Some understanding about matrix inversion and elementary transformation

$AA ^ {- 1} = E$
$(A | E) = (E | A ^ {-1})$

From simple `初等矩阵`to start

MATRIX: E through a matrix of elementary transformation of a matrix obtained.

There are three types of elementary matrix, for example in order to transform E:

The first MATRIX: E reversed the two rows (columns), referred to as $E_{ij}$

The second category MATRIX: Number k (! K = 0) by E i-th row (column), referred to as E_i(k)

MATRIX third category: the number of E k by the j row (column i) to the i-th row (column j), denoted as $E_ {ij} (a)$

Next, let's look at some of the nature of your elementary matrix and its inverse matrix:

$E_{ij}^{-1} = E_{ij}$
$E_i^{-1}(k) = E_i(1/k)$
$E_ {ij} ^ {- 1} (a) = E_ {ij} (- a)$
A matrix for the purposes of a class of elementary s row (column) conversion, A corresponds to the left (right) s-th category by MATRIX.

With this, we found that elementary elementary transformation matrix may be understood as one operation, while the inverse matrix MATRIX is actually the primary inverse transform matrix corresponding to the first elementary transformation . That matrix Aleft (and right) multiplied by a matrix of elementary Cand then left (right) multiplied $C^{-1}$ , the resulting matrix or A.

And we know that any reversible matrix A matrix of a unit E can be obtained through a series of elementary transformation. That series of elementary transforming effect on E get the A. Let us take it that A is an ordered set of elementary transformation of this series. So $A^{-1}$ what? $A^{-1}$ Each A is an ordered set of elementary transformation of the inverse transform is formed.

It $A ^ {- 1} A = E$ is very good to understand

$A ^ {- 1} A = E$ Can be written $A ^ {- 1} AE = E$ , is equivalent to the first embodiment of the E series of elementary transformation operation in set A, followed inverse operation of this series of operations, so that the resulting or E.

Look $(A | E) = (E | A ^ {-1})$

$(A | E) = (E | A ^ {-1})$ It is a common method of matrix inversion. The left side of Formula A is written in the form of augmentation (A | E) to the same operation simultaneously on the A and E.

Our goal is to transform A into E, A = AE , it is to transform E implemented a series of elementary operations set A, in the implementation of all its reverse operation $A^{-1}$ can return to E, while $A^{-1}$ they at the same time acting on E, so E becomes $A^{-1}$ .