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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.

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 \(I\).
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 \(A^{-1}\).