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Generalized Inverses : Theory and Applications. 2nd ed. 2003.
・ISBN 978-0-387-00293-4 hard EUR 99.99
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| 著者・編者 | Ben-Israel, Adi / Greville, Thomas N.E., |
|---|---|
| シリーズ | CMS Books in Mathematics |
| 出版社 | (Springer-Verlag New York Inc., US) |
| 出版年月 | 2003 |
| ページ数 | 420 pp. |
| 言語 | ENG |
| ニュース番号 | ＜M25-23425＞ |
解説
1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse, ?1 denoted by A , such that ?1 ?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus, ?1 ?1 (A ) = A, T ?1 ?1 T (A ) =(A ) , ? ?1 ?1 ? (A ) =(A ) , ?1 ?1 ?1 (AB) = B A , T ? where A and A , respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number ? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to ?,if Ax = ?x. ?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.