1. ROW OPERATION AND COLUMN OPERATIONS 1.1. Elementary Matrices, Row and Column Transformations. We have already studied in the last section that we can view the matrix multiplication as linear combi- nation of column vectors of the first matrix, or row vectors of the second matrix. And the coefficient of matrix multiplication is exactly given by other matrix. This shows that to understand matrix multiplication, we have to study the linear combination of row and column vectors. In this section, we will study the most basic linear combination of rows and columns, row and column transformation. 1.1.1. Elementary Row transformation. We have three types of row transformation. row switching This transformation swiches two row of matrix. 0 1 4 8 1 0 3 3 5 1 r1$r3 Switch the 1st and 3rd row of matrix @ 2 0 9 A −−−−! @ 2 0 9 A 3 3 5 1 4 8 row multiplying This transformation multiplies some row with a non-zero scalar λ 0 1 4 8 1 0 1 4 8 1 2×r2 Multiply the 2nd row of matrix by 2 : @ 2 0 9 A −−−! @ 4 0 18 A 3 3 5 3 3 5 row adding In this transformation, we multiply some row by a scalar, but add that into another row. 0 1 4 8 1 0 1 4 8 1 r3+2×r2 Add twice of the 2nd row to the 3rd row : @ 2 0 9 A −−−−−! @ 2 0 9 A 3 3 5 7 3 23 Caution: Write 2 × r instead of r × 2, the reason for that is simple, because scalar is 1 × 1 matrix. In this view, scalar can only appear in fromt of row vectors. Simillarly, we can define the column transformation in the same way. 1.1.2. column transformation. column switcing This transformation swiches two column of matrix. 0 1 4 8 1 0 8 4 1 1 c1$c3 Switch the 1st and 3rd column of matrix @ 2 0 9 A −−−−! @ 9 0 2 A 3 3 5 5 3 3 column multiplying This transformation multiplies some column with a non-zero scalar λ 0 1 4 8 1 0 1 8 8 1 c2×2 Multiply the 2nd column of matrix by 2 : @ 2 0 9 A −−−! @ 2 0 9 A 3 3 5 3 6 5 column adding In this transformation, we multiply some column by a scalar, but add that into another column. 1 0 1 4 8 1 0 1 4 16 1 c3+c2×2 Add twice of the 2nd column to the 3rd column : @ 2 0 9 A −−−−−! @ 2 0 9 A 3 3 5 3 3 11 1.1.3. Realization of elementary transformation by matrix multiplication. In the view of last section, row transformation is equivalent to left multiplication, column transformation is equivalent to right multiplica- tion. In order to make it precise, we define the following Elementary Matrices Definition 1.1.1 The Switching matrix is the matrix obtained by swapping ith and jth rows of unit matrix. Denote by Sij: 0 1 1 . B .. C B C i B 0 1 C B C B .. C Sij = B . C B C j B 1 0 C B . C @ .. A 1 Proposition 1.1 Left multiplyting switching matrix Sij will switch ith and jth rows of the matrix. 0 1 0 0 1 @ 0 0 1 A is obtained by switching 2nd and 3rd row of unit matrix, left multiplying 0 1 0 0 1 0 0 1 @ 0 0 1 A will do the same thing to rows of other matrix. As we compute using definition of 0 1 0 matrix multiplication: 0 1 0 0 1 0 1 2 3 1 0 1 2 3 1 @ 0 0 1 A @ 4 5 6 A = @ 7 8 9 A 0 1 0 7 8 9 4 5 6 0 1 2 3 1 The result is exactly switch 2nd and 3rd row of @ 4 5 6 A 7 8 9 Proposition 1.2 Right multiplying switching matrix Sij will switch jth and ith columns of the matrix. 2 0 1 0 0 1 @ 0 0 1 A is obtained by switching 2nd and 3rd column of unit matrix, right multiplying 0 1 0 0 1 0 0 1 @ 0 0 1 A will do the same thing to columns of other matrix. As we compute using definition 0 1 0 of matrix multiplication: 0 1 2 3 1 0 1 0 0 1 0 1 3 2 1 @ 4 5 6 A @ 0 0 1 A = @ 4 6 5 A 7 8 9 0 1 0 7 9 8 0 1 2 3 1 The result is exactly switch 2nd and 3rd column of @ 4 5 6 A 7 8 9 Definition 1.1.2 The Multiplying matrix is the matrix obtained by multiplying ith row by non-zero scalar λ of unit matrix. Denote by Mi(λ): i 0 1 1 . B .. C B C B C Mi(λ) = iB λ C B .. C @ . A 1 Proposition 1.3 Left multiplyting multiplying matrix Mi(λ) will multiplies i’th row of the matrix by λ. 0 1 0 0 1 @ 0 3 0 A is obtained by multiplying the 2nd row of unit matrix by 3, left multiplying 0 0 1 0 1 0 0 1 @ 0 3 0 A will do the same thing to rows of other matrix. As we compute using definition of 0 0 1 matrix multiplication: 0 1 0 0 1 0 1 2 3 1 0 1 2 3 1 @ 0 3 0 A @ 4 5 6 A = @ 12 15 18 A 0 0 1 7 8 9 7 8 9 3 0 1 2 3 1 The result is exactly multiplying the 2nd row of @ 4 5 6 A by 3 7 8 9 Proposition 1.4 Right multiplyting multiplying matrix Mi(λ) will multiplies i’th column of the matrix by λ. 0 1 0 0 1 @ 0 3 0 A is obtained by multiplying the 2nd column of unit matrix by 3, right multi- 0 0 1 0 1 0 0 1 plying @ 0 3 0 A will do the same thing to columns of other matrix. As we compute using 0 0 1 definition of matrix multiplication: 0 1 2 3 1 0 1 0 0 1 0 1 6 3 1 @ 4 5 6 A @ 0 3 0 A = @ 4 15 6 A 7 8 9 0 0 1 7 24 9 0 1 2 3 1 The result is exactly multiplying the 2nd column of @ 4 5 6 A by 3 7 8 9 Definition 1.1.3 The Addition matrix is the matrix obtained by add jth row by scalar λ to the ith row of unit matrix. Denote by Aij(λ): j 0 1 1 . B .. C B C iB 1 λ C B C B .. C Aij(λ) = B . C B C B 1 C B . C @ .. A 1 Proposition 1.5 Left multiplying addition matrix Mi(λ) will add λ times j’th row to the i’th row. 4 0 1 2 0 1 @ 0 1 0 A is obtained by adding twice of the 2nd row of unit matrix to 1st row, left mul- 0 0 1 0 1 2 0 1 tiplying @ 0 1 0 A will do the same thing to rows of other matrix. As we compute using 0 0 1 definition of matrix multiplication: 0 1 2 0 1 0 1 2 3 1 0 9 12 15 1 @ 0 1 0 A @ 4 5 6 A = @ 4 5 6 A 0 0 1 7 8 9 7 8 9 0 1 2 3 1 The result is exactly adding twice of the 2nd row to the 1st row of @ 4 5 6 A 7 8 9 Proposition 1.6 right multiplying addition matrix Mi(λ) will add λ times i’th column to the j’th column. 0 1 2 0 1 @ 0 1 0 A is obtained by adding twice of the 1st column of unit matrix to 2nd column, 0 0 1 0 1 2 0 1 right multiplying @ 0 1 0 A will do the same thing to columns of other matrix. As we 0 0 1 compute using definition of matrix multiplication: 0 1 2 3 1 0 1 2 0 1 0 1 4 3 1 @ 4 5 6 A @ 0 1 0 A = @ 4 13 6 A 7 8 9 0 0 1 7 22 9 0 1 2 3 1 The result is exactly adding twice of the 1st column to the 2nd column of @ 4 5 6 A 7 8 9 Caution: when Aij(m) serving at left, it is add m times j’th row to i’th row. But when it serving at right, it is add m times i’th column to j’th column. Keep in mind that the order is different. As previous example, we have seen there is an easy way to remember the operation. When left multi- plying, like AB, you think how can we get A by row transformation from unit matrix, and then to get the product, do the same row transformation to B. When right multiplying, like BA, you think how to get A by column transformation from unit matrix, and then to get the product, do the same column transformation to B. All the matrices we defined above Sij, Mi(λ), Aij(λ), are called Elementary matrices.