and any p, q ∈ (1, ∞) with p1 + 1q = 1,
1
1
|αβ| ≤ |α|p + |β|q .
p
q
Bhatia and Kittaneh [2], Ando [1] extended the classical arithmetic-geometric
mean inequality and Young’s inequality to n×n complex matrices, respectively.
This is Ando’s matrix-valued Young’s inequality: for any n × n complex matrices A and B, any p, q ∈ (1, ∞) with p1 + 1q = 1, there is unitary complex matrix
U such that
1
1
U |AB ∗ |U ∗ ≤ |A|p + |B|q .
p
q
Bhatia and Kittaneh’s result is the case of p = q = 2, i.e., Young’s inequality
recovers Bhatia and Kittaneh’s arithmetic-geometric-mean inequality, likewise,
Ando’s matrix version of Young’s inequality captures the Bhatia-Kittaneh matricial arithmetic-geometric-mean inequality.
We mention that Erlijman, Farenick and the author [8] proved Young’s inequality for compact operators.
This paper extends the Young’s inequality to n × n quaternion matrices and
examines the case where equality in the inequality holds.
The Quaternion Matrix-Valued
Young’s Inequality
Renying Zeng
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2.
Matrix-valued Young’s inequality: the
Quaternion Version
We use R, C, and H to denote the set of real numbers, the set of complex
numbers, and the set of quaternions, respectively.
For any z ∈ H, we have the unique representation z = a1 + bi + cj + dk,
where {1, i, j, k} is the basis of H. It is well-known that I is the multiplicative
identity of H, and 12 = i2 = j 2 = k 2 = −1, ij = k, ki = j, jk = i, and
ji = −k, ik = −j, kj = −i.
For each z = a1 + bi + cj + dk ∈ H, define the conjugate z̄ of z by
The Quaternion Matrix-Valued
Young’s Inequality
Renying Zeng
z̄ = a1 − bi − cj − dk.
Obviously we have z̄z = z z̄ = a2 + b2 + c2 + d2 . This implies that z̄z = z z̄ = 0
if and only if z = 0. So z is invertible in H if z 6= 0.
We note that as subalgebras of H, the meaning of conjugate in R, or C is as
usual (for any z ∈ R we have z̄ = z).
We can consider R and C as real subalgebras of H : R={a1 : a ∈ R}, and
C={a1 + bi : a, b ∈ R}.
We define the real representation ρ of H, i.e., ρ : H → M4 (R) by


a −b −c −d
 b a −d c 
,
ρ(z) = ρ(a1 + bi + cj + dk) = 
 c d
a −b 
d −c b
a
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J. Ineq. Pure and Appl. Math. 6(3) Art. 89, 2005
where z = a1 + bi + cj + dk ∈ H.
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Note that ρ(z̄) is the transpose of ρ(z).
From the real representation ρ of H, we define a faithful representation by
ρn : Mn (H) → M4n (R) as follows:
ρ(A) = ρn ([qst ]ns,t=1 ) = ([ρ(qst )]ns,t=1 )
for all matrices A = [qst ]ns,t=1 ∈ Mn (H).
We note that each ρn is an injective and homomorphism; and for all A ∈
Mn (H),
ρn (A∗ ) = ρn (A)∗ .
For the set Mn (F) of n × n matrices with entries from F, where F is R, C, or
H, we use A∗ to denote the conjugate transpose of A ∈ Mn (F).
We consider Mn (R) and Mn (H) as algebras over R, but Mn (C) as a complex
algebra.
Definition 2.1. The spectrum σ(A) of A ∈ Mn (F) is a subset of C that consists
of all the roots of the minimal monic annihilating polynomial f of A. We note
that if F = R or F = H, then f ∈ R[x]; but if F = H, then f ∈ C[x]. If
F = R or F = C, then the spectrum σ(A) is the set of eigenvalues of A. But if
F = H, then σ(A) is the set of eigenvalues of ρn (A). A is called Hermitian if
A = A∗ . A is said to be nonnegative definite if A is Hermitian and σ(A) are all
non-negative real numbers. A is said to be unitary if A∗ A = AA∗ = I, where I
is the identity matrix in Mn (F).
If A and B are Hermitian, we define A ≤ B or B ≥ A if B − A is nonnegative definite.
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Young’s Inequality
Renying Zeng
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For any Hermitian matrix A, λ1 (A) ≥ λ2 (A) ≥ · · · ≥ λn (A) are its eigenvalues, arranged in descending order; where the number of appearances of a
particular eigenvalue λ is equal to the dimension of the kernel of A − λI and is
known as the geometric multiplicity of λ.
Lemma 2.1 ([1]). If A, B ∈ Mn (C), and if p, q ∈ (1, ∞) with
there is a unitary U ∈ Mn (C) such that
1
p
+
1
q
= 1, then
1
1
U |AB ∗ |U ∗ ≤ |A|p + |B|q ,
p
q
where |A| denotes the nonnegative definite Hermitian matrix
The Quaternion Matrix-Valued
Young’s Inequality
Renying Zeng
1
|A| = (A∗ A) 2 .
Lemma 2.2 ([3]). Let Q ∈ Mn (H), then Q∗ Q is nonnegative definite. Furthermore, if A ∈ Mn (H) is nonnegative definite, then there are matrices U, D ∈
Mn (H) such that
(i) U is unitary and D is diagonal matrix with nonnegative diagonal entries
d1 , d2 , . . . , dn ;
(ii) U ∗ AU = D;
(iii) σ(A) = {d1 , d2 , . . . , dn };
(iv) If µ ∈ σ(A)appears tµ times on the diagonal of D , then the geometric
multiplicity of µ as an eigenvalue of ρn (A)is 4tµ .
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J. Ineq. Pure and Appl. Math. 6(3) Art. 89, 2005
Lemma 2.3. For any A, B ∈ Mn (H),
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(i) ρn (|A|) = |ρn (A)|;
(ii) ρn (|A|p ) = |ρn (A)|p for any nonnegative definite p;
(iii) ρn (|AB|) = |ρn (A)ρn (B)|.
1
The meaning of |A| is similar to that in Lemma 2.1, i.e., |A| = (A∗ A) 2 .
Proof. (i) Note that ρn : Mn (H) → M4n (R) is a homomorphism, if X ∈
Mn (H) is nonnegative definite, then there is a Y ∈ Mn (H) such that X = Y Y ∗ ,
so
The Quaternion Matrix-Valued
Young’s Inequality
ρn (X) = ρn (Y ∗ Y ) = ρn (Y ∗ ) · ρn (Y ) = ρn (Y )∗ · ρn (Y ) = |ρn (Y )|2,
Renying Zeng
which means that ρn (X) is also nonnegative definite. Hence, for any X ∈
Mn (H) we have (since ρn is a homomorphism),
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1 2
ρn (|X|) 2
1
1
= ρn (|X|) = ρn |X| 2 · |X| 2
2
1
= ρn |X| 2
1
1
So ρn (|X|) 2 = ρn |X| 2 . Therefore
1
.
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ρn (|A|) = (ρn (A∗ A)) 2 = (ρn (A∗ )ρn (A)) 2 = |ρn (A)|.
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We get (i).
(ii) For any nonnegative definite p,
ρn (|A|p ) = (ρn (|A|))p = |ρn (A)|p ,
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the first equality is because ρn : Mn (H) → M4n (R) is a homomorphism, and
the second equality is from (i).
(iii) Similar to (ii) we have
ρn (|AB|) = |ρn (AB)| = |ρn (A)ρn (B)|.
The proof is complete.
The following Theorem 2.4 is one of our main results.
Theorem 2.4. For any A, B ∈ Mn (H), any p, q ∈ (1, ∞) with p1 + 1q = 1, there
is a unitary U ∈ Mn (H), such that
1
1
U |AB ∗ |U ∗ ≤ |A|p + |B|q .
p
q
Proof. By Lemma 2.3 ρn (|AB ∗ |) = |ρn (A)ρn (B)∗ |, and
1 p 1 q
1
1
ρn
|A| + |B| = |ρn (A)|p + |ρn (B)|q .
p
q
p
q
Because real n × n matrices |ρn (A)ρn (B)∗ | and p1 |ρn (A)|p + 1q |ρn (B)|q are
nonnegative definite, from Linear Algebra there are n × n unitary matrices
V, W ∈ Mn (C) such that
1
1
∗
∗
p
q
V |ρn (A)ρn (B) |V = C and W
|ρn (A)| + |ρn (B)| W ∗ = D,
p
q
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Young’s Inequality
Renying Zeng
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J. Ineq. Pure and Appl. Math. 6(3) Art. 89, 2005
where C and D are diagonal matrices in M4n (R).
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Thus from Lemma 2.2(iv) one has
C = C1 ⊕ C2 ⊕ · · · ⊕ Cn
and
D = D1 ⊕ D2 ⊕ · · · ⊕ Dn
with Cs = diag{cs , cs , . . . , cs } and Ds = diag{ds , ds , . . . , ds }, where cs and ds
are nonnegative real numbers, s = 1, 2, . . . , n. By Lemma 2.2 (iii) we have
σ(|AB ∗ |) = {c1 , c2 , . . . , cs }
and
1
1
σ
|ρn (A)|p + |ρn (B)|q
p
q
Furthermore, Lemma 2.2 implies that
= {d1 , d2 , . . . , dn }.
C = C1 ⊕ C2 ⊕ · · · ⊕ Cn ≤ D = D1 ⊕ D2 ⊕ · · · ⊕ Dn .
Hence the equation above and Lemma 2.3 yield that
diag{c1 , c2 , . . . , cn } ≤ diag{d1 , d2 , . . . , dn }.
Thus from Lemma 2.2 (i) (ii) (iii) there are unitary matrices U1 , U2 ∈ Mn (H)
such that
1 p 1 q
∗
∗
U1 |AB |U1 ≤ U2
|A| + |B| U2∗ ,
p
q
then there is a unitary matrix U ∈ Mn (H) for which
1
1
U |AB ∗ |U ∗ ≤ |A|p + |B|q .
p
q
The proof is complete.
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Young’s Inequality
Renying Zeng
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3.
The Case of Equality
Hirzallah and Kittaneh [4] proved a result as follows.
Lemma 3.1. Let A, B ∈ Mn (C) be nonnegative definite. If p, q ∈ (1, ∞) with
1
+ 1q = 1, and if there exists unitary U ∈ Mn (C) such that
p
1
1
U |AB|U ∗ = Ap + B q
p
q
then B = Ap−1 .
The Quaternion Matrix-Valued
Young’s Inequality
We have the following result.
Renying Zeng
1
p
1
q
Theorem 3.2. For any A, B ∈ Mn (H), any p, q ∈ (1, ∞) with + = 1, there
is a unitary U ∈ Mn (H) such that
(3.1)
1
1
U |AB ∗ |U ∗ = |A|p + |B|q
p
q
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if and only if |B| = |A|p−1 .
Proof. The sufficiency. In fact, if |B| = |A|p−1 then
p−1
|ρn (B)| = ρn (|B|) = ρn (|A|
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p−1
) = |ρn (A)|
.
Write X = ρn (A), Y = ρn (B).
Suppose X = V |X|, Y = W |Y | are the polar decomposition of X, Y respectively, where V, W are 4n × 4n unitary complex matrices. Then from (3.1)
we have
|XY ∗ | = W ||X||Y ||W ∗ = W |X|p W ∗ .
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Simply computation yields
1
1
|X|p + |Y |q = |X|p .
p
q
So
1
1
W ∗ |XY ∗ |W = |X|p + |Y |q .
p
q
Since W is a unitary, using the notations in Theorem 2.4, this implies
C = C1 ⊕ C2 ⊕ · · · ⊕ Cn = D = D1 ⊕ D2 ⊕ · · · ⊕ Dn .
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Young’s Inequality
Renying Zeng
Hence Lemma 2.2 yields that
diag{c1 , c2 , . . . , cn } = diag{d1 , d2 , . . . , dn }.
Again, by Lemma 2.2, there is a unitary U ∈ Mn (H) such that
1
1
U |AB ∗ |U ∗ = |A|p + |B|q .
p
q
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The necessity. Assume there exists unitary U ∈ Mn (H) such that (3.1) holds,
i.e.
1
1
U |AB ∗ |U ∗ = |A|p + |B|q .
p
q
Then
1 p 1 q
∗
∗
|A| + |B| .
ρn (U |AB |U ) = ρn
p
q
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Writing X = ρn (A), Y = ρn (B), and T = ρn (U ), one gets
1
1
T |XY ∗ |T ∗ = |X|p + |Y |q .
p
q
This and Lemma 3.1 imply that
1
|Y | = (|X|p ) q = |X|,
which means
p−1
ρn (|B|) = ρn (|A|)
p−1
= ρn (|A|
).
Therefore (note that ρn : Mn (H) → M4n (R) is a faithful representation)
|B| = |A|p−1 .
This completes the proof.
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Young’s Inequality
Renying Zeng
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References
[1] T. ANDO, Matrix Young’s inequality, Oper. Theory Adv. Appl., 75 (1995),
33–38.
[2] R. BHATIA AND F. KITTANEH, On the singular values of a product of
operators, SIAM J. Matrix Anal. Appl., 11 (1990), 272–277.
[3] J.L. BRENNER, Matrices of quaternion, Pacific J. Math., 1 (1951), 329–
335.
[4] O. HIRZALLAH AND F. KITTANEH, Matrix Young inequalities for the
Hilbert-Schmidt norm, Linear Algebra and Application, 308 (2000), 77–
84.
[5] R.C. THOMPSON, Matrix-valued triangle inequality: quaternion version,
Linear and Multilinear Algebra, 25 (1989), 85–91
[6] R.C. THOMPSON, The case of equality in the matrix-valued triangle inequality, Pacific J. of Math., 82 (1979), 279–280.
[7] R.C. THOMPSON, Convex and concave functions of singular values of
matrix sums, Pacific J. Math., 66 (1976), 285–290.
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Young’s Inequality
Renying Zeng
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[8] J. ERLIJMAN, D.R. FARENICK AND R. ZENG, Young’s inequality in
compact operators, Oper. Theory. Adv. and Appl., 130 (2001), 171–184.
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