19
Aug 19

## Elementary transformations

Here we look at matrix representations of transformations called elementary.

Exercise 1. Let $e_{i}$ denote the $i$-th unit row vector and let $A$ be an arbitrary matrix. Then

a) premultiplication of $A$ by $e_{j}$ cuts out of $A$ the $j$-th row $A_{j}.$

b) Premultiplication of $A$ by $e_{i}^{T}e_{j}$ puts the row $A_{j}$ into the $i$-th row of the null matrix.

c) Premultiplication of $A$ by $I+ce_{i}^{T}e_{j}$ adds row $A_{j}$ multiplied by $c$ to row $A_{i},$ without changing the other rows of $A.$

d) The matrix $I+ce_{i}^{T}e_{j}$ has determinant 1.

Proof. a) It's easy to see that

$e_{j}A=\left( 0...0~1~0...0\right) \left(\begin{array}{ccc}a_{11} & ... & a_{1n} \\ ... & ... & ... \\a_{j1} & ... & a_{jn} \\... & ... & ... \\a_{n1} & ... & a_{nn}\end{array} \right) =\left(\begin{array}{ccc}a_{j1} & ... & a_{jn}\end{array}\right) =A_{j}.$

b) Obviously,

$e_{i}^{T}e_{j}A=\left(\begin{array}{c}0 \\... \\0 \\1 \\0 \\... \\0\end{array} \right) \left(\begin{array}{ccc}a_{j1} & ... & a_{jn}\end{array} \right) =\left(\begin{array}{ccc}0 & ... & 0 \\... & ... & ... \\a_{j1} & ... & a_{jn} \\... & ... & ... \\ 0 & ... & 0\end{array} \right) =\left(\begin{array}{c}\Theta \\A_{j} \\\Theta\end{array} \right)$

($A_{j}$ in the $i$-th row, $\Theta$ denotes null matrices of conformable dimensions)

c) $(I+ce_{i}^{T}e_{j})A=A+ce_{i}^{T}e_{j}A=\left(\begin{array}{c} A_{1} \\... \\A_{i} \\... \\A_{n}\end{array} \right) +\left(\begin{array}{c}\Theta \\... \\cA_{j} \\... \\\Theta\end{array} \right) =\left(\begin{array}{c}A_{1} \\... \\A_{i}+cA_{j} \\ ... \\A_{n}\end{array}\right) .$

d) The matrix $A=I+ce_{i}^{T}e_{j}$ has ones on the main diagonal and only one nonzero element $a_{ij}=c$ outside it. By the Leibniz formula it's determinant is 1.

The reader can easily solve the next

Exercise 2. a) Postmultiplication of $A$ by $e_{j}^{T}$ cuts out of $A$ the $j$-th column $A^{(j)}.$

b) Postmultiplication of $A$ by $e_{j}^{T}e_{i}$ puts the column $A^{(j)}$ into the $i$-th column of the null matrix.

c) Postmultiplication of $A$ by $I+ce_{j}^{T}e_{i}$ adds column $A^{(j)}$ multiplied by $c$ to column $A^{(i)},$ without changing the other columns of $A.$

d) The matrix $I+ce_{j}^{T}e_{i}=(I+ce_{i}^{T}e_{j})^{T}$ has determinant 1.

Exercise 3. a) Premultiplication of $A$ by

(1) $\left(\begin{array}{c}e_{1} \\... \\e_{j} \\... \\e_{i} \\... \\e_{n}\end{array} \right)$

permutes rows $A_{i},A_{j}.$

b) Postmultiplication of $A$ by the transpose of (1) permutes columns $A^{(i)},A^{(j)}.$

This is a general property of permutation matrices. Recall also that their determinants can be only $\pm 1.$

Definition. 1) Adding some row multiplied by a constant to another row or 2) adding some column multiplied by a constant to another column or 3) permuting rows or columns is called an elementary operation. Accordingly, matrices that realize them are called elementary matrices.