词条 | Elementary matrix |
释义 |
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group of invertible matrices. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss-Jordan elimination to further reduce the matrix to reduced row echelon form. Elementary row operationsThere are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):
If E is an elementary matrix, as described below, to apply the elementary row operation to a matrix A, one multiplies A by the elementary matrix on the left, EA. The elementary matrix for any row operation is obtained by executing the operation on the identity matrix. Row-switching transformationsThe first type of row operation on a matrix A switches all matrix elements on row i with their counterparts on row j. The corresponding elementary matrix is obtained by swapping row i and row j of the identity matrix. So TijA is the matrix produced by exchanging row i and row j of A. Properties
Row-multiplying transformationsThe next type of row operation on a matrix A multiplies all elements on row i by m where m is a non-zero scalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ith position, where it is m. So Di(m)A is the matrix produced from A by multiplying row i by m. Properties
Row-addition transformationsThe final type of row operation on a matrix A adds row i multiplied by a scalar m to row j. The corresponding elementary matrix is the identity matrix but with an m in the (j, i) position. So Lij(m)A is the matrix produced from A by adding m times row i to row j. Properties
See also
References{{See also|Linear algebra#Further reading}}
| last = Axler | first = Sheldon Jay | date = 1997 | title = Linear Algebra Done Right | publisher = Springer-Verlag | edition = 2nd | isbn = 0-387-98259-0 }}
| last = Lay | first = David C. | date = August 22, 2005 | title = Linear Algebra and Its Applications | publisher = Addison Wesley | edition = 3rd | isbn = 978-0-321-28713-7 }}
|last = Meyer |first = Carl D. |date = February 15, 2001 |title = Matrix Analysis and Applied Linear Algebra |publisher = Society for Industrial and Applied Mathematics (SIAM) |isbn = 978-0-89871-454-8 |url = http://www.matrixanalysis.com/DownloadChapters.html |deadurl = yes |archiveurl = https://web.archive.org/web/20091031193126/http://matrixanalysis.com/DownloadChapters.html |archivedate = 2009-10-31 |df = }}
| last = Poole | first = David | date = 2006 | title = Linear Algebra: A Modern Introduction | publisher = Brooks/Cole | edition = 2nd | isbn = 0-534-99845-3 }}
| last = Anton | first = Howard | date = 2005 | title = Elementary Linear Algebra (Applications Version) | publisher = Wiley International | edition = 9th }}
| last = Leon | first = Steven J. | date = 2006 | title = Linear Algebra With Applications | publisher = Pearson Prentice Hall | edition = 7th }}{{DEFAULTSORT:Elementary Matrix}} 1 : Linear algebra |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。