direct sum of matrices
Direct sum of matrices
Let be an matrix and be a matrix. By the direct sum of and , written , we mean the matrix of the form
where the ’s represent zero matrices. The on the top right is an matrix, while the on the bottom left is .
For example, if and , then
Remark. It is not hard to see that the operation on matrices is associative:
because both sides lead to
In fact, we can inductively define the direct sum of matrices unambiguously.
Direct sums of linear transformations
The direct sum of matrices is closely related to the direct sum of vector spaces and linear transformations. Let and be as above, over some field . We may view and as linear transformations and using the standard ordered bases. Then may be viewed as the linear transformation
using the standard ordered basis, such that
the restriction of to the subspace (embedded in ) is , and
the restriction of to is .
The above suggests that we can define direct sums on linear transformations. Let and be linear transformations, where and are finite dimensional vector spaces over some field such that . Then define such that for any ,
where . Based on this definition, it is not hard to see that
for any matrices and .
More generally, if is an ordered basis for , then extending the linear orders on , such that if and , then is an ordered basis for , and
|Title||direct sum of matrices|
|Date of creation||2013-03-22 17:36:48|
|Last modified on||2013-03-22 17:36:48|
|Last modified by||CWoo (3771)|