Let be a ring and an matrix over . is said to be left invertible if there is an matrix such that , where is the identity matrix. We call a left inverse of . Similarly, is right invertible if there is an matrix , called a right inverse of , such that , where is the identity matrix. If is both left invertible and right invertible, we say that is invertible. If is an associative ring, and 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 .
If is a division ring, then it can be shown that for any matrix over , is left invertible iff it is invertible iff it is right invertible. In addition, when is invertible, it is a square matrix. Furthermore, is a field iff for any square matrix (over ), is invertible implies that , 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 matrix
over the Hamiltonian quaternions is not invertible, as its determinant . It is interesting to note that, however, its transpose
is invertible, whose determinant is . 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 . This can be done as follows: given any , the matrix
is not invertible because its determinant is . Therefore, its transpose
is also not invertible, and its determinant is , whence is a field.
|Date of creation||2013-03-22 19:23:09|
|Last modified on||2013-03-22 19:23:09|
|Last modified by||CWoo (3771)|