As we all know, there are two forms of the orthogonality: one-sided or two-sided orthogonality. We call that $R\left(A\right)$ and $R\left(B\right)$ are orthogonal if $A^{*}B=0$; if $A{B}^{*}=0$, we call that $R\left(A^{*}\right)$ and $R\left(B^{*}\right)$ are orthogonal. And we call that $R\left(A^{*}\right)$ and $R\left(B\right)$ are orthogonal if $AB=0$. If $AB=0$ and $BA=0$, then A and B are orthogonal, denoted as $A\perp B$. Notice that, when $A^{\#}$ exists and $AB=0$, where $A^{\#}$ is group inverse of A, we have $A^{\#}B=A^{\#}A{A}^{\#}B={\left(A^{\#}\right)}^{2}AB=0$. And it is obvious that $A^{\#}B=0$ implies $AB=0$. Thus, when $A^{\#}$ exists, $A\perp B$ if and only if $A^{\#}B=0$ and $B{A}^{\#}=0$ (i.e. A and B are #-orthogonal, denoted as $A{\perp }_{\#}B$). And Hestenes [1] gave the concept of *-orthogonality: let $A,B\in {\mathbb{C}}^{m\times n}$, if $A^{*}B=0$ and $B{A}^{*}=0$ hold, then A is *-orthogonal to B, denoted by $A{\perp }_{*}B$. For matrices, Hartwig and Styan [2] gave that if the dagger additivity (i.e. ${(A+B)}^{†}=A^{†}+B^{†}$, where $A^{†}$ is Moore-penrose inverse of A) and the rank additivity (i.e. $rk(A+B)=rk\left(A\right)+rk\left(B\right)$), then A is *-orthogonal to B. Ferreyra and Malik [3] introduced the core and strongly core orthogonal matrices by using the core inverse. It is that let $A,B\in {\mathbb{C}}^{m\times n}$ with Ind$\left(A\right)\le 1$, where Ind$\left(A\right)$ is the index of A, if $A^{\overline{)\#}}B=0$ and $B{A}^{\overline{)\#}}=0$, then A is core orthogonal to B, denoted as $A{\perp }_{\overline{)\#}}B$. $A,B\in {\mathbb{C}}^{m\times n}$ with Ind$\left(A\right)\le 1$ and Ind$\left(B\right)\le 1$, are strongly core orthogonal matrices (denoted as $A{\perp }_{s,\overline{)\#}}B$) if $A{\perp }_{\overline{)\#}}B$ and $B{\perp }_{\overline{)\#}}A$. In [3],we can see that $A{\perp }_{s,\overline{)\#}}B$ implies ${(A+B)}^{\overline{)\#}}=A^{\overline{)\#}}+B^{\overline{)\#}}$ (core additivity).

In [4], Liu, Wang and Wang proved that $A,B\in {\mathbb{C}}^{n\times n}$ with Ind$\left(A\right)\le 1$ and Ind$\left(B\right)\le 1$ are strongly core orthogonal if and only if ${(A+B)}^{\overline{)\#}}=A^{\overline{)\#}}+B^{\overline{)\#}}$ and $A^{\overline{)\#}}B=0$ (or $B{A}^{\overline{)\#}}=0$) instead of $A{\perp }_{\overline{)\#}}B$, which is more concise than Theorem $7.3$ in [3]. And Ferreyra and Malik in [3] have proved that if A is strongly core orthogonal to B, then rk$(A+B)=$rk$\left(A\right)+$rk$\left(B\right)$ and ${(A+B)}^{\overline{)\#}}=A^{\overline{)\#}}+B^{\overline{)\#}}$. But whether the reverse holds is still an open question. In [4], Liu, Wang and Wang solved the problem completely. Furthermore, they also gave some new equivalent conditions for the strongly core orthogonality, which are related to the minus partial order and some Hermitian matrices.

On the basis of the core orthogonal matrix, Mosić, Dolinar, Kuzma and Marovt [5] extended the concept of the core orthogonality and present the new concept of the core-EP orthogonality. A is said to be core-EP orthogonal to B if $A^{\overline{)\mathrm{D}}}B=0$ and $B{A}^{\overline{)\mathrm{D}}}=0$, where $A^{\overline{)\mathrm{D}}}$ is core-EP inverse of A. A number of characterizations for core-EP orthogonality were proved in [5]. Applying the core-EP orthogonality, the concept and characterizations of the strongly core-EP orthogonality were introduced in [5].

In [7], Wang and Liu introduced the generalized core inverse (called the C-S inverse) and gave some properties and characterizations of the inverse. By the C-S inverse, a binary relation (denoted “ $A{\le }^{\overline{)\mathrm{S}}}B$ ”) and a partial order (called the C-S partial order and denoted “ $A{\le }^{\overline{)\mathrm{CS}}}B$ ”) are given.

Motivated by these ideas, we give the concepts of the C-S orthogonality and the strongly C-S orthogonality, and discuss their characterizations in this paper. The connection between the C-S partial order and the C-S orthogonality has been given. Moreover, we get some characterizing properties of the C-S orthogonal matrix when A is EP.

For $A,X\in {\mathbb{C}}^{n\times n}$, and k is the index of A, we consider the following equations:

(1) $AXA=A$;

(2) $XAX=X$;

(3) ${\left(AX\right)}^{*}=AX$;

(4) ${\left(XA\right)}^{*}=XA$;

(5) $AX=XA$;

(6) $X{A}^2=A$;

(7) $A{X}^2=X$;

(8) $A^{2}X=A$;

(9) $A{X}^2=X$;

(10) $X{A}^{k+1}=A^k$.

The set of all elements $X\in {\mathbb{C}}^{n\times n}$ which satisfies equations $\left(i\right),\left(j\right),...,\left(k\right)$ among Eqs (1)-(10) are denoted as $A\left\{i,j,...,k\right\}$. If there exists then it is called the Moore-Penrose inverse of A and $A^{†}$ is unique. It was introduced by Moore [6] and improved by Bjerhammar [8] and Penrose [9]. Furthermore, based on the Moore-Penrose inverse, it is known to us that it is EP if and only if $A{A}^{†}=A^{†}A$. If there exists then it is called the group inverse of A and $A^{\#}$ is unique [10]. If there exists then $A^{\overline{)\#}}$ is called the core inverse of A [11]. And if there exists then $A^{\overline{)\mathrm{d}}}$ is called the core-EP inverse of A[12]. Moreover, ${\mathbb{C}}^{\overline{)\mathrm{d}}}$ is the set of all core-EP invertible matrices of ${\mathbb{C}}^{n\times n}$.

Based on this, we review the concepts of partial orders [3]. The symbols ${{\mathbb{C}}_n}^{GM}$ and ${{\mathbb{C}}_n}^{EP}$ will stand for the subsets of ${\mathbb{C}}^{n\times n}$ consisting of group and EP matrices, respectively.

Then, the core-EP decomposition of A is And by applying Lemma $2.1$, Wang and Liu in [7] obtained the following canonical form for the C-S inverse of A:

Firstly, we give the concept of the C-S orthogonality.

If $A,B\in {\mathbb{C}}^{n\times n}$, then

Next, we study the range and null space of the matrices which are C-S orthogonal. Firstly, we give some characterizations of the C-S inverse as follows.

By (5) and (6), it is easy to get the following lemma.

In view of $\left(i\right)$ and $\left(ii\right)$ in Theorem $3.3$, we obtain $A^k{\perp }_{\overline{)\mathrm{S}}}{B}^{*}$ from $\left(v\right)$. Using Lemma $4.4$ in [3], we have that $\left(i\right)-\left(vii\right)$ in Theorem $3.3$ and $A^k{\perp }_{\overline{)\#}}B$ are equivalent, i.e. $A^k{\perp }_{\overline{)\mathrm{S}}}B$ and $A^k{\perp }_{\overline{)\#}}B$ are equivalent. And from Lemma $2.1$ in [5], it is seen that $A^k{\perp }_{\overline{)\#}}B$ is equivalent to $A{\perp }_{\overline{)\mathrm{D}}}B$ and $A{\perp }_{\overline{)\mathrm{D}}}{B}^{*}$. As a consequence of the theorem we have the following:

Using the core-EP decomposition, we obtain the following characterization of C-S orthogonal matrices.

Next, based on the C-S partial order, we get some relation between the C-S orthogonality and the C-S partial order.

When A is an $EP$ matrix, we have a more refined result which reduces to the well-known characterizations of the orthogonality in the usual sense.

The concept of the strongly C-S orthogonality is considered in this section as a relation which is symmetric but unlike the C-S orthogonality.