# 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.

## 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 ). 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

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

## 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 ).

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.

## 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 ), 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 )

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

## References

1. 1.

Perose, A: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406-413 (1955)

2. 2.

Berger, M: Geometry I. Springer, New York (1987)

3. 3.

Veljan, D: The sine theorem and inequalities for volume of simplices and determinants. Linear Algebra Appl. 219, 79-91 (1995)

4. 4.

Horn, RA, Johnson, CR: Matrix Analysis. Cambridge University Press, Cambridge (1988)

5. 5.

Golub, GH, Van Loan, CF: Matrix Computations, 2nd edn. Johns Hopkings University Press, Baltimore (1989)

6. 6.

Golub, GH, Van Loan, CF: Matrix Computations, 4th edn. Johns Hopkings University Press, Baltimore (2013)

7. 7.

Ben-Israel, A: A volume associated with $$m\times n$$ matrices. Linear Algebra Appl. 167, 87-111 (1992)

8. 8.

Ben-Israel, A: An application of the matrix volume in probability. Linear Algebra Appl. 321, 9-25 (2000)

9. 9.

Ben-Israel, A: The change of variables formula using matrix volume. SIAM J. Matrix Anal. Appl. 21, 300-312 (1999)

## Acknowledgements

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

## Author information

Authors

### Corresponding author

Correspondence to Yanping Zhou.

### 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.

## Rights and permissions 