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.
Anzeige
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.,
\({[z_{1},\ldots,z_{i-1},z_{i+1},\ldots,z_{m}]}\) denotes the facets of the m-parallellotope \({[z_{1},z_{2},\ldots,z_{m}]}\) for an \((m-1)\)-hyperplane,
\(z_{i}\) 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
$${[z_{1},z_{2},\ldots,z_{m}]}= \bigl\{ z\in \mathbb{R}^{n} \mid t_{1}z_{1}+ \cdots+t_{m}z_{m}, t_{i}\in [0,1],i=1,2,\ldots,m \bigr\} ; $$
$$\mathcal{H}_{i}=\operatorname{span}\{z_{1}, \ldots,z_{i-1},z_{i+1},\ldots,z_{m}\}. $$
$${[z_{1},z_{2},\ldots,z_{m}]}^{*}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\bigr]}. $$
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
where
\(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\)
are
m
edge vectors of the
m-parallellotope
\([z_{1},z_{2},\ldots,z_{m}]^{*}\).
$$\frac{z^{*}_{1}}{\|z_{1}\|^{2}}, \frac{z^{*}_{2}}{\|z_{2}\|^{2}}, \ldots, \frac {z^{*}_{m}}{\|z_{m}\|^{2}}, $$
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
where
\(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n}\)
are
n
edge vectors of the
n-parallellotope
\({[z_{1},z_{2},\ldots,z_{n}]^{*}}\).
$$\frac{z^{*}_{1}}{\|z_{1}\|^{2}},\frac{z^{*}_{1}}{\|z_{1}\|^{2}},\ldots,\frac{z^{*}_{n}}{\| z_{n}\|^{2}}, $$
Anzeige
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)
if
\(\|y_{i}\|\rightarrow0\), then
\(\|z_{i}\|\rightarrow+\infty\);
(2)
if
\({\langle\hat{y_{i},L}(i-1)\rangle}\rightarrow0\), then there is
k (\(1\leq k\leq n\)) such that
\(\|z_{k}\|\rightarrow+\infty\).
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
where
\(\operatorname{vol}([z_{1},\ldots,z_{n}])\)
denotes the volume of the parallellotope
\([z_{1},\ldots,z_{n}]\).
$$\begin{aligned}& \operatorname{vol} \bigl([z_{1},z_{2},\ldots,z_{n}]^{*} \bigr)\cdot \operatorname{vol} \bigl([z_{1},z_{2}, \ldots,z_{n}] \bigr)= \Biggl(\prod^{n}_{i=1} \|z_{i}\| \Biggr)^{2}, \end{aligned}$$
(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)
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 where \(\delta_{ij}\) is the Kronecker delta symbol.
$$g_{j}(z_{i})=\delta_{ij}\|z_{i} \|^{2}, \quad i,j=1,2,\ldots,m, $$
From Riesz’s representation theorem for the linear functional, we get \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) such that where \(\langle,\rangle\) is the ordinary inner product in \(\mathbb{R}^{n}\).
$$ \bigl\langle z_{i},z^{*}_{j}\bigr\rangle = \delta_{ij}\|z_{i}\|^{2},\quad i,j=1,2,\ldots,m, $$
(3.1)
Further, let by we have \(\alpha_{i}=0,i=1,2,\ldots,m\). This shows that \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) are linearly independent.
$$\sum^{m}_{j=1}\alpha_{j}z^{*}_{j}=0, \quad\alpha_{j}\in\mathbb{R}, $$
$$0=\Biggl\langle z_{i},\sum^{m}_{j=1} \alpha_{j}z^{*}_{j}\Biggr\rangle =\alpha_{i} \|z_{i}\|^{2}, $$
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 Thus, \(z_{1},z_{2},\ldots,z_{m}\) are altitude vectors of \({[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}]}\), i.e., This yields the desired m-parallellotope \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\). □
$$ z_{i}\perp{\bigl[z^{*}_{1},z^{*}_{2}, \ldots,z^{*}_{i-1},z^{*}_{i+1},\ldots,z^{*}_{m}\bigr]}. $$
(3.2)
$${[z_{1},z_{2},\ldots,z_{m}]^{*}}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\bigr]}. $$
Proof of Theorem 2.1
For a given \(m\times n\) matrix full row rank \(M=(c_{ij})_{m\times n}\), let By Lemma 3.1, we have an unique vector set \(\{z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\}\) such that
i.e., and \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\) are m edge vectors of the parallellotope \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\).
$$z_{i}=(c_{i1},c_{i2},\ldots,c_{in}), \quad i=1,2,\ldots,m. $$
$$ \bigl\langle z_{i},z^{*}_{j}\bigr\rangle = \delta_{ij}\|z_{i}\|^{2}, \quad i=1,2,\ldots,m; j=1,2,\ldots,n, $$
$$ \biggl\langle z_{i},\frac{z^{*}_{j}}{\|z_{i}\|^{2}}\biggr\rangle = \delta_{ij},\quad i=1,2,\ldots,m; j=1,2,\ldots,n, $$
(3.3)
Suppose and It follows from (3.3) that
$$d_{i}=\frac{z^{*}_{i}}{\|z_{i}\|^{2}}, \quad i=1,2,\ldots,m, $$
$$N=(d_{1},d_{2},\ldots,d_{m}). $$
$$MN= \begin{pmatrix} z_{1}\\ z_{2}\\ \vdots\\ z_{m} \end{pmatrix} (d_{1},d_{2}, \ldots,d_{m}) = \begin{pmatrix} 1 & & 0\\ &\ddots\\ 0& & 1 \end{pmatrix}. $$
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 By \(\|z_{i}\|^{2}=\|p_{i}\|^{2}+\|h_{i}\|^{2}\), it follows that
$$ \cos{\bigl\langle \hat{z_{i},L}(i-1)\bigr\rangle }= \frac{\langle z_{i},p_{i}\rangle}{\|z_{i}\|\|p_{i}\|}=\frac{\langle p_{i},p_{i}\rangle}{\|z_{i}\|\|p_{i}\|}=\frac{p_{i}}{\|z_{i}\|}. $$
(3.5)
$$\|h_{i}\|=\sqrt{\|z_{i}\|^{2}-\|p_{i} \|^{2}}=\|z_{i}\|\sin{\bigl\langle \hat{z_{i},L}(i-1) \bigr\rangle }. $$
From the definition of the volume of the parallellotope, we get (see [7‐9]) The proof of Lemma 3.2 is completed. □
$$ \operatorname{vol}\bigl({[z_{1},z_{2}, \ldots,z_{m}]}\bigr)= \prod^{m}_{i=1} \|h_{i}\|=\prod^{m}_{i=1} \|z_{i}\| \cdot\prod^{m}_{i=2} \sin{\bigl\langle \hat{z_{i},L}(i-1)\bigr\rangle }. $$
(3.6)
Proof of Theorem 2.3
From Theorem 2.1, it follows that
i.e.,
$$ \begin{pmatrix} y_{1}\\ y_{2}\\ \vdots\\ y_{n} \end{pmatrix} (z_{1},z_{2}, \ldots,z_{n} ) = \begin{pmatrix} \langle y_{1},z_{1}\rangle && 0\\ &\ddots\\ 0 & &\langle y_{1},z_{1}\rangle \end{pmatrix} = \begin{pmatrix} 1 & & 0\\ &\ddots\\ 0 & & 1 \end{pmatrix}, $$
(3.7)
$$\langle y_{i},z_{i}\rangle=1, \quad i=1,2,\ldots,n. $$
It follows from the Cauchy inequality that Thus the assertion (1) holds.
$$1=\bigl\vert \langle y_{i},z_{i}\rangle\bigr\vert \leq \|y_{i}\|\|z_{i}\|. $$
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 and the assertion (2) is given. □
$$0\leq\Biggl\vert \prod^{n}_{j=1}\sin{ \bigl\langle \hat{y_{j},L}(j-1)\bigr\rangle }\Biggr\vert \leq1 $$
$$\prod^{n}_{i=1}\|y_{i}\|\leq G, $$
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 From and the definition of the volume of parallellotopes, the equality (2.1) holds.
$$\begin{aligned}& \det \begin{pmatrix} \begin{pmatrix} \frac{z^{*}_{1}}{\|z_{1}\|^{2}}\\ \frac{z^{*}_{2}}{\|z_{2}\|^{2}}\\ \vdots\\ \frac{z^{*}_{n}}{\|z_{n}\|^{2}} \end{pmatrix} (z_{1},z_{2}, \ldots,z_{n}) \end{pmatrix} =1, \\& \det \begin{pmatrix} z^{*}_{1}\\ z^{*}_{2}\\ \vdots\\ z^{*}_{n} \end{pmatrix} \cdot \det(z_{1},z_{2}, \ldots,z_{n})= \Biggl(\prod^{n}_{i=1} \|z_{i}\| \Biggr)^{2}. \end{aligned}$$
$${[x_{1},x_{2},\ldots,x_{n}]^{*}}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n}\bigr]}, $$
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 Thus Taking together (2.1) and (3.11), the equality (2.2) holds. □
$$\det \begin{pmatrix} z^{*}_{1}\\ z^{*}_{2}\\ \vdots\\ z^{*}_{n} \end{pmatrix} \cdot \det\bigl(z^{**}_{1},z^{**}_{2}, \ldots,z^{**}_{n}\bigr)= \Biggl(\prod ^{n}_{i=1}\|z_{i}\| \Biggr)^{2}. $$
$$ \operatorname{vol} \bigl({[z_{1},z_{2}, \ldots,z_{n}]^{*}} \bigr)\cdot \operatorname{vol} \bigl({[z_{1},z_{2},\ldots,z_{n}]^{**}} \bigr) = \Biggl(\prod^{n}_{i=1}\bigl\Vert z^{*}_{i}\bigr\Vert \Biggr)^{2}. $$
(3.11)
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 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 \({[z_{1},z_{2},\ldots,z_{n}]^{**}}\).
$${[z_{1},z_{2},\ldots,z_{n}]^{**}}={ \bigl[z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n} \bigr]^{*}}. $$
Remark 1
By (3.10), we get From (3.9) and (3.12), we see that 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.
$$ \begin{pmatrix} \frac{z^{*}_{1}}{\|z_{1}\|^{2}}\\ \frac{z^{*}_{2}}{\|z_{2}\|^{2}}\\ \vdots\\ \frac{z^{*}_{n}}{\|z_{n}\|^{2}} \end{pmatrix} \left(\textstyle\begin{array}{@{}c@{}} \frac{\|z_{1}\|^{2}}{\|z^{*}_{1}\|^{2}}z^{**}_{1}, \frac{\|z_{2}\|^{2}}{\|z^{*}_{2}\|^{2}}z^{**}_{2}, \vdots, \frac{\|z_{n}\|^{2}}{\|z^{*}_{n}\|^{2}}z^{**}_{n} \end{array}\displaystyle \right) = \begin{pmatrix} 1 && 0\\ &\ddots\\ 0 && 1 \end{pmatrix}, $$
(3.12)
$$ z^{**}_{i}=\frac{\|z^{*}_{i}\|^{2}}{\|z_{i}\|^{2}}z_{i}, \quad i=1,2,\ldots,n. $$
(3.13)
Acknowledgements
The authors would like to acknowledge the support from the National Natural Science Foundation of China (11371239).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.