Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. V)
Point of post: This is a continuation of this post.
Problem: For which values of are the following matrices invertible? Find the inverses whenever possible.
a) This is invertible precisely when . Indeed, if then . And, if then serves as an inverse.
b) This is invertible precisely when . Indeed, if then . And, if then . If then serves as an inverse.
c) Assuming we’re discussing real matrices this is true precisely when . Indeed, if then . If then serves as an inverse.
d) This is invertible precisely when . Indeed, if then . Otherwise serves as an inverse.
a) It is easy to extend matrix theory to linear transformations between different vector spaces. Suppose that and are vector spaces over the same field . Let and be bases for and respectively, and . The matrix of is, by definition, the rectangular array of scalars defined by
Define addition and multiplication of rectangular matrices so as to generalize as many possible of the results of section 38.
b) Suppose that and are multipliable matrices. Partition into four rectangular blocks (top left, top right, bottom left, bottom right) and then partition similarly so that the number of columns in the top left part of is the same as the number of rows in the top left part of . If, in an obvious shorthand, these partitioned matrices are indicated by
c) Use subspaces and complements to express the result of b) in terms of linear transformations (instead of matrices).
a) We may define addition of rectangular matrices exactly the same. For multiplication, we may only (in any normal sense) define matrix multiplication of rectangular matrices and when the size of them are and respectively. From there we may define the multiplication to be the matrix whose general term is, unsurprisingly
One can check that with this definition all the axioms of an associative non-unital algebra are held.
b) This is just tedious computation, if someone has a dying desire for me to upload this, let me know.
c) Suppose that . Then, for some we can compute , where by thinking of and we see then that
from where, with the proper definitions of the result follows.
1. Halmos, Paul R. Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print
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