Raising the bar

Elementary transformations

Here we look at matrix representations of transformations called elementary.

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

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

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

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

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

Proof. a) It's easy to see that

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

b) Obviously,

$latex 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) $

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

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

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

The reader can easily solve the next

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

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

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

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

Exercise 3. a) Premultiplication of $latex A$ by

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

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

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

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

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.

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