Path: ...!fu-berlin.de!uni-berlin.de!not-for-mail From: ram@zedat.fu-berlin.de (Stefan Ram) Newsgroups: sci.physics.relativity Subject: Re: Vector notation? Date: 2 Aug 2024 10:54:50 GMT Organization: Stefan Ram Lines: 157 Expires: 1 Jul 2025 11:59:58 GMT Message-ID: References: Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit X-Trace: news.uni-berlin.de Ab8ZVlbzMaWBH8ok6Djywg9oAKusH8ow5Uz090HX4ku2JI Cancel-Lock: sha1:CmxsAVe2yKN7xw+tVKJTitPbZHg= sha256:weJnsMvkLeO9mq1pf3G0BNGOsTAxkhynW3o3giBXE34= X-Copyright: (C) Copyright 2024 Stefan Ram. All rights reserved. Distribution through any means other than regular usenet channels is forbidden. It is forbidden to publish this article in the Web, to change URIs of this article into links, and to transfer the body without this notice, but quotations of parts in other Usenet posts are allowed. X-No-Archive: Yes Archive: no X-No-Archive-Readme: "X-No-Archive" is set, because this prevents some services to mirror the article in the web. But the article may be kept on a Usenet archive server with only NNTP access. X-No-Html: yes Content-Language: en-US Bytes: 7752 Mikko wrote or quoted: >Matrices do not match very well with the needs of physics. Many physical >quantities require more general hypermatrices. But then one must be >very careful that the multiplicatons are done correctly. In the meantime, I have written about this for the case of a ( 0, 2 ) tensor, i.e., a bilinear form (such as "eta"). It turns out that for this case, a simple single rule for matrix multiplication suffices. To give it the right context, my following text starts with a small introduction into the linear algebra of vectors and forms and arrives at the actual matrix multiplication only near the end: (If one is not into bilinear algebra, one may stop reading now!) It's about the fact that the matrix representation of a ( 0, 2 )- tensor should actually be a row of rows, not a row of columns, as you often see in certain texts. A row of columns, on the other hand, would be suitable for a ( 1, 1 )-tensor. I got this from a text by Viktor T. Toth. All errors here are my own though. But since I want to start with the basics, this matrix representation will only be dealt with towards the end of this text, where impatient readers could of course jump to. In this text, I limit myself to real vector spaces R, R^1, R^2, etc. For a vector space R^n, let the set of indices be I := { i | 0 <= i < n }. Forms The structure-preserving mappings f into the field R are precisely the linear mappings of a vector to R. I call such a linear mapping f of a vector to R a /form/ or a /covector/. Let f_i be n forms. If the tuple ( f_i( v ))ieI of a vector v is equal to the tuple ( f_i( w ))ieI of a vector w if and only if v=w, I call the tuple ( f_i )ieI a /basis/ of the vector space. The numbers v^i := f_i( v ) are the /(contravariant) coordinates/ of the vector v in the basis ( f_i )ieM. I call the vector e_i, for which f_j( e_i ) is 1 for i=j and 0 for i<>j, the i-th /basis vector/ of the basis ( f_i( v ))ieI. If f is a form, then the tuple ( f( e_i ))ieI are the /(covariant) coordinates/ of the form f. Matrices We write the covariant coordinates f_i of a form f as a "horizontal" 1xn-matrix M( B, f ): ( f_0, f_1, ..., f_(n-1) ). The contravariant coordinates v^i of a vector v we write in a basis B as a "vertical" nx1-matrix M( B, v ): ( v^0 ) ( v^1 ) ( . . . ) ( v^( n-1 )). The application f( v ) of a form to a vector then results from the matrix multiplication M( B, f )X M( B, v ). Rule For The Matrix Multiplication X .-----------------------------------------------------------------. | The /multiplication X/ of a 1xn-matrix with an nx1-matrix is | | a sum with n summands, where the summand i is the product of | | the column i of the first matrix with the row i of the second | | matrix. | '-----------------------------------------------------------------' ( 0, 2 )-Tensors We also call the forms (covectors) "( 0, 1 )-tensors" to express that they make a scalar out of 0 covectors and one vector linearly. Accordingly, a /( 0, 2 )-tensor/ is a bilinear mapping (bilinear form) that makes a scalar out of 0 covectors and /two/ vectors. Matrix representation of ( 0, 2 )-tensors According to Viktor T. Toth, for us, the matrix representation of a ( 0, 2 )-tensor f is a horizontal 1xn-matrix M( B, f ), whose individual components are horizontal 1xn-matrices of scalars. The scalar at position j of component i of M( B, f ) is f( e^i, e^j ), where the superscripts here do not indicate components of e but a basis vector. (PS: Here I am not sure about the correct order "f( e^i, e^j )" or "f( e^j, e^i )", but this is a technical detail.) Let's now look at the case n=3 and see how we calculate the application of such a tensor f to two vectors v and w with the matrix representations! ( v^0 ) ( w^0 ) ( (f_00,f_01,f_02)(f_10,f_11,f_12)(f_20,f_21,f_22) ) X ( v^1 ) X ( w^1 ) ( v^2 ) ( w^2 ) We start with the first product: ( v^0 ) ( (f_00,f_01,f_02) (f_10,f_11,f_12) (f_20,f_21,f_22) ) X ( v^1 ) ( v^2 ). According to our rule for the matrix multiplication X, this is the sum v^0*(f_00,f_01,f_02)+v^1*(f_10,f_11,f_12)+v^2*(f_20,f_21,f_22)= (v^0*f_00,v^0*f_01,v^0*f_02)+ (v^1*f_10,v^1*f_11,v^1*f_12)+ (v^2*f_20,v^2*f_21,v^2*f_22)= (v^0*f_00+v^1*f_10+v^2*f_20, v^0*f_01+v^1*f_11+v^2*f_21, v^0*f_02+v^1*f_12+v^2*f_22). This is again a "horizontal" 1xn-matrix (written vertically here because it does not fit on one line), which can be multiplied by the vertical nx1-matrix for w according to our rules for matrix multiplication X: (v^0*f_00+v^1*f_10+v^2*f_20, v^0*f_01+v^1*f_11+v^2*f_21, ( w^0 ) v^0*f_02+v^1*f_12+v^2*f_22) X ( w^1 ) ( w^2 ). According to our rule for the matrix multiplication X, this results in the number w^0*(v^0*f_00+v^1*f_10+v^2*f_20)+ w^1*(v^0*f_01+v^1*f_11+v^2*f_21)+ w^2*(v^0*f_02+v^1*f_12+v^2*f_22). So, the multiplication of the given matrix representation of a ( 0, 2 )-tensor with the matrix representations of two vectors correctly results in a /number/ using the single uniform rule for the matrix multiplication X. In the literature (especially on special relativity), the "Minkowski metric", which is a (0,2)-tensor, is written as a row of /columns/. The application to two vectors would then be: ( f_00, f_01, f_02 ) ( v^0 ) ( w^0 ) ( f_10, f_11, f_12 ) ( v^1 ) ( w^1 ) ( f_20, f_21, f_22 ) ( v^2 ) ( w^2 ) = ( f_00 * v^0 + f_01 * v1 + f_02 * v^2 ) ( w^0 ) ( f_10 * v^0 + f_11 * v1 + f_12 * v^2 ) ( w^1 ) ( f_20 * v^0 + f_21 * v1 + f_22 * v^2 ) ( w^2 ) Now the product of /two column vectors/ appears, which is not defined as a matrix multiplication! (Matrix multiplication is not the same as the dot product of two vectors.)