Linear Algebra

220 Chapter Three. Maps Between Spaces3.13 Example From the left, the action of multiplication by a diagonal matrixis to rescales the rows.( ) ( ) ( )2 0 2 1 4 −1 4 2 8 −2=0 −1 −1 3 4 4 1 −3 −4 −4From the right such a matrix rescales the columns.( ) ⎛1 2 1 ⎝ 3 0 0⎞( )0 2 0 ⎠ 3 4 −2=2 2 26 4 −40 0 −2The second generalization of identity matrices is that we can put a single onein each row and column in ways other than putting them down the diagonal.3.14 Definition A permutation matrix is square and is all zeros except for asingle one in each row and column.3.15 Example From the left these matrices permute rows.From the right they permute columns.⎛⎝ 0 0 1⎞ ⎛1 0 0⎠⎝ 1 2 3⎞ ⎛4 5 6⎠ = ⎝ 7 8 9⎞1 2 3⎠0 1 0 7 8 9 4 5 6⎛⎝ 1 2 3⎞ ⎛4 5 6⎠⎝ 0 0 1⎞ ⎛1 0 0⎠ = ⎝ 2 3 1⎞5 6 4⎠7 8 9 0 1 0 8 9 7We finish this subsection by applying these observations to get matrices thatperform Gauss’ method and Gauss-Jordan reduction.3.16 Example We have seen how to produce a matrix that will rescale rows.Multiplying by this diagonal matrix rescales the second row of the other by afactor of three.⎛⎝ 1 0 0⎞ ⎛⎠ ⎝ 0 2 1 1⎞ ⎛⎠ = ⎝ 0 2 1 1⎞⎠0 3 00 0 10 1/3 1 −11 0 2 00 1 3 −31 0 2 0We have seen how to produce a matrix that will swap rows. Multiplying by thispermutation matrix swaps the first and third rows.⎛⎝ 0 0 1⎞ ⎛0 1 0⎠⎝ 0 2 1 1⎞ ⎛0 1 3 −3⎠ = ⎝ 1 0 2 0⎞0 1 3 −3⎠1 0 0 1 0 2 0 0 2 1 1