# More on semipositive tensor and tensor complementarity problem

R. Deba,1 and A. K. Dasb,2
_a Jadavpur University, Kolkata , 700 032, India._
_b Indian Statistical Institute, 203 B. T. Road, Kolkata, 700 108, India._
_1 Email<EMAIL_ADDRESS>
_2 Email<EMAIL_ADDRESS>

###### Abstract

In recent years several classes of structured matrices are extended to classes
of tensors in the context of tensor complementarity problem. The tensor
complementarity problem is a class of nonlinear complementarity problem where
the involved functions are special polynomials defined by a tensor.
Semipositive and strictly semipositive tensors play an important role in the
study of the tensor complementarity problem. The article considers some
important properties of semipositive tensor. We establish invariance property
of semipositive tensor. We prove necessary and sufficient conditions for a
tensor to be semipositive tensor. A relation between even order row diagonal
semipositive tensor and its majorization matrix is proposed.

Keywords: Tensor complementarity problem, semipositive tensor, semimonotone
matrix, null vector, majorization matrix.

AMS subject classifications: 90C33, 47H60, 15A69, 46G25.

11footnotetext: Corresponding author

##  Introduction

A tensor is a multidimensional array which is a natural extension of matrices.
A real tensor of order $r$ and dimension $n$,
$\mathscr{M}=(m_{i_{1}...i_{r}})$ is a multidimensional array of entries
$m_{i_{1}...i_{r}}\in\mathbb{R}$ where $i_{j}\in I_{n}$ with $j\in I_{r}.$
Here $I_{n}$ denotes the set $I_{n}=\\{1,2,...,n\\}.$ $\mathbb{T}_{r,n}$
denotes the set of real tensors of order $m$ and dimension $n.$ Tensors have
many application in science and engineering. The common applications are found
in electromagnetism, continuum mechanics, quantum mechanics and quantum
computing, spectral hypergraph theory, diffusion tensor imaging, image
authenticity verification problem, optimization theory and in many other
areas. In optimization theory the tensor complementarity problem (TCP) was
proposed by Song and Qi [40], which is a natural extension of the linear
complementarity problem and a subclass of the nonlinear complementarity
problem where the involved functions are defined by a tensor. Given a square
matrix $A$ of order $n$ with real entries and an $n$ dimensional vector $q$,
the linear complementarity problem [2] is to find $n$ dimensional vectors $w$
and $z$ satisfying

$z\geq 0,\;\;w=Az+q\geq 0,\;\;z^{T}w=0.$ (1.1)

The problem is denoted by LCP$(q,A)$ and the solution set of LCP$(q,A)$ is
denoted by SOL$(q,A).$ Several matrix classes arising in linear
complementarity problem are important due to computational point of view. For
details see [25], [28], [23], [27], [12], [14], [29], [6]. For details of game
theory see [20], [26], [31], [22], [21], [5] and for details of QMOP see [19].
Even several matrix classes arise during the study of Lemke’s algorithm as
well as principal pivot transform. For details see [18], [17] [24], [4], [30],
[12], [14], [13].

Given a tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ and an $n$ dimensional real
vector $q$, the tensor complementarity problem is to find $n$ dimensional
vectors $\omega$ and $u$ satisfying

$u\geq 0,\;\;\omega=\mathscr{M}u^{r-1}+q\geq 0,\;\;u^{T}\omega=0.$ (1.2)

The problem is denoted by TCP$(q,\mathscr{M}).$ The solution set of
TCP$(q,\mathscr{M})$ is denoted by SOL$(q,\mathscr{M}).$ Tensor
complementarity problem arises in optimization theory, game theory and in
other areas. Many practical problems can be modeled as forms of equivalent
tensor complementarity problem like a class of multi-person noncooperative
game [9], hypergraph clustering problem and traffic equilibrium problem [10].
Tensor classes play an important role to study the theory of TCP. In recent
years many structured tensors are developed and studied in context of TCP. For
details see [34, 37, 38, 41, 32, 16]. In linear complementarity problem the
class of semimonotone matrices play an important role. Several matrix
theoretic properties and properties related to solution of linear
complementarity problem involving semimonotone matrices were studied in
literature. For details see [15, 33, 7, 3, 42]. Song and Qi [37] extended the
class of semimonotone matrices to semipositive tensors in context of tensor
complementarity problem. Song and Qi [37] proved that a strictly semipositive
tensor is a $Q$ tensor. Song and Qi [39] showed that for a (strictly)
semipositive tensor $\mathscr{A}$ the TCP$(q,\mathscr{M})$ has a unique
solution for every $q>0\;(q\geq 0).$ Many other tensor theoretic properties as
well as properties of solution of TCP with semipositive tensor were discussed.
For details see [37, 39, 41, 8, 43].

In this paper, we study some tensor theoretic properties of semipositive
tensor as well as strictly semipositive tensor. We discuss some necessary and
sufficient condition for (strictly) semipositive tensor. We prove an
equivalence in connection with the majorization matrix of an even order row
diagonal (strictly) semipositive tensor.

The paper is organised as follows. Section 2 contains some basic notations and
results. In Section 3, we investigate some tensor theoretic properties of
semipositive tensor. We establish invariance property of semipositive tensor.
We propose necessary and sufficient condition for semipositive tensor. We
establish connection between an even order row diagonal semipositive tensor
and its majorization matrix.

##  Preliminaries

Now we introduce some basic notations used in this paper. We consider vectors,
matrices and tensors with real entries. For any positive integer $n,$ let
$I_{n}$ denote the set $\\{1,2,...,n\\}$. Let $\mathbb{R}^{n}$ denote the
$n$-dimensional Euclidean space and
$\mathbb{R}^{n}_{+}=\\{u\in\mathbb{R}^{n}:u\geq 0\\}$,
$\mathbb{R}^{n}_{++}=\\{u\in\mathbb{R}^{n}:u>0\\}$. Any vector
$u\in\mathbb{R}^{n}$ is a column vector and $u^{T}$ denotes the row transpose
of $u.$ A diagonal matrix $D=[d_{ij}]_{n\times
n}=diag(d_{1},\;d_{2},\;...,\;d_{n})$ is defined as
$d_{ij}=\left\\{\begin{array}[]{ll}d_{i}&;\;\forall\;i=j,\\\
0&;\;\forall\;i\neq j.\end{array}\right.$

######  definition 2.1:

[33, 1] A matrix $M\in\mathbb{R}^{n\times n}$ is said to be semimonotone, if
for every $0\neq z\geq 0,\;\exists$ an index $k\in I_{n}$ such that $z_{k}>0$
and $(Mz)_{k}\geq 0.$

######  definition 2.2:

[33, 1] A matrix $M\in\mathbb{R}^{n\times n}$ is said to be strictly
semimonotone, if for every $0\neq z\geq 0,\;\exists$ an index $k\in I_{n}$
such that $z_{k}>0$ and $(Mz)_{k}>0.$

Let $r$th order $n$ dimensional real tensor $\mathscr{M}=(m_{i_{1}...i_{r}})$
be a multidimensional array of entries $m_{i_{1}...i_{r}}\in\mathbb{R}$ where
$i_{j}\in I_{n}$ with $j\in I_{r}$. $\mathbb{T}_{r,n}$ denotes the set of real
tensors of order $r$ and dimension $n.$ Any
$\mathscr{M}=(m_{i_{1}...i_{r}})\in\mathbb{T}_{r,n}$ is called a symmetric
tensor, if the entries $m_{i_{1}...i_{r}}$ are invariant under any permutation
of their indices. $\mathbb{S}_{r,n}$ denotes the collection of all symmetric
tensors of order $r$ and dimension $n$ where $r$ and $n$ are two given
positive integers with $r,n\geq 2$. An identity tensor of order $r,$
$\mathscr{I}=(\delta_{i_{1}...i_{r}})\in\mathbb{T}_{r,n}$ is defined as
follows:
$\delta_{i_{1}...i_{r}}=\left\\{\begin{array}[]{ll}1&:\;i_{1}=...=i_{r}\\\
0&:\;else\end{array}\right..$ Let $\mathscr{O}$ denote the zero tensor where
each entry of $\mathscr{O}$ is zero. For any $u\in\mathbb{R}^{n}$ and
$p\in\mathbb{R}$, let $u^{[p]}$ denote the vector
$(u_{1}^{p},\;u_{2}^{p},\;...,\;u_{n}^{p})^{T}.$ For
$\mathscr{M}\in\mathbb{T}_{r,n}$ and
$u\in\mathbb{R},\;\mathscr{M}u^{r-1}\in\mathbb{R}^{n}$ is a vector defined by

$(\mathscr{M}u^{r-1})_{i}=\sum_{i_{2},...,i_{r}=1}^{n}m_{ii_{2}...i_{r}}u_{i_{2}}\cdot\cdot\cdot
u_{i_{r}},\mbox{ for all }i\in I_{n}$

and $\mathscr{M}u^{r}\in\mathbb{R}$ is a scalar defined by

$u^{T}\mathscr{M}u^{r-1}=\mathscr{M}u^{m}=\sum_{i_{1},...,i_{r}=1}^{n}m_{i_{1}...i_{r}}u_{i_{1}}\cdot\cdot\cdot
u_{i_{r}}.$

For a tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ a vector $u\in\mathbb{R}^{n}$ is
said to be a null vector of $\mathscr{M},$ if $\mathscr{M}u^{r-1}=0.$

The general product of tensors was introduced by Shao [35]. Let $\mathscr{A}$
and $\mathscr{B}$ be two $n$ dimensional tensor of order $p\geq 2$ and $r\geq
1,$ respectively. The product $\mathscr{A}\mathscr{B}$ is an $n$ dimensional
tensor $\mathscr{C}$ of order $((p-1)(r-1))+1$ with entries

$c_{j\beta_{1}\cdots\beta_{p-1}}=\sum_{j_{2},\cdots,j_{p}\in
I_{n}}a_{jj_{2}\cdots j_{p}}b_{j_{2}\beta_{1}}\cdots b_{j_{p}\beta_{p-1}},$

where $j\in I_{n}$, $\beta_{1},\cdots,\beta_{p-1}\in I_{n}^{r-1}$.

######  definition 2.3:

[34, 38] Given $\mathscr{M}=(m_{i_{1}...i_{r}})\in\mathbb{T}_{r,n}$ and an
index set $J\subseteq I_{n}$ with $|J|=l,\;1\leq l\leq n,$ a principal
subtensor of $\mathscr{M}$ is denoted by $\mathscr{M}^{J}_{l}$ and is defined
as $\mathscr{M}^{J}_{l}=(m_{i_{1}...i_{r}}),\;\forall\;i_{1},i_{2},...i_{r}\in
J.$

######  definition 2.4:

[41] Given $\mathscr{M}\in\mathbb{T}_{r,n}$ and $q\in\mathbb{R}^{n}$, a vector
$u$ is said to be (strictly) feasible solution of TCP$(q,\mathscr{M}),$ if
$u\geq 0\;(>0)$ and $\mathscr{M}u^{r-1}+q\geq 0\;(>0)$.

######  definition 2.5:

[41] Given $\mathscr{M}\in\mathbb{T}_{r,n}$ and $q\in\mathbb{R}^{n}$,
TCP$(q,\mathscr{M})$ is said to be (strictly) feasible if a (strictly)
feasible vector exists.

######  definition 2.6:

[41] Given $\mathscr{M}\in\mathbb{T}_{r,n}$ and $q\in\mathbb{R}^{n}$,
TCP$(q,\mathscr{M})$ is said to be solvable if there exists a feasible vector
$u$ satisfying $u^{T}(\mathscr{M}u^{r-1}+q)=0$ and $u$ is said to be a
solution of the TCP$(q,\mathscr{M})$.

######  definition 2.7:

[38] A tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ is said to be a
$P_{0}(P)$-tensor, if for each $u\in\mathbb{R}^{n}\backslash\\{0\\}$, there
exists an index $i\in I_{n}$ such that $u_{i}\neq 0$ and
$u_{i}(\mathscr{M}u^{r-1})_{i}\geq 0\;(>0)$.

######  definition 2.8:

[37, 41] A tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ is said to be a
$R_{0}$-tensor if the TCP$(0,\mathscr{M})$ has unique $0$ solution.

######  definition 2.9:

[37, 41] A tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ is said to be a $R$-tensor
if it is a $R_{0}$-tensor and the TCP$(e,\mathscr{M})$ has unique solution
$0,$ for $e=(1,1,...,1)^{T}.$

######  definition 2.10:

[11] A tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ is said to be a $Q$-tensor if
the TCP$(q,\mathscr{M})$ is solvable for all $q\in\mathbb{R}^{n}$.

i.e., $u\geq 0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\omega=\mathscr{M}u^{r-1}+q\geq 0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
u^{T}\omega=0.$

######  definition 2.11:

[36] The $i$th row subtensor of $\mathscr{M}\in\mathbb{T}_{r,n}$ is denoted by
$R_{i}(\mathscr{M})$ and its entries are given as
$(R_{i}(\mathscr{M}))_{i_{2}...i_{r}}=(m_{ii_{2}...i_{r}})$, where $i_{j}\in
I_{n}$ and $2\leq j\leq r.$ $\mathscr{M}$ is said to be a row diagonal tensor
if all its row subtensors $R_{1}(\mathscr{M}),...,R_{n}(\mathscr{M})$ are
diagonal tensors.

######  definition 2.12:

[36] Given $\mathscr{M}=(m_{i_{1}...i_{r}})\in\mathbb{T}_{r,n},$ the
majorization matrix of $\mathscr{M}$ is a matrix of order $n,$ denoted by
$\tilde{\mathscr{M}}$ and is defined as
$\tilde{\mathscr{M}}_{ij}=(m_{ij...j}),$ $\forall\;i,j\in I_{n}.$

######  lemma 2.1:

[36] Let $\mathscr{M}\in\mathbb{T}_{r,n}$ and $\tilde{\mathscr{M}}$ be the
majorization matrix of $\mathscr{M}.$ Then $\mathscr{M}$ is row diagonal
tensor if and only if $\mathscr{M}=\tilde{\mathscr{M}}\mathscr{I},$ where
$\mathscr{I}$ is the identity tensor of order $r$ and dimension $n.$

######  theorem 2.1:

[39] Let $\mathscr{M}=(m_{i_{1},...,i_{r}})\in\mathbb{T}_{r,n}.$ Then each
principal subtensor of a (strictly) semipositive tensor is (strictly)
semipositive tensor.

######  theorem 2.2:

[39] Let $\mathscr{M}=(m_{i_{1},...,i_{r}})\in\mathbb{T}_{r,n}.$ The following
statements are equivalent:
(a) $\mathscr{M}$ is semipositive tensor.
(b) The TCP$(q,\mathscr{M})$ has a unique solution for every $q>0.$
(c) For every index set $J\subseteq I_{n}$ with $|J|=r,$ the system
$\mathscr{M}^{J}_{r}u_{J}^{r-1}<0,\;\;u_{J}\geq 0$ has no solution where,
$u_{J}\in\mathbb{R}^{r}.$

######  theorem 2.3:

[39] Let $\mathscr{M}=(m_{i_{1},...i_{r}})\in\mathbb{T}_{r,n.}$ The following
statements are equivalent:
(a) $\mathscr{M}$ is strictly semipositive tensor.
(b) The TCP$(q,\mathscr{M})$ has a unique solution for every $q\geq 0.$
(c) For every index set $J\subseteq I_{n}$ with $|J|=r,$ the system
$\mathscr{M}^{J}_{r}u_{J}^{r-1}\leq 0,\;\;u_{J}\geq 0,\;\;u_{J}\neq 0$ has no
solution where, $u_{J}\in\mathbb{R}^{r}.$

##  Main results

At the outset, we define semipositive tensor along with an example.
Subsequently, we establish the invariance property of semipositive tensor with
respect to diagonal matrix as well as permutation matrix.

######  definition 3.13:

[37] A tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ is said to be (strictly)
semipositive tensor if for each $u\in\mathbb{R}^{n}_{+}\backslash\\{0\\}$,
there exists an index $l\in I_{n}$ such that $u_{l}>0$ and
$(\mathscr{M}u^{r-1})_{l}\;\geq 0\;(>0)$.

Here we give an example of a semipositive tensor.

######  example 3.1:

Let $\mathscr{M}\in\mathbb{T}_{4,3}$ such that
$m_{1211}=1,\;m_{1233}=2,\;m_{1323}=-1,\;m_{1223}=-3,\;m_{1232}=4,\;m_{2211}=1,\;m_{2223}=-3,\;m_{2322}=5,\;m_{3232}=-1,\;m_{3322}=-3,\;m_{3313}=-2$
and all other entries of $\mathscr{M}$ are zero. For
$u=\left(\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\end{array}\right)\in\mathbb{R}^{3}$ we have
$\mathscr{M}u^{3}=\left(\begin{array}[]{c}u_{1}^{2}u_{2}+u_{2}u_{3}^{2}+u_{2}^{2}u_{3}\\\
u_{1}^{2}u_{2}+2u_{2}^{2}u_{3}\\\
-2u_{1}u_{3}^{2}-4u_{2}^{2}u_{3}\end{array}\right).$ Then $\mathscr{M}$ is a
semipositive tensor.

######  theorem 3.1:

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ and $D\in\mathbb{R}^{n\times n}$ be a
positive diagonal matrix. Then $\mathscr{M}$ is (strictly) semipositive tensor
if and only if $D\mathscr{M}$ is (strictly) semipositive tensor.

###### Proof.

Let $D=diag(d_{1},...,d_{n})$ be a positive diagonal matrix. Then the diagonal
entries $d_{i}>0,\;\forall\;i\in I_{n}.$ Suppose
$\mathscr{M}\in\mathbb{T}_{r,n}$ is a (strictly) semipositive tensor. Then for
$0\neq u\geq 0\;\exists$ an index $l$ such that $u_{l}>0$ and
$(\mathscr{M}u^{r-1})_{l}\geq 0\;(>0).$ We show that $D\mathscr{M}$ is
(strictly) semipositive tensor. Since $d_{l}>0$ we obtain
$(D\mathscr{M}u^{r-1})_{l}=d_{l}(\mathscr{M}u^{r-1})_{l}\geq 0\;(>0).$ Thus
$D\mathscr{M}$ is (strictly) semipositive tensor.

Conversely, suppose $D\mathscr{M}$ is (strictly) semipositive tensor. Now $D$
is positive diagonal matrix so $D^{-1}$ exists and $D^{-1}$ is again a
positive diagonal matrix. Therefore $D^{-1}(D\mathscr{M})=\mathscr{M}$ is a
(strictly) semipostive tensor. ∎

######  corollary 3.1:

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ be a semipositive tensor and
$D\in\mathbb{R}^{n\times n}$ be a nonnegative diagonal matrix. Then
$D\mathscr{M}$ is semipositive tensor.

######  remark 3.1:

The converse of the above corollary is not true in general.

######  example 3.2:

Let $\mathscr{M}\in\mathbb{T}_{4,2}$ be such that
$m_{1111}=1,\;m_{1121}=-1,\;m_{1112}=-3,\;m_{1222}=-1,\;m_{1211}=1,\;m_{2121}=-3,\;m_{2222}=2$
and all other entries of $\mathscr{M}$ are zero. Then for
$u=\left(\begin{array}[]{c}u_{1}\\\ u_{2}\end{array}\right)\in\mathbb{R}^{2}$
we have
$\mathscr{M}u^{3}=\left(\begin{array}[]{c}u_{1}^{3}-3u_{1}^{2}u_{2}-u_{2}^{3}\\\
-3u_{1}^{2}u_{2}+2u_{2}^{3}\end{array}\right).$ Clearly for
$u=\left(\begin{array}[]{c}1\\\ 1\end{array}\right)$ we obtain
$\mathscr{M}u^{3}=\left(\begin{array}[]{c}-3\\\ -1\end{array}\right).$ This
implies $\mathscr{M}$ is not a semipositive tensor.

Now we consider two nonnegative diagonal matrices
$D_{1}=\left(\begin{array}[]{cc}0&0\\\ 0&3\end{array}\right)$ and
$D_{2}=\left(\begin{array}[]{cc}2&0\\\ 0&0\end{array}\right).$ Let
$D_{1}\mathscr{M}=\mathscr{B}$ and $D_{2}\mathscr{M}=\mathscr{C}.$ Then
$\mathscr{B}=(b_{ijkl})\in\mathbb{T}_{4,2}$ with $b_{2121}=-9,\;b_{2222}=6$
and all other entries of $\mathscr{B}$ are zero. For
$u=\left(\begin{array}[]{c}u_{1}\\\ u_{2}\end{array}\right)\in\mathbb{R}^{2}$
we have $\mathscr{B}u^{3}=\left(\begin{array}[]{c}0\\\
-9u_{1}^{2}u_{2}+6u_{2}^{3}\end{array}\right).$ Clearly $\mathscr{B}$ is a
semipositive tensor.

Also $\mathscr{C}=(c_{ijkl})\in\mathbb{T}_{4,2}$ with
$c_{1111}=2,\;c_{1121}=-2,\;c_{1112}=-6,\;c_{1222}=-2,\;c_{1211}=2$ and all
other entries of $\mathscr{C}$ are zero. Then for
$u=\left(\begin{array}[]{c}u_{1}\\\ u_{2}\end{array}\right)\in\mathbb{R}^{2}$
we have
$\mathscr{C}u^{3}=\left(\begin{array}[]{c}2u_{1}^{3}-6u_{1}^{2}u_{2}-2u_{2}^{3}\\\
0\end{array}\right).$ Clearly $\mathscr{C}$ is a semipositive tensor.

######  theorem 3.2:

Let $P\in\mathbb{R}^{n\times n}$ be a permutation matrix also let
$\mathscr{M}\in\mathbb{T}_{r,n}.$ Then the tensor $\mathscr{M}$ is (strictly)
semipositive tensor if and only if $P\mathscr{M}P^{T}$ is (strictly)
semipositive tensor.

###### Proof.

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ be a (strictly) semipositive tensor. Let
$P\in\mathbb{R}^{n\times n}$ be a permutation matrix such that for any vector
$v\in\mathbb{R}^{n},$ $(Pv)_{k}=v_{\sigma(k)}$ where $\sigma$ is a permutation
on the set of indices $I_{n}.$ Let $0\neq u\geq 0$ where $u\in\mathbb{R}^{n}.$
Let $v=P^{T}u.$ Then $v\geq 0.$ As $\mathscr{M}$ is a (strictly) semipositive
tensor $\exists\;l$ such that $v_{l}>0$ and $(\mathscr{M}v^{r-1})_{l}\geq
0\;(>0).$ Thus
$(P\mathscr{M}P^{T}u^{r-1})_{\sigma(l)}=(\mathscr{M}P^{T}u^{r-1})_{l}=(\mathscr{M}v^{r-1})_{l}\geq
0\;(>0).$ Also $u_{\sigma(l)}=(P^{T}u)_{l}=v_{l}>0.$ Therefore
$P\mathscr{M}P^{T}$ is (strictly) semipositive tensor.

Conversely, suppose $P\mathscr{M}P^{T}$ is (strictly) semipositive tensor. We
have $\mathscr{M}=P(P\mathscr{M}P^{T})P^{T}$, since $P$ is a permutation
matrix. Hence by the forward part of the proof $\mathscr{M}$ is (strictly)
semipositive tensor. ∎

Now we prove the necessary and sufficient conditions for a tensor to be
semipositive tensor.

######  theorem 3.3:

Let $\mathscr{M},\mathscr{N}\in\mathbb{T}_{r,n}$ where $\mathscr{M}$ is a
(strictly) semipositive tensor and $\mathscr{N}\geq\mathscr{O}$. Then
$\mathscr{M}+\mathscr{N}$ is (strictly) semipositive tensor.

###### Proof.

Suppose $\mathscr{M}\in\mathbb{T}_{r,n}$ is (strictly) semipositive tensor.
Then for $0\neq u\geq 0\;\exists\;l\in I_{n}$ such that $u_{l}>0$ and
$(\mathscr{M}u^{r-1})_{l}\geq 0\;(>0).$ Let $\mathscr{N}\in\mathbb{T}_{r,n}$
where $\mathscr{N}\geq\mathscr{O}.$ Then for $0\neq u\in\mathbb{R}^{n}_{+}$ we
have $(\mathscr{N}u^{r-1})_{i}\geq 0,\;\forall\;i\in I_{n}.$ Therefore
$((\mathscr{M}+\mathscr{N})u^{r-1})_{l}=(\mathscr{M}u^{r-1})_{l}+(\mathscr{N}u^{r-1})_{l}\geq
0\;(>0).$ Hence, $\mathscr{M}+\mathscr{N}$ is (strictly) semipositive tensor.
∎

Here we discuss a necessary and sufficient condition for (strictly)
semipositive tensor.

######  theorem 3.4:

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ be a tensor such that all its proper
principal subtensors are (strictly) semipositive tensor. $\mathscr{M}$ is
(strictly) semipositive tensor if and only if for all diagonal tensors
$\mathscr{D}\in\mathbb{T}_{r,n}$ with
$\mathscr{D}>\mathscr{O}\;(\geq\mathscr{O})$, the tensor
$\mathscr{M}+\mathscr{D}$ does not have a positive null vector.

###### Proof.

Let there be a diagonal tensor $\mathscr{D}\in\mathbb{T}_{r,n}$ with
$\mathscr{D}>\mathscr{O}\;(\geq\mathscr{O})$ such that
$\mathscr{M}+\mathscr{D}$ has a positive null vector. i.e., There exists
$0<u\in\mathbb{R}^{n}$ such that

$\displaystyle(\mathscr{M}+\mathscr{D})u^{r-1}=0$
$\displaystyle\implies\mathscr{M}u^{r-1}+\mathscr{D}u^{r-1}=0$
$\displaystyle\implies\mathscr{M}u^{r-1}=-\mathscr{D}u^{r-1}<0\;(\leq 0),$

which shows that $\mathscr{M}$ is not (strictly) semipositive tensor.

Conversely, suppose $\mathscr{M}$ is not (strictly) semipositive tensor. We
show that there exists a diagonal tensor $\mathscr{D}\in\mathbb{T}_{r,n}$ with
$\mathscr{D}>\mathscr{O}\;(\geq\mathscr{O})$ such that the tensor
$\mathscr{M}+\mathscr{D}$ has a positive null vector. Since $\mathscr{M}$ is
not (strictly) semipositive tensor (but all proper principal subtensors are),
$\exists\;u>0$ such that $\mathscr{M}u^{r-1}<0\;(\leq 0).$ Now we construct
diagonal tensor $\mathscr{D}$ with diagonal entries being defined as
$d_{i...i}=-\frac{(\mathscr{M}u^{r-1})_{i}}{u_{i}^{r-1}}.$ Then
$\mathscr{D}>\mathscr{O}\;(\geq\mathscr{O})$ and
$\mathscr{M}u^{r-1}=-\mathscr{D}u^{r-1}.$ Therefore
$\mathscr{M}u^{r-1}+\mathscr{D}u^{r-1}=(\mathscr{M}+\mathscr{D})u^{r-1}=0.$
Thus $\mathscr{M}+\mathscr{D}$ has a positive null vector, since $u>0.$ ∎

Here we propose a necessary and sufficient condition for a semipositive tensor
to be a strictly semipositive tensor.

######  theorem 3.5:

A tensor $\mathscr{M}\in\mathbb{T}_{r,n}$ is semipositive tensor if and only
if for every $\delta>0,$ the tensor $\mathscr{M}+\delta\mathscr{I}$ is
strictly semipositive tensor.

###### Proof.

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ be a semipositive tensor. The for every
$0\neq u\geq 0,\;\exists$ an index $l$ such that $u_{l}>0$ and
$(\mathscr{M}u^{r-1})_{l}\geq 0.$ Therefore
$[(\mathscr{M}+\delta\mathscr{I})u^{r-1}]_{l}=(\mathscr{M}u^{r-1})_{l}+\delta
u^{r-1}_{l}>0.$ Thus $\mathscr{M}+\delta\mathscr{I}$ is a strictly
semipositive tensor.

Conversely, suppose that $\mathscr{M}+\delta\mathscr{I}$ is a strictly
semipositive tensor for each $\delta>0.$ For arbitrarily chosen $0\neq u\geq
0$ consider a sequence $\\{\delta_{k}\\}$, where $\delta_{k}>0$ and
$\delta_{k}$ converges to zero. Then for each $k,\;\exists\;l_{k}\in I_{n}$
such that $u_{l_{k}}>0$ and
$[(\mathscr{M}+\delta\mathscr{I})u^{r-1}]_{l_{k}}>0.$ This implies
$\exists\;l\in I_{n}$ such that $u_{l}>0$ and
$(\mathscr{M}u^{r-1})_{l}+\delta_{k}u^{r-1}_{l}=[(\mathscr{M}+\delta_{k}\mathscr{I})u^{r-1}]_{l}>0$
for infinitely many $\delta_{k}.$ Since $\delta_{k}\rightarrow 0$ when
$k\rightarrow\infty$ we conclude that $\mathscr{M}u^{r-1}_{l}\geq 0.$ Hence
$\mathscr{M}$ is semipositive tensor. ∎

Let $I$ denote the identity matrix of order $n$ and $[0,I]$ denote all
$n\times n$ diagonal matrices whose diagonal entries are in $[0,1].$ Now we
establish a necessary and sufficient condition for strictly semipositive
tensor.

######  theorem 3.6:

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ and $r$ is even. $\mathscr{M}$ is
strictly semipositive tensor if and only if $\forall\;G\in[0,I]$ the tensor
$G\mathscr{I}+(I−G)\mathscr{M}$ has no nonzero nonnegative null vector.

###### Proof.

Let $\mathscr{M}$ be a strictly semipositive tensor and
$\forall\;G\in[0,I],\;\exists\;0\neq u\geq 0$ which is a null vector of the
tensor $G\mathscr{I}+(I−G)\mathscr{M}.$ i.e.
$(G\mathscr{I}+(I−G)\mathscr{M})u^{r-1}=0.$ Let $v=\mathscr{M}u^{r-1}.$ As
$\mathscr{M}$ is strictly semipositive tensor $\exists\;l\in I_{n}$ such that
$u_{l}>0$ and $v_{l}=(\mathscr{M}u^{r-1})_{l}>0.$ Now

$\displaystyle[G\mathscr{I}+(I-G)\mathscr{M}]u^{r-1}=0$ $\displaystyle\implies
g_{ll}u_{l}^{r-1}+(1-g_{ll})(\mathscr{M}u^{r-1})_{l}=0$ $\displaystyle\implies
g_{ll}(u_{l}^{r-1}-v_{l})=-v_{l}.$

Note that if $u_{l}^{r-1}=v_{l}$ then $v_{l}=0$ which contradicts the fact
that $v_{l}>0.$ Therefore $u_{l}^{r-1}-v_{l}\neq 0$ and
$g_{ll}=\frac{-v_{l}}{u_{l}^{r-1}-v_{l}}.$ Now if $u_{l}^{r-1}-v_{l}>0$ then
we have $-v_{l}\geq 0,$ since $g_{ll}\geq 0.$ Therefore $v_{l}\leq 0,$ which
is again a contradiction, since $v_{l}>0$. Let $u_{l}^{r-1}-v_{l}<0.$ Now
since $g_{ll}\leq 1,$ we have $\frac{-v_{l}}{u_{l}^{r-1}-v_{l}}\leq 1\implies
u_{l}^{r-1}-v_{l}\leq-v_{l}\implies u_{l}\leq 0$ (since $r$ is even) which
contradicts the fact that $u_{l}>0.$ Hence no nonzero nonnegative null vector
exists.

Conversely, let $\mathscr{M}$ be not a strictly semipositive tensor. Then
$\exists\;0\neq u\geq 0$ such that $u_{i}(\mathscr{M}u^{r-1})_{i}\leq
0\;\forall\;i\in I_{n}.$ Let $v=\mathscr{M}u^{r-1}.$ Then for $u_{l}>0$ we
have either $v_{l}=0$ or $v_{l}<0.$ Note that if $u_{l}=0,$ then there are
three choices for $v_{l}$, they are $v_{l}=0,\;v_{l}>0\text{ or }v_{l}<0.$ Now
we define diagonal a matrix $G$ with the diagonal elements as
$g_{ii}=\left\\{\begin{array}[]{cc}0&\text{if }u_{i}>0,\;v_{i}=0\\\
\frac{-v_{i}}{u_{i}^{r-1}-v_{i}}&\text{if }u_{i}>0,\;v_{i}<0\\\ 0&\text{if
}u_{i}=0,\;v_{i}=0\\\ 1&\text{if }u_{i}=0,\;v_{i}\lessgtr
0\end{array}\right..$ Note that if $u_{i}>0$ and $v_{i}<0,$ then
$0\leq\frac{-v_{i}}{u_{i}^{r-1}-v_{i}}\leq 1.$ If not, then there are two
possibilities either $\frac{-v_{i}}{u_{i}^{r-1}-v_{i}}<0$ or
$\frac{-v_{i}}{u_{i}^{r-1}-v_{i}}>1.$ Now $\frac{-v_{i}}{u_{i}^{r-1}-v_{i}}<0$
implies $v_{i}>0$, since $u_{i}^{r-1}-v_{i}>0.$ This leads to a contradiction.
Again if $\frac{-v_{i}}{u_{i}^{r-1}-v_{i}}>1$ then we have
$-v_{i}>u_{i}^{r-1}-v_{i}\implies u_{i}^{r-1}<0,$ which is again a
contradiction. Thus we have $0\leq g_{ii}\leq 1$ and
$[(I-G)\mathscr{M}u^{r-1}]_{i}=(1-g_{ii})v_{i}=-g_{ii}u_{i}^{r-1}=-(G\mathscr{I}u^{r-1})_{i}.$
Therefore $[G\mathscr{I}+(I-G)\mathscr{M}]u^{r-1}=0.$ i.e. There exists a
diagonal matrix $G$ with diagonal entries from $[0,1]$ such that the tensor
$[G\mathscr{I}+(I-G)\mathscr{M}]$ has a nonzero nonnegative null vector. ∎

Here we establish a relation between an even order row diagonal (strictly)
semipositive tensor and its majorization matrix.

######  theorem 3.7:

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ be a row diagonal tensor of even order.
$\mathscr{M}$ is (strictly) semipositive tensor if and only if
$\tilde{\mathscr{M}}\in\mathbb{R}^{n\times n}$ is a (strictly) semimonotone
matrix.

###### Proof.

Let $\mathscr{M}\in\mathbb{T}_{r,n}$ be an even order row diagonal tensor.
i.e., $r$ is even. Let $\tilde{\mathscr{M}}\in\mathbb{R}^{n\times n}$ be the
majorization matrix of the tensor $\mathscr{M}.$ Then for $u\in\mathbb{R}^{n}$
we have
$\mathscr{M}u^{r-1}=\tilde{\mathscr{M}}\mathscr{I}u^{r-1}=\tilde{\mathscr{M}}u^{[r-1]}.$
Let $\mathscr{M}$ be a (strictly) semipositive tensor. We prove that
$\tilde{\mathscr{M}}$ is (strictly) semimonotone matrix. Now for $0\neq v\geq
0,$ $v^{[\frac{1}{r-1}]}$ exists uniquely with $0\neq v^{[\frac{1}{r-1}]}\geq
0,$ since $r$ is even. Let $u=v^{[\frac{1}{r-1}]}.$ Since $\mathscr{M}$ is a
(strictly) semipositive tensor, for $0\neq u\geq 0\;\exists\;l$ such that
$u_{l}>0$ and $(\mathscr{M}u^{r-1})_{l}=(\tilde{\mathscr{M}}u^{[r-1]})_{l}\geq
0\;(>0)$ This implies $u_{l}^{r-1}>0$ and
$(\tilde{\mathscr{M}}u^{[r-1]})_{l}\geq 0\;(>0).$ i.e., For $0\neq v\geq
0\;\exists\;l$ such that $v_{l}=u_{l}^{r-1}>0$ and
$(\tilde{\mathscr{M}}v)_{l}=(\tilde{\mathscr{M}}u^{[r-1]})_{l}\geq 0\;(>0).$
This implies that $\tilde{\mathscr{M}}$ is a (strictly) semimonotone matrix.

Conversely, let $\tilde{\mathscr{M}}$ be a (strictly) semimonotone matrix.
Since $\mathscr{M}=\tilde{\mathscr{M}}\mathscr{I},$ for $x\in\mathbb{R}^{n}$
we have

$(\mathscr{M}x^{r-1})_{i}=(\tilde{\mathscr{M}}x^{[r-1]})i,\;\forall\;i\in
I_{n}.$ (3.1)

We choose $x\in\mathbb{R}^{n}$ such that $0\neq x\geq 0$ and construct the
vector $y=x^{[r-1]}.$ Then $0\neq y\geq 0.$ Since $\tilde{\mathscr{M}}$ is a
(strictly) semimonotone matrix $\exists\;t\in I_{n}$ such that $y_{t}>0$ and
$(\tilde{\mathscr{M}}y)_{t}\geq 0\;(>0).$ i.e., For $0\neq x\geq
0\;\exists\;t\in I_{n}$ such that $x^{r-1}_{t}>0$ and
$(\mathscr{M}x^{r-1})_{t}\geq 0\;(>0)$ by equation 3.1. Therefore for $0\neq
x\geq 0\;\exists\;t\in I_{n}$ such that $x_{t}>0$ (since $r$ is even) and
$(\mathscr{M}x^{r-1})_{t}\geq 0\;(>0).$ Hence $\mathscr{M}$ is (strictly)
semipositive tensor. ∎

Not all even order semipositive tensors are row diagonal. Here we give an
example of a semipositive tensor which is not row diagonal tensor.

######  example 3.3:

Let $\mathscr{M}\in\mathbb{T}_{4,3}$ be such that
$m_{1122}=2,\;m_{1131}=2,\;m_{2211}=1,\;m_{2112}=-4,\;m_{2322}=-1,\;m_{3232}=-1,\;m_{3322}=1,\;m_{3313}=3$
and all other entries of $\mathscr{M}$ are zero. Then for
$u=\left(\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\end{array}\right)\in\mathbb{R}^{3}$ we have
$\mathscr{M}u^{3}=\left(\begin{array}[]{c}2u_{1}u_{2}^{2}+2u_{1}^{2}u_{3}\\\
-3u_{1}^{2}u_{2}-u_{2}^{2}u_{3}\\\ 3u_{1}u_{3}^{2}\end{array}\right).$ The
tensor $\mathscr{M}$ is a semipositive tensor of even order but $\mathscr{M}$
is not a row diagonal tensor.

##  Conclusion

In this article, we establish some tensor theoretic properties of semipositive
tensor and strictly semipositive tensor. We show invariance property of
(strictly) semipositive tensor with respect to diagonal matrix and permutation
matrix. We show that $\mathscr{M}\in\mathbb{T}_{r,n}$ is semipositive tensor
if and only if for every $\delta>0,$ the tensor
$\mathscr{M}+\delta\mathscr{I}$ is strictly semipositive tensor. We propose a
necessary and sufficient conditions for (strictly) semipositive tensor.
Furthermore, we prove that an even order row diagonal tensor is (strictly)
semipositive tensor if and only if its majorization matrix is (strictly)
semimonotone matrix.

##  Acknowledgment

The author R. Deb is thankful to the Council of Scientific $\&$ Industrial
Research (CSIR), India, Junior Research Fellowship scheme for financial
support.

## References

  * [1] RW Cottle.  On a problem in linear inequalities.  Journal of the London Mathematical Society, 1(1):378–384, 1968\.
  * [2] RW Cottle, JS Pang, and RE Stone.  The linear complementarity problem.  SIAM, 2009.
  * [3] RA Danao.  A note on ${E′}$-matrices.  Linear algebra and its applications, 259:299–305, 1997.
  * [4] AK Das.  Properties of some matrix classes based on principal pivot transform.  Annals of Operations Research, 243(1):375–382, 2016.
  * [5] AK Das, R Jana, and Deepmala.  On generalized positive subdefinite matrices and interior point algorithm.  In International Conference on Frontiers in Optimization: Theory and Applications, pages 3–16. Springer, 2016.
  * [6] AK Das, R Jana, and Deepmala.  Finiteness of criss-cross method in complementarity problem.  In International Conference on Mathematics and Computing, pages 170–180. Springer, 2017.
  * [7] BC Eaves.  The linear complementarity problem.  Management science, 17(9):612–634, 1971.
  * [8] Qi Guo, Meng-Meng Zheng, and Zheng-Hai Huang.  Properties of ${S}$-tensors.  Linear and Multilinear Algebra, 67(4):685–696, 2019.
  * [9] Zheng-Hai Huang and Liqun Qi.  Formulating an n-person noncooperative game as a tensor complementarity problem.  Computational Optimization and Applications, 66(3):557–576, 2017\.
  * [10] Zheng-Hai Huang and Liqun Qi.  Tensor complementarity problems—part III: applications.  Journal of Optimization Theory and Applications, 183(3):771–791, 2019.
  * [11] Zheng-Hai Huang, Yun-Yang Suo, and Jie Wang.  On ${Q}$-tensors.  arXiv preprint arXiv:1509.03088, 2015.
  * [12] R Jana, AK Das, and A Dutta.  On hidden ${Z}$-matrix and interior point algorithm.  Opsearch, 56(4):1108–1116, 2019.
  * [13] R Jana, AK Das, and S Sinha.  On processability of Lemke’s algorithm.  Applications & Applied Mathematics, 13(2), 2018.
  * [14] R Jana, A Dutta, and AK Das.  More on hidden ${Z}$-matrices and linear complementarity problem.  Linear and Multilinear Algebra, 69(6):1151–1160, 2021.
  * [15] S Karamardian.  The complementarity problem.  Mathematical programming, 2(1):107–129, 1972.
  * [16] Ziyan Luo, Liqun Qi, and Naihua Xiu.  The sparsest solutions to ${Z}$-tensor complementarity problems.  Optimization letters, 11(3):471–482, 2017.
  * [17] SR Mohan, SK Neogy, and AK Das.  More on positive subdefinite matrices and the linear complementarity problem.  Linear Algebra and its Applications, 338(1-3):275–285, 2001.
  * [18] SR Mohan, SK Neogy, and AK Das.  On the classes of fully copositive and fully semimonotone matrices.  Linear Algebra and its Applications, 323(1-3):87–97, 2001.
  * [19] SR Mohan, SK Neogy, and AK Das.  A note on linear complementarity problems and multiple objective programming.  Mathematical programming, 100(2):339–344, 2004.
  * [20] P Mondal, S Sinha, SK Neogy, and AK Das.  On discounted ARAT semi-markov games and its complementarity formulations.  International Journal of Game Theory, 45(3):567–583, 2016.
  * [21] SK Neogy, RB Bapat, and AK Das.  Optimization models with economic and game theoretic applications.  Annals of Operations Research, 243(1):1–3, 2016.
  * [22] SK Neogy and AK Das.  Linear complementarity and two classes of structured stochastic games.  Operations Research with Economic and Industrial Applications: Emerging Trends, eds: SR Mohan and SK Neogy, Anamaya Publishers, New Delhi, India, pages 156–180, 2005.
  * [23] SK Neogy and AK Das.  On almost type classes of matrices with ${Q}$-property.  Linear and Multilinear Algebra, 53(4):243–257, 2005.
  * [24] SK Neogy and AK Das.  Principal pivot transforms of some classes of matrices.  Linear algebra and its applications, 400:243–252, 2005.
  * [25] SK Neogy and AK Das.  Some properties of generalized positive subdefinite matrices.  SIAM journal on matrix analysis and applications, 27(4):988–995, 2006.
  * [26] SK Neogy and AK Das.  Mathematical programming and game theory for decision making, volume 1.  World Scientific, 2008.
  * [27] SK Neogy and AK Das.  On singular ${N_{0}}$-matrices and the class ${Q}$.  Linear algebra and its applications, 434(3):813–819, 2011.
  * [28] SK Neogy and AK Das.  On weak generalized positive subdefinite matrices and the linear complementarity problem.  Linear and Multilinear Algebra, 61(7):945–953, 2013.
  * [29] SK Neogy, AK Das, and RB Bapat.  Modeling, computation and optimization, volume 6.  World Scientific, 2009.
  * [30] SK Neogy, AK Das, and A Gupta.  Generalized principal pivot transforms, complementarity theory and their applications in stochastic games.  Optimization Letters, 6(2):339–356, 2012.
  * [31] SK Neogy, AK Das, S Sinha, and A Gupta.  On a mixture class of stochastic game with ordered field property.  In Mathematical programming and game theory for decision making, pages 451–477. World Scientific, 2008.
  * [32] K Palpandi and Sonali Sharma.  Tensor complementarity problems with finite solution sets.  Journal of Optimization Theory and Applications, 190(3):951–965, 2021.
  * [33] JS Pang.  On ${Q}$-matrices.  Mathematical Programming, 17(1):243–247, 1979.
  * [34] Liqun Qi.  Eigenvalues of a real supersymmetric tensor.  Journal of Symbolic Computation, 40(6):1302–1324, 2005.
  * [35] Jia-Yu Shao.  A general product of tensors with applications.  Linear Algebra and its applications, 439(8):2350–2366, 2013.
  * [36] Jiayu Shao and Lihua You.  On some properties of three different types of triangular blocked tensors.  Linear Algebra and its Applications, 511:110–140, 2016.
  * [37] Yisheng Song and Liqun Qi.  Properties of tensor complementarity problem and some classes of structured tensors.  arXiv preprint arXiv:1412.0113, 2014.
  * [38] Yisheng Song and Liqun Qi.  Properties of some classes of structured tensors.  Journal of Optimization Theory and Applications, 165(3):854–873, 2015.
  * [39] Yisheng Song and Liqun Qi.  Tensor complementarity problem and semi-positive tensors.  Journal of Optimization Theory and Applications, 169(3):1069–1078, 2016.
  * [40] Yisheng Song and Liqun Qi.  Properties of tensor complementarity problem and some classes of structured tensors.  Annals of Applied Mathematics, 2017.
  * [41] Yisheng Song and Gaohang Yu.  Properties of solution set of tensor complementarity problem.  Journal of Optimization Theory and Applications, 170(1):85–96, 2016\.
  * [42] MJ Tsatsomeros and M Wendler.  Semimonotone matrices.  Linear Algebra and its Applications, 578:207–224, 2019.
  * [43] Ya-nan Zheng and Wei Wu.  On a class of semi-positive tensors in tensor complementarity problem.  Journal of Optimization Theory and Applications, 177(1):127–136, 2018.