Items related to A Treatise on Universal Algebra Volume 1; With Applications
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1898 Excerpt: ...into a point in the subregion of the null region defined by b,... b. Call this subregion the subregion of the null region associated with the vacuous region of the first species. 153. Symmetrical Matrices. (1) In general, if a-and y be any two elements and f any matrix, (x fy) is not equal to (y $x). In order to obtain the conditions which must hold for these expressions to be equal, let the matrix be a'j_?" a' where, according to the notation 6i, 62,... 6r of § 141 (4), a(, = o„e1 + ae2 +... + avfe,. In other words the matrix is ( a„, aa,... a,, ). I «m » I Then, supposing that... e, are a set of normal elements at unit normal intensities cf. §§ 109 (3) and 110 (1), (e„ fe.) = (e, a„) = a„ (e„e„) = a„, and (e. (be,) = (e, a,) = a,, (e„ e.) = a.,. Hence, if the required condition holds, = a„. (2) Thus the matrix with the desired property is a matrix symmetrical about its leading diagonal when the elements of the denominator form a Symmetrical matrices are considered by Grassmann cf. Aiudehnungslehre von 1862, § 391; bat his use ol supplements implicitly implies a purely imaginary, self-normal quadric. Hence his conclusions are limited to those of subsection (7). normal system (at unit normal intensities) with respect to the quadric chosen as the self-normal quadric. Let such matrices be called symmetrical with respect to the normal systems, or, more shortly, symmetrical matrices. (3) If /x out of the v latent roots of a symmetrical matrix be distinct and not zero, so that at least (i points Cj, c,,... c, can be found with the property )cp = 7pc» then the fi points c,, c.2... cM corresponding to different latent roots 7n 7a. 7m are mutually normal. For let x-f...
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- PublisherRarebooksclub.com
- Publication date2012
- ISBN 10 1231231637
- ISBN 13 9781231231630
- BindingPaperback
- Number of pages164
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