Aug 19

Elementary transformations

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.

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