A geometrical interpretation of the inverse matrix Zhou and He Journal of Inequalities and Applications (2016) 2016 257 DOI 10 1186/s13660 016 1198 6 R E S E A R C H Open Access A geometrical interpre[.]
Zhou and He Journal of Inequalities and Applications (2016) 2016:257 DOI 10.1186/s13660-016-1198-6 RESEARCH Open Access A geometrical interpretation of the inverse matrix Yanping Zhou* and Binwu He * Correspondence: zhouyp@i.shu.edu.cn Department of Mathematics, Shanghai University, Shanghai, 200444, China 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 MSC: 15A15; 52A20 Keywords: parallellotope; inverse matrix; generalized inverse Introduction and notations Let Rn denote an n-dimensional real Euclidean vector space, for a nonzero n × vector x ∈ Rn , the generalized inverse of x, denoted by x+ , has the geometrical interpretation that xT is divided by x , that is, x+ = xT /x , where xT 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 , z , , zm ] be the m-dimensional parallellotope with m linearly independent vectors z , z , , zm as its edge vectors, i.e., [z , z , , zm ] = z ∈ Rn | t z + · · · + tm zm , ti ∈ [, ], i = , , , m ; [z , , zi– , zi+ , , zm ] denotes the facets of the m-parallellotope [z , z , , zm ] for an (m – )-hyperplane, Hi = span{z , , zi– , zi+ , , zm } zi is the altitude vector on facet [z , , zi– , zi+ , , zm ] (see [, ]) with the orthogonal component of zi with respect to Hi If [z , z , , zm ]∗ denotes the m-parallellotope constructed by m linearly independent vectors z , z , , zm as its altitude vectors, then we will ∗ , exclusive such that show that there exist z∗ , z∗ , , zm ∗ [z , z , , zm ]∗ = z∗ , z∗ , , zm © 2016 Zhou and He 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 Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of Main results Our main results are the following theorems Theorem . If M is a matrix with full row (column) rank and z , z , , zm is its row (column) vectors, then the right (left) inverse of the matrix M is the matrix whose column (row) vectors are z∗ z∗ z∗ , , , m , z z zm ∗ are m edge vectors of the m-parallellotope [z , z , , zm ]∗ where z∗ , z∗ , , zm Corollary . If M is nonsingular n × n matrix and z , z , , zn is its row (column) vectors, then the inverse of the matrix M is the matrix whose column (row) vectors are z∗ zn∗ z∗ , , , , z z zn where z∗ , z∗ , , zn∗ are n edge vectors of the n-parallellotope [z , z , , zn ]∗ We may say roughly if the [z , z , , zm ] (z , z , , zm as edge vectors) is the geometrical interpretation of the matrix M, then [z , z , , zm ]∗ (z , z , , zm as altitude vectors) is one of the M– We list some basic facts to state the following theorems We write L(i), for the linear subspace spanned by z , z , , zi , zi ∈ Rn ( ≤ i ≤ n) Let z,ˆL be the angle between vector z and linear subspace L, where if z ∈/ L, then z,ˆL is the angle between z and the orthogonal projection of z on L, denoted by z|L , i.e., z|L = ((L⊥ + x) ∩ L) If z ∈ L, then z,ˆL = Theorem . Suppose y , y , , yn are n row vectors of the matrix M, and z , z , , zn are column vectors of the matrix M– , () if yi → , then zi → +∞; () if yiˆ, L(i – ) → , then there is k ( ≤ k ≤ n) such that zk → +∞ Theorem . 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 × n real matrices M, N Theorem . Let [z , z , , zn ]∗∗ be the parallellotope structured by the edge vectors of [z , z , , zn ]∗ as altitude vectors Then vol [z , z , , zn ] ∗ n · vol [z , z , , zn ] = zi , (.) i= n ∗ z /zi , vol [z , z , , zn ] / vol [z , x , , zn ] = i ∗∗ i= where vol([z , , zn ]) denotes the volume of the parallellotope [z , , zn ] The proofs of the theorems will be given in Section (.) Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of Proofs of the theorems Given m linearly independent vectors z , z , , zm in Rn , if we structure an m-parallellotope [z , z , , zm ] by them as edge vectors, then [z , z , , zm ] has m linearly independent altitude vectors Conversely, for any given m linearly independent vectors z , z , , zm , can we structure an m-parallellotope by them as m altitude vectors? The following lemma gives an affirmative answer Lemma . If {z , z , , zm } (m ≥ ) is a given set of linearly independent vectors in Rn , then there is an m-parallellotope [z , z , , zm ]∗ whose m altitude vectors are z , z , , zm Proof If z , z , , zm are linearly independent, then we have m linear functionals g , g , , gm such that gj (zi ) = δij zi , i, j = , , , m, where δij is the Kronecker delta symbol ∗ From Riesz’s representation theorem for the linear functional, we get z∗ , z∗ , , zm such that ∗ zi , zj = δij zi , i, j = , , , m, (.) where , is the ordinary inner product in Rn Further, let m αj zj∗ = , αj ∈ R, j= by = zi , m αj zj∗ = αi zi , j= ∗ are linearly independent we have αi = , i = , , , m This shows that z∗ , z∗ , , zm Now, we prove that z , z , , zm are altitude vectors of the m-parallellotope [z∗ , z∗ , , ∗ ∗ ∗ ] are z∗ , z∗ , , zm ) zm ] (the edge vectors of [z∗ , z∗ , , zm ∗ ∗ ∗ ∗ ∗ ∗ Suppose that [z , z , , zi– , zi+ , , zm ] are the facets of [z∗ , z∗ , , zm ] From zi ⊥zj∗ (j = i), we have ∗ ∗ ∗ , zi+ , , zm zi ⊥ z∗ , z∗ , , zi– (.) ∗ ], i.e., Thus, z , z , , zm are altitude vectors of [z∗ , z∗ , , zm ∗ [z , z , , zm ]∗ = z∗ , z∗ , , zm This yields the desired m-parallellotope [z , z , , zm ]∗ Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of Proof of Theorem . For a given m × n matrix full row rank M = (cij )m×n , let zi = (ci , ci , , cin ), i = , , , m ∗ } such that By Lemma ., we have an unique vector set {z∗ , z∗ , , zm ∗ zi , zj = δij zi , i = , , , m; j = , , , n, i.e., zi , zj∗ zi = δij , i = , , , m; j = , , , n, (.) ∗ are m edge vectors of the parallellotope [z , z , , zm ]∗ and z∗ , z∗ , , zm Suppose di = zi∗ , zi i = , , , m, and N = (d , d , , dm ) It follows from (.) that ⎛ ⎞ ⎛ z ⎜z ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ MN = ⎜ ⎟ (d , d , , dm ) = ⎝ ⎝ ⎠ zm ⎞ ⎟ ⎟ ⎠ Thus, the matrix N is the inverse of the matrix M, and the column vectors d , d , , dm of the matrix N are the edge vectors of [z , z , , zm ]∗ divided by z , z , , zm , respectively Together with Theorem . and taking M for an n × n matrix with full rank, we have Corollary . Here, we will complete the proof of Theorem . The following lemma will be required Lemma . For L(i) the linear subspace spanned by z , z , , zi , i = , , , m (≤ n), if vol([z , z , , zm ]) is the volume of the parallellotope [z , z , , zm ] (see []), we have m m zi · sin ziˆ, L(i – ) vol [z , z , , zm ] = i= (.) i= Proof Assume that hi , pi are the orthogonal component and orthogonal projection of zi with respect to L(i – ), respectively (i = , , m, h = z , p = ) Since zi cos zi ,ˆpi = pi , we have pi , pi pi zi , pi cos ziˆ, L(i – ) = = = zi pi zi pi zi (.) Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of By zi = pi + hi , it follows that hi = zi – pi = zi sin ziˆ, L(i – ) From the definition of the volume of the parallellotope, we get (see [–]) m m m vol [z , z , , zm ] = hi = zi · sin ziˆ, L(i – ) i= i= (.) i= The proof of Lemma . is completed Proof of Theorem . From Theorem ., it follows that ⎛ ⎞ ⎛ y y , z ⎜y ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ (z , z , , zn ) = ⎜ ⎜.⎟ ⎝ ⎝.⎠ yn ⎞ ⎛ ⎟ ⎜ ⎟=⎜ ⎠ ⎝ y , z ⎞ ⎟ ⎟, ⎠ (.) i.e., yi , zi = , i = , , , n It follows from the Cauchy inequality that = yi , zi ≤ yi zi Thus the assertion () holds Let {y , y , , yn } and {z , z , , zn } in Lemma . From (.), we get n yi · i= n n n sin yiˆ, L(i – ) · zj · sin zjˆ, L(j – ) = i= j= (.) j= From n ˆ ≤ sin yj , L(j – ) ≤ j= and n yi ≤ G, i= the assertion () is given Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of Proof of Theorem . Together with Theorem ., we get ⎛ z∗ ⎞ z ⎜ z∗ ⎟ ⎜ ⎟ ⎜ z ⎟ ⎛ ⎜ ⎜ ⎟ (z , z , , zn ) = ⎜ ⎝ ⎜ ⎟ ⎝ ⎠ zn∗ zn ⎞ ⎟ ⎟ ⎠ (.) Thus ⎛⎛ ⎞ z∗ ⎞ z ⎜⎜ z ∗ ⎟ ⎜⎜ ⎟ ⎜⎜ z ⎟ ⎟ ⎟ ⎟ det ⎜⎜ ⎟ (z , z , , zn )⎟ = , ⎜⎜ ⎟ ⎟ ⎝⎝ ⎠ ⎠ zn∗ zn ⎛ ∗⎞ z n ⎜z ∗ ⎟ ⎜ ⎟ ⎟ zi det ⎜ ⎜ ⎟ · det(z , z , , zn ) = ⎝.⎠ i= zn∗ From [x , x , , xn ]∗ = z∗ , z∗ , , zn∗ , and the definition of the volume of parallellotopes, the equality (.) holds Assume that {z∗∗ , z∗∗ , , zn∗∗ } is a set of the edge vectors of [z , z , , zn ]∗∗ Together with Theorem ., we get ⎛ ∗⎞ ⎛ z ⎜z ∗ ⎟ ⎜ ⎜ ⎟ z∗∗ z∗∗ ∗∗ zn ⎜ ⎜.⎟ ⎜ ⎟ z∗ , z∗ , , zn∗ = ⎝ ⎝.⎠ ∗ zn ⎞ ⎟ ⎟ ⎠ (.) If follows from (.) that ⎛ ∗⎞ z n ⎜z ∗ ⎟ ⎜ ⎟ ∗∗ ∗∗ ∗∗ ⎟ ⎜ det ⎜ ⎟ · det z , z , , zn = zi ⎝ ⎠ i= zn∗ Thus vol [z , z , , zn ] ∗ n ∗∗ ∗ z · vol [z , z , , zn ] = i (.) i= Taking together (.) and (.), the equality (.) holds Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of For {z , z , , zn }, from Lemma ., [z , z , , zn ]∗ is structured by them as altitude vectors Denote [z , z , , zn ]∗ by z∗ , z∗ , , zn∗ Let ∗ [z , z , , zn ]∗∗ = z∗ , z∗ , , zn∗ Thus Theorem . denotes the relationship of volumes about [z , z , , zn ], [z , z , , zn ]∗ , and [z , z , , zn ]∗∗ Remark By (.), we get ⎛ z∗ ⎞ ⎛ z ⎜ z∗ ⎟ ⎜ ⎟ ⎜ ⎜ z ⎟ z ∗∗ z ∗∗ zn ∗∗ =⎜ ⎜ ⎟ ⎝ ⎜ ⎟ z∗ z , z∗ z , , zn∗ zn ⎝ ⎠ zn∗ zn ⎞ ⎟ ⎟, ⎠ (.) From (.) and (.), we see that zi∗∗ = zi∗ zi , zi i = , , , n (.) By (.), we can see that [z , z , , zn ]∗∗ and [z , z , , zn ] are two parallellotopes and their edge vectors are of the same direction 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 Acknowledgements The authors would like to acknowledge the support from the National Natural Science Foundation of China (11371239) Received: 26 April 2016 Accepted: October 2016 References Perose, A: A generalized inverse for matrices Proc Camb Philos Soc 51, 406-413 (1955) Berger, M: Geometry I Springer, New York (1987) Veljan, D: The sine theorem and inequalities for volume of simplices and determinants Linear Algebra Appl 219, 79-91 (1995) Horn, RA, Johnson, CR: Matrix Analysis Cambridge University Press, Cambridge (1988) Golub, GH, Van Loan, CF: Matrix Computations, 2nd edn Johns Hopkings University Press, Baltimore (1989) Golub, GH, Van Loan, CF: Matrix Computations, 4th edn Johns Hopkings University Press, Baltimore (2013) Ben-Israel, A: A volume associated with m × n matrices Linear Algebra Appl 167, 87-111 (1992) Ben-Israel, A: An application of the matrix volume in probability Linear Algebra Appl 321, 9-25 (2000) Ben-Israel, A: The change of variables formula using matrix volume SIAM J Matrix Anal Appl 21, 300-312 (1999) ... (2013) Ben-Israel, A: A volume associated with m × n matrices Linear Algebra Appl 167, 87-111 (1992) Ben-Israel, A: An application of the matrix volume in probability Linear Algebra Appl 321, 9-25... disturbances (see [–]) Utilizing the geometrical interpretation of the inverse matrix, we have the following relational volume formulas of parallellotopes for the n × n real matrices M, N Theorem... denotes the volume of the parallellotope [z , , zn ] The proofs of the theorems will be given in Section (.) Zhou and He Journal of Inequalities and Applications (2016) 2016:257 Page of Proofs