invertible matrix
Let R be a ring and M an m×n matrix over R. M is said to be left invertible if there is an n×m matrix such that NM=In, where In is the n×n identity matrix. We call N a left inverse
of M. Similarly, M is right invertible if there is an n×m matrix P, called a right inverse of M, such that MP=Im, where Im is the m×m identity matrix. If M is both left invertible and right invertible, we say that M is invertible
. If R is an associative ring, and M is invertible, then it has a unique left and a unique right inverse, and they are in fact equal, we call this matrix the inverse
of M.
If R is a division ring, then it can be shown that for any matrix M over R, M is left invertible iff it is invertible iff it is right invertible. In addition, when M is invertible, it is a square matrix. Furthermore, R is a field iff for any square matrix M (over R), M is invertible implies that MT, its transpose
, is invertible as well. Invertibility of matrices over a division ring can also be determined by quantities known as ranks and determinants
. It can be shown that a matrix over a division ring is invertible iff its left row rank (or right column rank) is full iff its determinant is non-zero. For example, the 2×2 matrix
(1jik) |
over the Hamiltonian quaternions is not invertible, as its determinant k-ji=0. It is interesting to note that, however, its transpose
(1ijk) |
is invertible, whose determinant is 2k≠0. The relationship between determinants and matrix invertibility can also be used to prove the following: preservation of matrix invertibility upon matrix transposition implies commutativity of division ring D. This can be done as follows: given any a,b∈D, the 2×2 matrix
(abba1) |
is not invertible because its determinant is 0. Therefore, its transpose
(abab1) |
is also not invertible, and its determinant is 0=ab-ba, whence D is a field.
Title | invertible matrix |
---|---|
Canonical name | InvertibleMatrix |
Date of creation | 2013-03-22 19:23:09 |
Last modified on | 2013-03-22 19:23:09 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 15-01 |
Classification | msc 15A09 |
Classification | msc 15A33 |