1 Introduction and notations

Let \(\mathbb{R}^{n}\) denote an n-dimensional real Euclidean vector space, for a nonzero \(n\times1\) vector \(x\in{\mathbb{R}^{n}}\), the generalized inverse of x, denoted by \(x^{+}\), has the geometrical interpretation that \(x^{T}\) is divided by \(\|x\|^{2}\), that is, \(x^{+}=x^{T}/\|x\|^{2}\), where \(x^{T}\) is the transpose of x (see [1]). A natural question is whether a similar geometrical interpretation holds for the inverse of a matrix.

In this paper, using a new method to structure a m-dimensional parallellotope, the geometrical interpretation of the inverse matrix and the generalized inverse of a matrix with full column rank or full row rank are given.

Let \({[z_{1},z_{2},\ldots,z_{m}]}\) be the m-dimensional parallellotope with m linearly independent vectors \(z_{1},z_{2},\ldots,z_{m}\) as its edge vectors, i.e.,

$${[z_{1},z_{2},\ldots,z_{m}]}= \bigl\{ z\in \mathbb{R}^{n} \mid t_{1}z_{1}+ \cdots+t_{m}z_{m}, t_{i}\in [0,1],i=1,2,\ldots,m \bigr\} ; $$

\({[z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{m}]}\) denotes the facets of the m-parallellotope \({[z_{1},z_{2},\ldots,z_{m}]}\) for an \((m-1)\)-hyperplane,

$$\mathcal{H}_{i}=\operatorname{span}\{z_{1}, \ldots,z_{i-1},z_{i+1},\ldots,z_{m}\}. $$

\(z_{i}\) is the altitude vector on facet \({[z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{m}]}\) (see [2, 3]) with the orthogonal component of \(z_{i}\) with respect to \(\mathcal{H}_{i}\). If \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\) denotes the m-parallellotope constructed by m linearly independent vectors \(z_{1},z_{2},\ldots,z_{m}\) as its altitude vectors, then we will show that there exist \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\), exclusive such that

$${[z_{1},z_{2},\ldots,z_{m}]}^{*}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\bigr]}. $$

2 Main results

Our main results are the following theorems.

Theorem 2.1

If M is a matrix with full row (column) rank and \(z_{1},z_{2},\ldots,z_{m}\) is its row (column) vectors, then the right (left) inverse of the matrix M is the matrix whose column (row) vectors are

$$\frac{z^{*}_{1}}{\|z_{1}\|^{2}}, \frac{z^{*}_{2}}{\|z_{2}\|^{2}}, \ldots, \frac {z^{*}_{m}}{\|z_{m}\|^{2}}, $$

where \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) are m edge vectors of the m-parallellotope \([z_{1},z_{2},\ldots,z_{m}]^{*}\).

Corollary 2.2

If M is nonsingular \(n\times n\) matrix and \(z_{1},z_{2},\ldots,z_{n}\) is its row (column) vectors, then the inverse of the matrix M is the matrix whose column (row) vectors are

$$\frac{z^{*}_{1}}{\|z_{1}\|^{2}},\frac{z^{*}_{1}}{\|z_{1}\|^{2}},\ldots,\frac{z^{*}_{n}}{\| z_{n}\|^{2}}, $$

where \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n}\) are n edge vectors of the n-parallellotope \({[z_{1},z_{2},\ldots,z_{n}]^{*}}\).

We may say roughly if the \([z_{1},z_{2},\ldots,z_{m}]\) (\(z_{1},z_{2},\ldots,z_{m}\) as edge vectors) is the geometrical interpretation of the matrix M, then \([z_{1},z_{2},\ldots,z_{m}]^{*}\) (\(z_{1},z_{2},\ldots,z_{m}\) as altitude vectors) is one of the \(M^{-1}\).

We list some basic facts to state the following theorems.

We write \(L(i)\), for the linear subspace spanned by \(z_{1},z_{2},\ldots,z_{i}, z_{i}\in\mathbb{R}^{n}\) (\(1\leq i\leq n\)). Let \(\hat{\langle z,L\rangle}\) be the angle between vector z and linear subspace L, where if \(z\notin L\), then \(\hat{\langle z,L\rangle}\) is the angle between z and the orthogonal projection of z on L, denoted by \(z|_{L}\), i.e., \(z|_{L}=((L^{\bot}+x)\cap L)\). If \(z\in L\), then \(\hat{\langle z,L\rangle}=0\).

Theorem 2.3

Suppose \(y_{1},y_{2},\ldots,y_{n}\) are n row vectors of the matrix M, and \(z_{1},z_{2},\ldots,z_{n}\) are column vectors of the matrix \(M^{-1}\),

  1. (1)

    if \(\|y_{i}\|\rightarrow0\), then \(\|z_{i}\|\rightarrow+\infty\);

  2. (2)

    if \({\langle\hat{y_{i},L}(i-1)\rangle}\rightarrow0\), then there is k (\(1\leq k\leq n\)) such that \(\|z_{k}\|\rightarrow+\infty\).

Theorem 2.3 will be required in the study of matrix disturbances (see [4–6]).

Utilizing the geometrical interpretation of the inverse matrix, we have the following relational volume formulas of parallellotopes for the \(n\times n\) real matrices \(M,N\).

Theorem 2.4

Let \([z_{1},z_{2},\ldots,z_{n}]^{**}\) be the parallellotope structured by the edge vectors of \([z_{1},z_{2},\ldots,z_{n}]^{*}\) as altitude vectors. Then

$$\begin{aligned}& \operatorname{vol} \bigl([z_{1},z_{2},\ldots,z_{n}]^{*} \bigr)\cdot \operatorname{vol} \bigl([z_{1},z_{2}, \ldots,z_{n}] \bigr)= \Biggl(\prod^{n}_{i=1} \|z_{i}\| \Biggr)^{2}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \operatorname{vol} \bigl([z_{1},z_{2},\ldots,z_{n}]^{**} \bigr)/ \operatorname{vol} \bigl({[z_{1},x_{2}, \ldots,z_{n}]} \bigr)= \Biggl(\prod^{n}_{i=1}{ \bigl\| z^{*}_{i}\bigr\| }/{\|z_{i}\| } \Biggr)^{2}, \end{aligned}$$
(2.2)

where \(\operatorname{vol}([z_{1},\ldots,z_{n}])\) denotes the volume of the parallellotope \([z_{1},\ldots,z_{n}]\).

The proofs of the theorems will be given in Section 3.

3 Proofs of the theorems

Given m linearly independent vectors \(z_{1},z_{2},\ldots,z_{m}\) in \({\mathbb{R}^{n}}\), if we structure an m-parallellotope \([z_{1},z_{2},\ldots,z_{m}]\) by them as edge vectors, then \([z_{1},z_{2},\ldots,z_{m}]\) has m linearly independent altitude vectors. Conversely, for any given m linearly independent vectors \(z_{1},z_{2},\ldots,z_{m}\), can we structure an m-parallellotope by them as m altitude vectors? The following lemma gives an affirmative answer.

Lemma 3.1

If \(\{z_{1},z_{2},\ldots,z_{m}\} \) (\(m\geq2\)) is a given set of linearly independent vectors in \(\mathbb{R}^{n}\), then there is an m-parallellotope \([z_{1},z_{2},\ldots,z_{m}]^{*}\) whose m altitude vectors are \(z_{1},z_{2},\ldots,z_{m}\).

Proof

If \(z_{1},z_{2},\ldots,z_{m}\) are linearly independent, then we have m linear functionals \(g_{1},g_{2},\ldots, g_{m}\) such that

$$g_{j}(z_{i})=\delta_{ij}\|z_{i} \|^{2}, \quad i,j=1,2,\ldots,m, $$

where \(\delta_{ij}\) is the Kronecker delta symbol.

From Riesz’s representation theorem for the linear functional, we get \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) such that

$$ \bigl\langle z_{i},z^{*}_{j}\bigr\rangle = \delta_{ij}\|z_{i}\|^{2},\quad i,j=1,2,\ldots,m, $$
(3.1)

where \(\langle,\rangle\) is the ordinary inner product in \(\mathbb{R}^{n}\).

Further, let

$$\sum^{m}_{j=1}\alpha_{j}z^{*}_{j}=0, \quad\alpha_{j}\in\mathbb{R}, $$

by

$$0=\Biggl\langle z_{i},\sum^{m}_{j=1} \alpha_{j}z^{*}_{j}\Biggr\rangle =\alpha_{i} \|z_{i}\|^{2}, $$

we have \(\alpha_{i}=0,i=1,2,\ldots,m\). This shows that \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) are linearly independent.

Now, we prove that \(z_{1},z_{2},\ldots,z_{m}\) are altitude vectors of the m-parallellotope \([z^{*}_{1},z^{*}_{2},\ldots, z^{*}_{m}]\) (the edge vectors of \([z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}]\) are \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\)).

Suppose that \([z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{i-1},z^{*}_{i+1},\ldots,z^{*}_{m}]\) are the facets of \({[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}]}\). From \(z_{i}\bot z^{*}_{j} \) (\(j\neq i\)), we have

$$ z_{i}\perp{\bigl[z^{*}_{1},z^{*}_{2}, \ldots,z^{*}_{i-1},z^{*}_{i+1},\ldots,z^{*}_{m}\bigr]}. $$
(3.2)

Thus, \(z_{1},z_{2},\ldots,z_{m}\) are altitude vectors of \({[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}]}\), i.e.,

$${[z_{1},z_{2},\ldots,z_{m}]^{*}}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\bigr]}. $$

This yields the desired m-parallellotope \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\). □

Proof of Theorem 2.1

For a given \(m\times n\) matrix full row rank \(M=(c_{ij})_{m\times n}\), let

$$z_{i}=(c_{i1},c_{i2},\ldots,c_{in}), \quad i=1,2,\ldots,m. $$

By Lemma 3.1, we have an unique vector set \(\{z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\}\) such that

$$ \bigl\langle z_{i},z^{*}_{j}\bigr\rangle = \delta_{ij}\|z_{i}\|^{2}, \quad i=1,2,\ldots,m; j=1,2,\ldots,n, $$

i.e.,

$$ \biggl\langle z_{i},\frac{z^{*}_{j}}{\|z_{i}\|^{2}}\biggr\rangle = \delta_{ij},\quad i=1,2,\ldots,m; j=1,2,\ldots,n, $$
(3.3)

and \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) are m edge vectors of the parallellotope \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\).

Suppose

$$d_{i}=\frac{z^{*}_{i}}{\|z_{i}\|^{2}}, \quad i=1,2,\ldots,m, $$

and

$$N=(d_{1},d_{2},\ldots,d_{m}). $$

It follows from (3.3) that

$$MN= \begin{pmatrix} z_{1}\\ z_{2}\\ \vdots\\ z_{m} \end{pmatrix} (d_{1},d_{2}, \ldots,d_{m}) = \begin{pmatrix} 1 & & 0\\ &\ddots\\ 0& & 1 \end{pmatrix}. $$

Thus, the matrix N is the inverse of the matrix M, and the column vectors \(d_{1},d_{2},\ldots,d_{m}\) of the matrix N are the edge vectors of \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\) divided by \(\|z_{1}\|^{2},\|z_{2}\|^{2},\ldots,\|z_{m}\|^{2}\), respectively.

Together with Theorem 2.1 and taking M for an \(n\times n\) matrix with full rank, we have Corollary 2.2.

Here, we will complete the proof of Theorem 2.3. The following lemma will be required. □

Lemma 3.2

For \(L(i)\) the linear subspace spanned by \(z_{1},z_{2},\ldots,z_{i}, i=1,2,\ldots,m\) (≤n), if \(\operatorname{vol}({[z_{1},z_{2},\ldots,z_{m}]})\) is the volume of the parallellotope \({[z_{1},z_{2},\ldots,z_{m}]}\) (see [7]), we have

$$ \operatorname{vol}\bigl({[z_{1},z_{2}, \ldots,z_{m}]}\bigr)=\prod^{m}_{i=1} \|z_{i}\|\cdot\prod^{m}_{i=2}\sin{ \bigl\langle \hat{z_{i},L}(i-1)\bigr\rangle }. $$
(3.4)

Proof

Assume that \(h_{i},p_{i}\) are the orthogonal component and orthogonal projection of \(z_{i}\) with respect to \(L(i-1)\), respectively \((i=2,\ldots ,m,h_{1}=z_{1},p_{1}=0)\). Since \(\|z_{i}\|\cos{\langle \hat{z_{i},p_{i}}\rangle}=\|p_{i}\|\), we have

$$ \cos{\bigl\langle \hat{z_{i},L}(i-1)\bigr\rangle }= \frac{\langle z_{i},p_{i}\rangle}{\|z_{i}\|\|p_{i}\|}=\frac{\langle p_{i},p_{i}\rangle}{\|z_{i}\|\|p_{i}\|}=\frac{p_{i}}{\|z_{i}\|}. $$
(3.5)

By \(\|z_{i}\|^{2}=\|p_{i}\|^{2}+\|h_{i}\|^{2}\), it follows that

$$\|h_{i}\|=\sqrt{\|z_{i}\|^{2}-\|p_{i} \|^{2}}=\|z_{i}\|\sin{\bigl\langle \hat{z_{i},L}(i-1) \bigr\rangle }. $$

From the definition of the volume of the parallellotope, we get (see [7–9])

$$ \operatorname{vol}\bigl({[z_{1},z_{2}, \ldots,z_{m}]}\bigr)= \prod^{m}_{i=1} \|h_{i}\|=\prod^{m}_{i=1} \|z_{i}\| \cdot\prod^{m}_{i=2} \sin{\bigl\langle \hat{z_{i},L}(i-1)\bigr\rangle }. $$
(3.6)

The proof of Lemma 3.2 is completed. □

Proof of Theorem 2.3

From Theorem 2.1, it follows that

$$ \begin{pmatrix} y_{1}\\ y_{2}\\ \vdots\\ y_{n} \end{pmatrix} (z_{1},z_{2}, \ldots,z_{n} ) = \begin{pmatrix} \langle y_{1},z_{1}\rangle && 0\\ &\ddots\\ 0 & &\langle y_{1},z_{1}\rangle \end{pmatrix} = \begin{pmatrix} 1 & & 0\\ &\ddots\\ 0 & & 1 \end{pmatrix}, $$
(3.7)

i.e.,

$$\langle y_{i},z_{i}\rangle=1, \quad i=1,2,\ldots,n. $$

It follows from the Cauchy inequality that

$$1=\bigl\vert \langle y_{i},z_{i}\rangle\bigr\vert \leq \|y_{i}\|\|z_{i}\|. $$

Thus the assertion (1) holds.

Let \(\{y_{1},y_{2},\ldots,y_{n}\}\) and \(\{z_{1},z_{2},\ldots,z_{n}\}\) in Lemma 3.2. From (3.7), we get

$$ \Biggl(\prod^{n}_{i=1} \|y_{i}\|\cdot\prod^{n}_{i=1}\sin{ \bigl\langle \hat{y_{i},L}(i-1)\bigr\rangle } \Biggr)\cdot \Biggl(\prod ^{n}_{j=1}\|z_{j}\|\cdot\prod ^{n}_{j=1}\sin{\bigl\langle \hat{z_{j},L}(j-1)\bigr\rangle } \Biggr)=1. $$
(3.8)

From

$$0\leq\Biggl\vert \prod^{n}_{j=1}\sin{ \bigl\langle \hat{y_{j},L}(j-1)\bigr\rangle }\Biggr\vert \leq1 $$

and

$$\prod^{n}_{i=1}\|y_{i}\|\leq G, $$

the assertion (2) is given. □

Proof of Theorem 2.4

Together with Theorem 2.1, we get

$$ \begin{pmatrix} \frac{z^{*}_{1}}{\|z_{1}\|^{2}}\\ \frac{z^{*}_{2}}{\|z_{2}\|^{2}}\\ \vdots\\ \frac{z^{*}_{n}}{\|z_{n}\|^{2}} \end{pmatrix} (z_{1},z_{2}, \ldots,z_{n}) = \begin{pmatrix} 1 && 0\\ &\ddots\\ 0 && 1 \end{pmatrix}. $$
(3.9)

Thus

$$\begin{aligned}& \det \begin{pmatrix} \begin{pmatrix} \frac{z^{*}_{1}}{\|z_{1}\|^{2}}\\ \frac{z^{*}_{2}}{\|z_{2}\|^{2}}\\ \vdots\\ \frac{z^{*}_{n}}{\|z_{n}\|^{2}} \end{pmatrix} (z_{1},z_{2}, \ldots,z_{n}) \end{pmatrix} =1, \\& \det \begin{pmatrix} z^{*}_{1}\\ z^{*}_{2}\\ \vdots\\ z^{*}_{n} \end{pmatrix} \cdot \det(z_{1},z_{2}, \ldots,z_{n})= \Biggl(\prod^{n}_{i=1} \|z_{i}\| \Biggr)^{2}. \end{aligned}$$

From

$${[x_{1},x_{2},\ldots,x_{n}]^{*}}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n}\bigr]}, $$

and the definition of the volume of parallellotopes, the equality (2.1) holds.

Assume that \(\{z^{**}_{1},z^{**}_{2},\ldots,z^{**}_{n}\}\) is a set of the edge vectors of \({[z_{1},z_{2},\ldots,z_{n}]^{**}}\). Together with Theorem 2.1, we get

$$ \begin{pmatrix} z^{*}_{1}\\ z^{*}_{2}\\ \vdots\\ z^{*}_{n} \end{pmatrix} \left(\textstyle\begin{array}{@{}c@{}} \frac{z^{**}_{1}}{\|z^{*}_{1}\|^{2}}, \frac{z^{**}_{2}}{\|z^{*}_{2}\|^{2}}, \vdots, \frac{z^{**}_{n}}{\|z^{*}_{n}\|^{2}} \end{array}\displaystyle \right) = \begin{pmatrix} 1 && 0\\ &\ddots\\ 0 && 1 \end{pmatrix}. $$
(3.10)

If follows from (3.10) that

$$\det \begin{pmatrix} z^{*}_{1}\\ z^{*}_{2}\\ \vdots\\ z^{*}_{n} \end{pmatrix} \cdot \det\bigl(z^{**}_{1},z^{**}_{2}, \ldots,z^{**}_{n}\bigr)= \Biggl(\prod ^{n}_{i=1}\|z_{i}\| \Biggr)^{2}. $$

Thus

$$ \operatorname{vol} \bigl({[z_{1},z_{2}, \ldots,z_{n}]^{*}} \bigr)\cdot \operatorname{vol} \bigl({[z_{1},z_{2},\ldots,z_{n}]^{**}} \bigr) = \Biggl(\prod^{n}_{i=1}\bigl\Vert z^{*}_{i}\bigr\Vert \Biggr)^{2}. $$
(3.11)

Taking together (2.1) and (3.11), the equality (2.2) holds. □

For \(\{z_{1},z_{2},\ldots,z_{n}\}\), from Lemma 3.1, \({[z_{1},z_{2},\ldots,z_{n}]^{*}}\) is structured by them as altitude vectors. Denote \({[z_{1},z_{2},\ldots,z_{n}]^{*}}\) by \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n}\).

Let

$${[z_{1},z_{2},\ldots,z_{n}]^{**}}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n} \bigr]^{*}}. $$

Thus Theorem 2.4 denotes the relationship of volumes about \({[z_{1},z_{2},\ldots,z_{n}]}\), \({[z_{1},z_{2},\ldots,z_{n}]^{*}}\), and \({[z_{1},z_{2},\ldots,z_{n}]^{**}}\).

Remark 1

By (3.10), we get

$$ \begin{pmatrix} \frac{z^{*}_{1}}{\|z_{1}\|^{2}}\\ \frac{z^{*}_{2}}{\|z_{2}\|^{2}}\\ \vdots\\ \frac{z^{*}_{n}}{\|z_{n}\|^{2}} \end{pmatrix} \left(\textstyle\begin{array}{@{}c@{}} \frac{\|z_{1}\|^{2}}{\|z^{*}_{1}\|^{2}}z^{**}_{1}, \frac{\|z_{2}\|^{2}}{\|z^{*}_{2}\|^{2}}z^{**}_{2}, \vdots, \frac{\|z_{n}\|^{2}}{\|z^{*}_{n}\|^{2}}z^{**}_{n} \end{array}\displaystyle \right) = \begin{pmatrix} 1 && 0\\ &\ddots\\ 0 && 1 \end{pmatrix}, $$
(3.12)

From (3.9) and (3.12), we see that

$$ z^{**}_{i}=\frac{\|z^{*}_{i}\|^{2}}{\|z_{i}\|^{2}}z_{i}, \quad i=1,2,\ldots,n. $$
(3.13)

By (3.13), we can see that \({[z_{1},z_{2},\ldots,z_{n}]^{**}}\) and \({[z_{1},z_{2},\ldots,z_{n}]}\) are two parallellotopes and their edge vectors are of the same direction.