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 matrixMathworldPlanetmath. We call N a left inverseMathworldPlanetmath 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 invertiblePlanetmathPlanetmath. 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 inverseMathworldPlanetmathPlanetmathPlanetmath 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 matrixMathworldPlanetmath. Furthermore, R is a field iff for any square matrix M (over R), M is invertible implies that MT, its transposeMathworldPlanetmath, is invertible as well. Invertibility of matrices over a division ring can also be determined by quantities known as ranks and determinantsMathworldPlanetmath. 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


over the Hamiltonian quaternions is not invertible, as its determinant k-ji=0. It is interesting to note that, however, its transpose


is invertible, whose determinant is 2k0. The relationship between determinants and matrix invertibility can also be used to prove the following: preservation of matrix invertibility upon matrix transpositionMathworldPlanetmath implies commutativity of division ring D. This can be done as follows: given any a,bD, the 2×2 matrix


is not invertible because its determinant is 0. Therefore, its transpose


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