Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈0,1, write Aα(G) for the matrixAα(G)=αD(G)+(1−α)A(G). Let α0(G) be the smallest α ...for which Aα(G) is positive semidefinite. It is known that α0(G)≤1/2. The main results of this paper are:(1)if G is d-regular thenα0=−λmin(A(G))d−λmin(A(G)), where λmin(A(G)) is the smallest eigenvalue of A(G);(2)G contains a bipartite component if and only if α0(G)=1/2;(3)if G is r-colorable, then α0(G)≥1/r.
Let A:B be the parallel sum of the positive semidefinite matrices A and B. In this paper, we first establish an operator inequality involving A:B and then apply it to obtain some norm upper bounds ...for the perturbations of A:B. Our results lead to considerable improvements of the well-known inequalities due to Anderson and Duffin (1969) 2. Examples are also presented to illustrate the sharpness of the obtained norm upper bounds.
This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is infXtr(AˆXHAX) subject to BˆXHBX=I (the ...identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is established in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kovač-Striko and Veselić (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix.
We provide new classes of nonseparable univariate and multivariate space-time covariance functions that extend the Gneiting class. In particular, we prove that the spatial generator of the Gneiting ...class does not need to be completely monotone and can be replaced with the radial profile of a continuous positive semidefinite function in a finite dimensional Euclidean space, and we weaken the assumptions on the temporal margins so as to include more general behaviors than that of the original Gneiting construction. Our findings allow for covariance functions that are compactly-supported in space and/or nonmonotone in time, i.e., they offer versatility to model complex features commonly observed on data indexed with space-time coordinates.
We establish a trace inequality of symplectic matrices via a more general trace minimization theorem. As a consequence, we derive another proof of a result in R. Bhatia, T. Jain, On symplectic ...eigenvalues of positive-definite matrices, J. Math. Phys., 56 (2015),112201..
Let Ai and Bi be positive definite matrices for all i=1,…,m. It is shown that|||∑i=1m(Ai2♯Bi2)r|||≤|||((∑i=1mAi)rp2(∑i=1mBi)rp(∑i=1mAi)rp2)1p|||, for all unitarily invariant norms, where p>0 and r≥1 ...such that rp≥1. This gives an affirmative answer to a conjecture posed by Dinh, Ahsani and Tam. The preceding inequality directly leads to a recent result of Audenaert in 2015.
We give refinements of the classical Young inequality for positive real numbers and we use these refinements to establish improved Young and Heinz inequalities for matrices.
Let A and B be
$ n\times n $
n
×
n
positive semidefinite matrices,
$ f:0,\infty )\rightarrow \lbrack 0,\infty ) $
f
:
0
,
∞
)
→
0
,
∞
)
be an increasing convex function with
$ f\left ( 0\right ) =0 ...$
f
(
0
)
=
0
, and let
$ \Phi : \mathbb {M}_{n}( \mathbb {C} )\rightarrow \mathbb {M}_{n}( \mathbb {C} ) $
Φ
:
M
n
(
C
)
→
M
n
(
C
)
be a positive unital linear map. It is shown that
$$\begin{align*} & \prod_{j=1}^{k} s_{j}\left( f^{4}\left( \Phi \left( \frac{A+B}{2}\right) \right) \right)\\ & \quad \leq 2^{-4} \prod_{j=1}^{k} \left( s_{j}\left( \Phi^{2}\left( f^{2}\left( A\right) \right) +\Phi \left( f^{2}\left( A\right) \right) \right) +\left\Vert \Phi \left( f^{2}\left( B\right) \right) \right\Vert +1\right) \\ & \quad\quad \times \left( s_{j}\left( \Phi^{2}\left( f^{2}\left( B\right) \right) +\Phi \left( f^{2}\left( B\right) \right) \right) +\left\Vert \Phi \left( f^{2}\left( A\right) \right) \right\Vert +1\right) \end{align*}$$
∏
j
=
1
k
s
j
(
f
4
(
Φ
(
A
+
B
2
)
)
)
≤
2
−
4
∏
j
=
1
k
(
s
j
(
Φ
2
(
f
2
(
A
)
)
+
Φ
(
f
2
(
A
)
)
)
+
||
Φ
(
f
2
(
B
)
)
||
+
1
)
×
(
s
j
(
Φ
2
(
f
2
(
B
)
)
+
Φ
(
f
2
(
B
)
)
)
+
||
Φ
(
f
2
(
A
)
)
||
+
1
)
for
$ k=1,2,\ldots,n $
k
=
1
,
2
,
...
,
n
, where
$ s_{j}\left ( T\right ) $
s
j
(
T
)
and
$ \left \Vert T\right \Vert $
||
T
||
are the
$ j{\rm th} $
j
th
singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given.
The classical Hadamard-Fischer-Koteljanskii inequality is an inequality between principal minors of positive definite matrices. In this work, we present an extension of the ...Hadamard-Fischer-Koteljanskii inequality, that is inspired by the inclusion-exclusion formula for sets. We formulate necessary and sufficient conditions for the inequality to hold. We describe general structures of the collection of index sets involved. In analyzing these structures, a graph-theoretical property that applies to bipartite graphs is found. We establish that if the vertices of a bipartite graph satisfy simple conditions, then the bipartite graph contains a vertex subgraph which is a cycle or a complete subgraph missing a matching. This result is reminiscent of the Hall's marriage theorem for bipartite graphs.
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