# A geometrical interpretation of the inverse matrix

## Abstract

Utilizing a new method to structure parallellotopes, a geometrical interpretation of the inverse matrix is given, which includes the generalized inverse of full column rank or a full row rank matrices. Further, some relational volume formulas of parallellotopes are established.

## 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]). AÂ 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 $$\|z_{k}\|\rightarrow+\infty$$.

TheoremÂ 2.3 will be required in the study of matrix disturbances (see [4â€“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 SectionÂ 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 Rieszâ€™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 $${[z_{1},z_{2},\ldots,z_{m}]^{*}}$$.â€ƒâ–¡

### Proof of TheoremÂ 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 LemmaÂ 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 TheoremÂ 2.1 and taking M for an $$n\times n$$ matrix with full rank, we have CorollaryÂ 2.2.

Here, we will complete the proof of TheoremÂ 2.3. The following lemma will be required.â€ƒâ–¡

### Lemma 3.2

For $$L(i)$$ the linear subspace spanned by $$z_{1},z_{2},\ldots,z_{i}, i=1,2,\ldots,m$$ (â‰¤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 [7â€“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 LemmaÂ 3.2 is completed.â€ƒâ–¡

### Proof of TheoremÂ 2.3

From TheoremÂ 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 LemmaÂ 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.â€ƒâ–¡

### Proof of TheoremÂ 2.4

Together with TheoremÂ 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 TheoremÂ 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.â€ƒâ–¡

For $$\{z_{1},z_{2},\ldots,z_{n}\}$$, from LemmaÂ 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 TheoremÂ 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.

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## Acknowledgements

The authors would like to acknowledge the support from the National Natural Science Foundation of China (11371239).

## Author information

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Correspondence to Yanping Zhou.

### Competing interests

The authors declare that they have no competing interests.

### Authorsâ€™ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Zhou, Y., He, B. A geometrical interpretation of the inverse matrix. J Inequal Appl 2016, 257 (2016). https://doi.org/10.1186/s13660-016-1198-6