Uniqueness of identity elements for rectangular matrices

  • Thread starter Bipolarity
  • Start date
  • #1
775
1

Main Question or Discussion Point

Let A be the set of [itex] n \times n [/itex] matrices. Then the identity element of this set under matrix multiplication is the identity matrix and it is unique. The proof follows from the monoidal properties of multiplication of square matrices.

But if the matrix is not square, the left and right identities are not equivalent; they are both identity matrices, but have a different size.

How do you know that the left-identity is unique, and that the right-identity is unique?
So given an [itex] m \times n [/itex] matrix A, how do you know that the only matrix satisfying [itex] AI = A [/itex] for all A is the [itex] n \times n [/itex] identity matrix?

Is this even true? Could I possibly find multiple right-identity elements?

BiP
 

Answers and Replies

  • #2
22,097
3,281
Let ##B## be an identity element. Let ##A## be the ##m\times n## matrix with a 1 on the ##i##th row and the ##j##th column and zero everywhere else. You know that ##AB = A##. This gives a condition on ##B##. Which one? What if you vary ##i## and ##j##?
 

Related Threads on Uniqueness of identity elements for rectangular matrices

  • Last Post
Replies
1
Views
1K
Replies
7
Views
620
  • Last Post
Replies
6
Views
2K
Replies
1
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
642
Replies
6
Views
3K
Replies
3
Views
9K
Replies
3
Views
1K
Replies
8
Views
4K
Top