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

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.