Editing Invertible matrix (section)

== Applications ==
For most practical applications, it is not necessary to invert a matrix to solve a [[system of linear equations]]; however, for a unique solution, it is necessary for the matrix involved to be invertible.

Decomposition techniques like [[LU decomposition]] are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.

=== Regression/least squares ===
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy and is found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.<ref>{{cite journal |first1=Lin |last1=Lin |first2=Jianfeng |last2=Lu |first3=Lexing |last3=Ying |first4=Roberto |last4=Car |first5=Weinan |last5=E |title=Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems |journal=Communications in Mathematical Sciences |volume=7 |issue=3 |year=2009 |pages=755–777 |doi=10.4310/CMS.2009.v7.n3.a12 |doi-access=free }}</ref>

=== Matrix inverses in real-time simulations ===
Matrix inversion plays a significant role in [[computer graphics]], particularly in [[3D graphics]] rendering and [[3D simulations]]. Examples include screen-to-world [[ray casting]], world-to-subspace-to-world object transformations, and physical simulations.

=== Matrix inverses in MIMO wireless communication ===
Matrix inversion also plays a significant role in the [[MIMO]] (Multiple-Input, Multiple-Output) technology in [[wireless communications]]. The MIMO system consists of ''N'' transmit and ''M'' receive antennas. Unique signals, occupying the same [[frequency band]], are sent via ''N'' transmit antennas and are received via ''M'' receive antennas. The signal arriving at each receive antenna will be a [[linear combination]] of the ''N'' transmitted signals forming an ''N''&nbsp;×&nbsp;''M'' transmission matrix '''H'''. It is crucial for the matrix '''H''' to be invertible so that the receiver can figure out the transmitted information.