{"product_id":"generalized-inverses-theory-and-applications-von-adi-ben-israel-thomas-n-e-greville","title":"Generalized Inverses","description":"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.\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9780387002934\"\u003e\u003ch3\u003eTheory and Applications\u003c\/h3\u003e\u003c\/div\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9781441918147\"\u003e\u003ch3\u003eTheory and Applications\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Hardcover - 9780387002934","offer_id":50827735622,"sku":"9780387002934","price":106.99,"currency_code":"EUR","in_stock":true},{"title":"Softcover - 9781441918147","offer_id":39415018160221,"sku":"9781441918147","price":74.89,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/e37bc9de-7a83-4051-8a13-2b50b73f9cbd.jpg?v=1772170891","url":"https:\/\/shop.autorenwelt.de\/en\/products\/generalized-inverses-theory-and-applications-von-adi-ben-israel-thomas-n-e-greville","provider":"Autorenwelt Shop","version":"1.0","type":"link"}