# 1 1 1 1 r th row s th row 2 row scaling matrix e is

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 ··· 0 0 0 ··· 1 0 0 ··· 0 0 0 0 ··· 0 1 0 ··· 0 0 0 ··· 0 0 0 0 ··· 0 0 0 ··· 0 0 1 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 ··· 0 0 0 ··· 0 0 0 ··· 1 r th row s th row . 2. (row scaling matrix) E is obtained from the identity matrix I m by replacing the row vector I ( r, · ) m with αI ( r, · ) m for some choice of non-zero scalar 0 negationslash = α F and some choice of positive integer r ∈{ 1 , 2 , . . . , m } . I.e., E = I m + ( α 1) E rr = 1 0 ··· 0 0 0 ··· 0 0 1 ··· 0 0 0 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 ··· 1 0 0 ··· 0 0 0 ··· 0 α 0 ··· 0 0 0 ··· 0 0 1 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 ··· 0 0 0 ··· 1 r th row , where E rr is the matrix having “ r, r entry” equal to one and all other entries equal to zero. (Recall that E rr was defined in Section 12.2.1 as a standard basis vector for the vector space F m × m .)
12.3. SOLVING LINEAR SYSTEMS BY FACTORING THE COEFFICIENT MATRIX 183 3. (row combination, a.k.a. “row sum”, matrix) E is obtained from the identity matrix I m by replacing the row vector I ( r, · ) m with I ( r, · ) m + αI ( s, · ) m for some choice of scalar α F and some choice of positive integers r, s ∈{ 1 , 2 , . . ., m } . I.e., in the case that r < s , E = I m + αE rs = 1 0 0 ··· 0 0 0 ··· 0 0 0 ··· 0 0 1 0 ··· 0 0 0 ··· 0 0 0 ··· 0 0 0 1 ··· 0 0 0 ··· 0 0 0 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 ··· 1 0 0 ··· 0 0 0 ··· 0 0 0 0 ··· 0 1 0 ··· 0 α 0 ··· 0 0 0 0 ··· 0 0 1 ··· 0 0 0 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 ··· 0 0 0 ··· 1 0 0 ··· 0 0 0 0 ··· 0 0 0 ··· 0 1 0 ··· 0 0 0 0 ··· 0 0 0 ··· 0 0 1 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 ··· 0 0 0 ··· 0 0 0 ··· 1 r th row s th column where E rs is the matrix having “ r, s entry” equal to one and all other entries equal to zero. ( E rs was also defined in Section 12.2.1 as a standard basis vector for F m × m .) The “elementary” in the name “elementary matrix” comes from the correspondence be- tween these matrices and so-called “elementary operations” on systems of equations. In particular, each of the elementary matrices is clearly invertible (in the sense defined in Sec- tion 12.2.3), just as each “elementary operation” is itself completely reversible. We illustrate this correspondence in the following example. Example 12.3.5. Define A , x , and b by A = 2 5 3 1 2 3 1 0 8 , x = x 1 x 2 x 3 , and b = 4 5 9 .
184 CHAPTER 12. SUPPLEMENTARY NOTES We illustrate the correspondence between elementary matrices and “elementary” operations on the system of linear equations corresponding to the matrix equation Ax = b , as follows.