- Research
- Open access
- Published:
Some inequalities for coneigenvalues
Journal of Inequalities and Applications volume 2015, Article number: 15 (2015)
Abstract
In this manuscript, we present some inequalities for coneigenvalues which extend some classical relations between eigenvalues and singular values.
1 Introduction
Let \(M_{n}(\mathbb{C})\) be the space of complex \(n\times n\) matrices and \(\mathbb{R}^{+}\) be nonnegative real numbers. For any \(A\in M_{n}(\mathbb{C})\), the conjugate transpose of A is denoted by \(A^{*}\), i.e., \(A^{*}=\overline{A}^{T}=\overline{A^{T}}\). \(A^{T}\) stands for the transpose of A. The real part of A is denoted by \(\Re(A)= \frac{A+A^{*}}{2}\). If \(A^{*}A=AA^{*}\), we call A normal. If \(A=A^{*}\), A is Hermitian. Here, for \(A\in M_{n}(\mathbb{C})\), \(\lambda^{\downarrow }(A)=(\lambda^{\downarrow}_{1}(A),\ldots,\lambda^{\downarrow }_{n}(A))\) (\(\sigma^{\downarrow}(A)=(\sigma^{\downarrow }_{1}(A),\ldots,\sigma^{\downarrow}_{n}(A))\)) represents the eigenvalues (singular values) of A in decreasing order, \(\lambda ^{\downarrow}_{1}(A)\geq\cdots\geq\lambda^{\downarrow}_{n}(A)\) (\(\sigma^{\downarrow}_{1}(A)\geq\cdots\geq\sigma^{\downarrow }_{n}(A)\)). For \(a\in\mathbb{R}^{+}\), \([a]\) means an integral part of a.
For \(A\in M_{n}(\mathbb{C})\), we define \(B=\overline{A}A\). Thus \(\lambda(B)=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}\) is symmetric with respect to the real axis and the negative eigenvalues of B (if any) are of even algebraic multiplicity. Therefore, the definition of coneigenvalue is given below.
Definition 1
([1, p.301])
The coneigenvalues of \(A\in M_{n}(\mathbb{C})\) are n scalars \(\mu_{1},\mu_{2},\ldots,\mu _{n}\) obtained as follows:
-
1.
If \(\lambda_{k}\in\lambda(B)\) does not lie on the negative real semi-axis, then the corresponding coneigenvalue \(\mu_{k}\) is defined as the square root of \(\lambda_{k}\) with a nonnegative real part. The multiplicity of \(\mu_{k}\) is set equal to that of \(\lambda_{k}\).
-
2.
With a real negative \(\lambda_{k}\in\lambda(B)\), we associate two conjugate purely imaginary coneigenvalues (i.e., the two square roots of \(\lambda_{k}\)). The multiplicity of each coneigenvalue is set equal to half the multiplicity of \(\lambda_{k}\).
For \(A\in M_{n}(\mathbb{C})\), the vector of its coneigenvalues will be denoted by
The definition of conjugate-normal is presented as follows.
Definition 2
([2])
A matrix \(A \in M_{n}(\mathbb{C})\) is said to be conjugate-normal if
Complex symmetric, skew-symmetric, and unitary matrices are special subclasses of conjugate-normal matrices. For the properties and characterizations of this kind of matrices, readers are referred to [3].
Next, we review two known properties about coneigenvalues.
Define the matrix \(\widehat{A}=\bigl( {\scriptsize\begin{matrix} 0 & A \cr \overline{A} & 0\end{matrix}} \bigr)\).
Proposition 3
([4])
If \(\mu_{1}, \mu_{2},\ldots, \mu_{n}\) are the coneigenvalues of the matrix \(A\in M_{n}(\mathbb{C})\), then
Proposition 4
([4])
Let A be a conjugate-normal matrix. Then the coneigenvalues of the matrices \(\frac{A+A^{T}}{2}\) and \(\frac{A-A^{T}}{2}\) are the real and imaginary parts, respectively, of the coneigenvalues of A.
2 Main results
The purpose of this paper is to extend the property of the relations between eigenvalues and singular values to that of the relations between coneigenvalues and singular values.
A celebrated result due to Fan and Hoffman [5, p.63] is given in the first lemma.
Lemma 1
([5, p.63])
Let \(A\in M_{n}(\mathbb{C})\), then
The following lemmas state the relation between eigenvalues and singular values.
Lemma 2
([6, p.175])
Let \(A\in M_{n}(\mathbb{C})\) have ordered singular values \(\sigma _{1}(A)\geq\cdots\geq\sigma_{n}(A) \geq0\) and eigenvalues \({\lambda_{1}(A),\ldots,\lambda_{n}(A)}\) ordered so that \(|\lambda _{1}(A)|\geq\cdots\geq|\lambda_{n}(A)|\). Then, for any real-valued function f such that \(\varphi(t)=f(e^{t})\) is increasing and convex on the interval \([\sigma_{n}(A), \sigma_{1}(A)]\),
Lemma 3
([6, p.183, Problem 14])
Let \(A\in M_{n}(\mathbb{C})\). \(|\lambda_{1}(A)\cdots\lambda _{k}(A)|=\sigma_{1}(A)\cdots\sigma_{k}(A)\) for all \(k=1,\ldots, n\) if and only if A is normal.
Lemma 4
([6, p.185, Problem 17])
Let \(A\in M_{n}(\mathbb{C})\) be given,
The following lemma is about the property of conjugate-normal.
Lemma 5
([3, Theorem 4])
Let \(A\in M_{n}(\mathbb{C})\). Then A is conjugate-normal if and only if \(\widehat{A}\) is normal.
Now we extend the property of the relations between eigenvalues and singular values in the previous lemmas to that of the relations between coneigenvalues and singular values in the following theorems.
Theorem 6
Let \(A\in M_{n}(\mathbb{C})\). Then
Proof
Define
Let \(\lambda^{\downarrow}\bigl( {\scriptsize\begin{matrix} 0 & \frac{A+A^{T}}{2} \cr \frac{\overline{A+A^{T}}}{2} & 0\end{matrix}} \bigr) \) be the vector obtained by rearranging the coordinates of \(\lambda\bigl( {\scriptsize\begin{matrix} 0 & \frac{A+A^{T}}{2} \cr \frac{\overline{A+A^{T}}}{2} & 0\end{matrix}} \bigr)\) in decreasing order. That is,
By Proposition 3, \(\lambda^{\downarrow}\bigl( {\scriptsize\begin{matrix} 0 & \frac{A+A^{T}}{2} \cr \frac{\overline{A+A^{T}}}{2} & 0\end{matrix}} \bigr)\) is denoted by
In the same way, we define the singular value vector of \(\sigma \bigl( {\scriptsize\begin{matrix} 0 & A \cr \overline{ A} & 0\end{matrix}} \bigr) \) as
Therefore,
So the component of the vector in (2.1) is less than that of the vector in (2.2), respectively.
That is,
□
Theorem 7
Let \(A\in M_{n}(\mathbb{C})\) and a real-valued function f be such that the function \(\varphi(t)=f(e^{t})\) is increasing and convex on the interval \([\sigma_{n}(A), \sigma_{1}(A)]\) where singular values of A are ordered by \(\sigma_{1}(A)\geq\cdots\geq\sigma_{n}(A) \geq 0\), then
Proof
For the vector \(\lambda(\widehat{A})=\lambda \bigl( {\scriptsize\begin{matrix} 0 & A \cr \overline{A}& 0\end{matrix}} \bigr) = (\mu(A),-\mu(A))\), let \(|\lambda^{\downarrow} (\widehat{A})| \) be the vector obtained by rearranging the coordinates of \(|\lambda(\widehat{A})|\) in decreasing order. Thus
By Proposition 3, \(|\lambda^{\downarrow}(\widehat{A})| \) is denoted by
In the same way, we define the singular value vector of \(\sigma ( {\scriptsize\begin{matrix} 0 & A \cr \overline{ A} & 0\end{matrix}} ) \) as
By Lemma 2, we have that
By (2.3) and (2.4), inequality (2.5) is equivalent to the following inequality:
□
Theorem 8
Let \(A\in M_{n}(\mathbb{C})\). \(|\mu_{1}(A)\cdots\mu _{2i-1}(A)|=\sigma_{1}(A)\cdots\sigma_{2i-1}(A)\) for all \(i=1, 2,\ldots, n\) if and only if A is conjugate-normal (i.e., \(AA^{*}=\overline{A^{*}A}\)).
Proof
⟸ As A is conjugate-normal, thus by Lemma 5 we know that \(\widehat{A}\) is normal: Since \(\widehat{A}=\bigl( {\scriptsize\begin{matrix} 0 & A \cr \overline{ A} & 0\end{matrix}} \bigr)\), it is easy to obtain that
and
That is, \(\widehat{ A}^{*}\widehat{A}=\widehat{A}\widehat{ A}^{*}\). Thus, by Lemma 3, we have that
Furthermore,
So, by (2.6) and (2.7), we have that
⟹ By Lemma 3 and Lemma 5, the result is obvious. □
Theorem 9
Let \(A\in M_{n}(\mathbb{C})\). \(|\mu_{1}(A)\cdots\mu_{2i-1}(A)|\leq \sigma_{1}(A^{m})^{\frac{1}{m}}\cdots\sigma_{2i-1}(A^{m})^{\frac {1}{m}}\) for all \(i=1, 2,\ldots, n\), \(m=1,2,\ldots\) .
Proof
Let
and
By Lemma 4, we have that
for all \(k=1,\ldots, 2n\), \(m=1, 2\ldots\) .
By (2.8) and (2.9), we have that
for all \(i=1, 2,\ldots, n\); \(m=1,2,\ldots\) . □
The relations between eigenvalues or singular values are very active. It is expected that more results on coneigenvalues will be attractive in the future.
References
Horn, RA, Johnson, CR: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Ikramov, KD: On pseudo-eigenvalues and singular numbers of a complex square matrix. Zap. Nauč. Semin. POMI 334, 111-120 (2006) (in Russian). Translation in J. Math. Sci. (N.Y.) 141, 1639-1642 (2007)
Faßender, H, Ikramov, KD: Conjugate-normal matrices: a survey. Linear Algebra Appl. 429, 1425-1441 (2008)
De Sterck, H, Lin, M: Some majorization inequalities for coneigenvalues. Electron. J. Linear Algebra 23, 669-677 (2012)
Zhan, X: Matrix Theory. Higher Education Press, Beijing (2008) (in Chinese)
Horn, RA, Johnson, CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Acknowledgements
The author is grateful to Dr. Limin Zou for fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author has read the manuscript carefully.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Fu, X. Some inequalities for coneigenvalues. J Inequal Appl 2015, 15 (2015). https://doi.org/10.1186/s13660-014-0539-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-014-0539-6