mathematics-physics-wiki/docs/en/mathematics/linear-algebra/matrices/elementary-matrices.md

Elementary matrices

Definition: an elementary matrix is defined as an identity matrix with exactly one elementary row operation undergone.

1. An elementary matrix of type 1 E_1 is obtained by changing two rows I.
2. An elementary matrix of type 2 E_2 is obtained by multiplying a row of I by a nonzero constant.
3. An elementary matrix of type 3 E_3 is obtained from I by adding a multiple of one row to another row.

For example the elementary matrices could be given by

E_1 = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1\end{pmatrix}, \qquad E_2 = \begin{pmatrix} 1 & 0 & 0\ 0 & 1 & 0\ 0 & 0 & 3\end{pmatrix}, \qquad E_3 = \begin{pmatrix}1 & 0 & 3\ 0 & 1 & 0\ 0 & 0 & 1\end{pmatrix}.

Theorem: if E is an elementary matrix, then E is nonsingular and E^{-1} is an elementary matrix of the same type.

??? note "Proof:"

If $E$ is the elementary matrix of type 1 formed from $I$ by interchanging the $i$th and $j$th rows, then $E$ can be transfomred back into $I$ by interchanging these same rows again. Therefore, $EE = I$ and hence $E$ is its own inverse.

IF $E$ is the elementray matrix of type 2 formed by multiplying the $i$th row of $I$ by a nonzero scalar $\alpha$ then $E$ can be transformed into the identity matrix by multiplying either its $i$th row or its $i$th column by $1/\alpha$. 

If $E$ is the elemtary matrix of type 3 formed from $I$ by adding $m$ times the $i$th row to the $j$th row then $E$ can be transformed back into $I$ either by subtracting $m$ times the $i$th row from the $j$th row or by subtracting $m$ times the $j$th column from the $i$th column.

Definition: a matrix B is row equivalent to a matrix A if there exists a finite sequence E_1, E_2, \dots, E_K of elementary matrices with k \in \mathbb{N} such that

B = E_k E_{k-1} \cdots E_1 A.

It may be observed that row equivalence is a reflexive, symmetric and transitive relation.

Theorem: let A be an n \times n matrix, the following are equivalent

1. A is nonsingular,
2. A\mathbf{x} = \mathbf{0} has only the trivial solution \mathbf{0},
3. A is row equivalent to I.

??? note "Proof:"

Let $A$ be a nonsingular $n \times n$ matrix and $\mathbf{\hat x}$ is a solution of $A \mathbf{x} = \mathbf{0}$ then

$$
    \mathbf{\hat x} = I \mathbf{\hat x} = (A^{-1} A)\mathbf{\hat x} = A^{-1} (A \mathbf{\hat x}) = A^{-1} \mathbf{0} = \mathbf{0}.
$$

Let $U$ be the row echelon form of $A$. If one of the diagonal elements of $U$ were 0, the last row of $U$ would consist entirely of zeros. But then $A \mathbf{x} = \mathbf{0}$ would have a nontrivial solution. Thus $U$ must be a strictly triangular matrix with diagonal elements all equal to 1. It then follows that $I$ is the reduced row echelon form of $A$ and hence $A$ is row equivalent to $I$.

If $A$ is row equivalent to $I$ there exists elementary matrices $E_1, E_2, \dots, E_k$ with $k \in \mathbb{N}$ such that 

$$
    A = E_k E_{k-1} \cdots E_1 I = E_k E_{k-1} \cdots E_1.
$$

Since $E_i$ is invertible for $i \in \{1, \dots, k\}$ the product $E_k E_{k-1} \cdots E_1$ is also invertible, hence $A$ is nonsingular.

If A is nonsingular then A is row equivalent to I and hence there exists elemtary matrices E_1, \dots, E_k such that

E_k E_{k-1} \cdots E_1 A = I,

multiplyting both sides on the right by A^{-1} obtains

E_k E_{k-1} \cdots E_1 = A^{-1}

a method for computing A^{-1}.