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A geometrical interpretation of the inverse matrix
Journal of Inequalities and Applications volumeÂ 2016, ArticleÂ number:Â 257 (2016)
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 ndimensional 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 mdimensional 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 mdimensional parallellotope with m linearly independent vectors \(z_{1},z_{2},\ldots,z_{m}\) as its edge vectors, i.e.,
\({[z_{1},\ldots,z_{i1},z_{i+1},\ldots,z_{m}]}\) denotes the facets of the mparallellotope \({[z_{1},z_{2},\ldots,z_{m}]}\) for an \((m1)\)hyperplane,
\(z_{i}\) is the altitude vector on facet \({[z_{1},\ldots,z_{i1},z_{i+1},\ldots,z_{m}]}\) (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 mparallellotope 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
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 mparallellotope \([z_{1},z_{2},\ldots,z_{m}]^{*}\).
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
where \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{n}\) are n edge vectors of the nparallellotope \({[z_{1},z_{2},\ldots,z_{n}]^{*}}\).
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 \({\langle\hat{y_{i},L}(i1)\rangle}\rightarrow0\), 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
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 mparallellotope \([z_{1},z_{2},\ldots,z_{m}]\) 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 mparallellotope 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 mparallellotope \([z_{1},z_{2},\ldots,z_{m}]^{*}\) 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.
From Rieszâ€™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 \(\mathbb{R}^{n}\).
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.
Now, we prove that \(z_{1},z_{2},\ldots,z_{m}\) are altitude vectors of the mparallellotope \([z^{*}_{1},z^{*}_{2},\ldots, z^{*}_{m}]\) (the edge vectors of \([z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}]\) are \(z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{m}\)).
Suppose that \([z^{*}_{1},z^{*}_{2},\ldots,z^{*}_{i1},z^{*}_{i+1},\ldots,z^{*}_{m}]\) 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 mparallellotope \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\).â€ƒâ–¡
Proof of TheoremÂ 2.1
For a given \(m\times n\) matrix full row rank \(M=(c_{ij})_{m\times n}\), let
By LemmaÂ 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 \({[z_{1},z_{2},\ldots,z_{m}]^{*}}\).
Suppose
and
It follows from (3.3) that
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
Proof
Assume that \(h_{i},p_{i}\) are the orthogonal component and orthogonal projection of \(z_{i}\) with respect to \(L(i1)\), 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
From the definition of the volume of the parallellotope, we get (see [7â€“9])
The proof of LemmaÂ 3.2 is completed.â€ƒâ–¡
Proof of TheoremÂ 2.3
From TheoremÂ 2.1, it follows that
i.e.,
It follows from the Cauchy inequality that
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
From
and
the assertion (2) is given.â€ƒâ–¡
Proof of TheoremÂ 2.4
Together with TheoremÂ 2.1, we get
Thus
From
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
If follows from (3.10) that
Thus
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
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
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.
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Acknowledgements
The authors would like to acknowledge the support from the National Natural Science Foundation of China (11371239).
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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/s1366001611986
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DOI: https://doi.org/10.1186/s1366001611986