By Secchi S.
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Extra resources for A note on closed geodesics for a class of non-compact Riemannian manifolds
For example, . The operation of multiplying a matrix by a scalar has these basic properties: All four are readily proved by appealing to the definition. In the case of the second, we have (α + β)A = [(α + β)aij] = [αaij + βaij] = [αaij] + [βaij] = αA + βA. We leave it as an exercise to the reader to prove the other laws in a similar fashion. 7 The Multiplication of Matrices Frequently in the mathematical treatment of a problem, the work can be simplified by the introduction of new variables. Translations of axes, effected by equations of the form and rotations of axes, effected by are the most familiar examples.
Again, if A–1 and B–1 exist, then if and only if , so that the second equation gives the unique solution for W of the first. Note the importance of distinguishing between pre– and postmultiplication here. 16 The Product of a Row Matrix into a Column Matrix The product of a row matrix into a column matrix is a matrix of order 1: . 13. Now the set of all 1 × 1 matrices whose elements belong to is isomorphic to . In fact, if α and β are any two scalars of , we have, for the corresponding 1 × 1 matrices, [α] + [β] = [α + β] and [α][β] = [αβ], so that the matrices behave exactly like scalars with respect to addition and multiplication.
Consider finally the sum , which we can rewrite as . The sum in parentheses is the element in row i and column k2 of A2. Hence the second sum represents the element in row i and column j of A2 · A. 12 Symmetric, Skew-Symmetric, and Hermitian Matrices A symmetric matrix is a square matrix A such that A = A T. A skew-symmetric matrix is a square matrix A such that A = –A T. These definitions may also be stated in terms of the individual elements: A is symmetric if and only if aij = aji for all pairs of subscripts; it is skew-symmetric if and only if aij = –aji for all pairs of subscripts.
A note on closed geodesics for a class of non-compact Riemannian manifolds by Secchi S.