Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below assume that. + B B + + r( s) ( rs ) ( r + s) r + s r( + B) r + rb ( + B) + C + ( B + C) ( BC) ( B ) C I m I n ( B + C) B + C ( B + C) B + C r( B) ( r ) B ( rb) Page of 9
Example : Find "C" where C + B. a a a a a a B b b b b b b To find a particular entry for "C", just add the corresponding entries of "" and "B". That is, the ij th of "C" is c ij a ij + b ij. + B a a a a a a b + b b b b b C ( a + b ) a + b ( a + b ) a + b ( a + b ) a + b Example : Find "C" where C r. a a a a a a Here "r" is a scalar. The product of a scalar and a matrix is a matrix all of whose entries are the products of that scalar and the corresponding entry of the given matrix. That is, the ij th of "C" is c ij ra ij. Page of 9
C r a a a a a a ra ra ra ra ra ra Example : Find "C" where C Columns Method. Busing Rows times a a a a a a B b b b b b b The entry, c ij, for "C" is the dot product of the i th row of "" with j th column of "B". Suppose "" is mxn and "B" is nxm. c ij a i b j + a i b j +... + a in b nj n c ij k a ik b kj In this example, m and n. C b a a a b a a a b b b b Page of 9
C Row Col ( B) Row Col ( B) Row Col ( B) Row Col ( B) Example : Find "C" where C Column Method. a a a a a a B Busing Matrix times b b b b b b C Col ( B) Col ( B) Columns of "C" are Linear Combinations of the columns of "" Example 5: Find "C" where C Matrix Method. a a a a a a B Busing Row times b b b b b b Page of 9
C Row B Row B Rows of "C" are Linear Combinations of the rows of "B". Example 6: Find "C" where C Row Method. Busing Column times a a a a a a B b b b b b b C Col k Row k ( B) k "C" is the sum of columns of "" times corresponding row of "B". Definition: If " " is "n x m", then its transpose is "m x n" denoted by, T, and is created by interchanging the rows and columns of " " so that: a T. ij a ji a a a a a a T a a a a a a Page 5 of 9
Example 8: Prove that: ( B ) T B T T "m x n" & "B" is "n x p". where "" is C Bis "m x p" and C T ( B ) T is "p x m". n c ij k a ik b kj ( c T ) ij c ji n a jk b ki k ( a T ) kj a jk ( b T ) ik b ki B T T k n ( b T ) ik ( a T ) kj Page 6 of 9
Definition: If " " is a square "n x n", then "" is said to be invertible or to have an inverse if there exists a matrix "C" such that C and C. The matrix "C" is I n denoted as. ccordingly, and I n I n I n. There is an algorithm that you are already well equipped to use in finding the inverse of a matrix. Here is the basis for that algorithm. Recall the matix times the column method shown in example above. Let B, then B I n. ( b...b n ) e... e n This invites us to consider "n" matrix equations, each of the form: b k e k, where k,..., n. We could solve these equations and find all of the columns, b k, of the matrix "B" and use them to assemble "B", which is what we are seeking, namely:. But since we need to perform the same row operations to solve each of those "n" equations separately, why not solve them all at the same time by augmenting "" with "I n ". Page 7 of 9
Example 9: Find if it exists. Form the augmented matrix, ( I ). ( I ) Step # :Replace row with the sum of row and times row. (Fixes the "" position.) Step # :Replace row with the sum of row and times row. (Fixes the "" position.) 8 Page 8 of 9
Step # :Replace row with the sum of row and times row. (Fixes the "" position.) 7 Step # :Replace row with the sum of row and times row. (Fixes the "" position.) 8 7 Step # 5:Replace row with the sum of row and times row. (Fixes the "" position.) 8 7 Page 9 of 9
Step # 6:Relace row with times row. (Fixes the "" position.) 8 7 b 8 7 b b B b b b 8 7 Definition: n Elementary Matrix, "E", is a matrix that is obtained by performing a single row operation on the Identity Matrix. Page of 9
Example : Determine the Elementary matrices corresponding to each of the six individual steps in Example 9 above. Step # :Replace row with the sum of row and times row. I E Step # :Replace row with the sum of row and times row. I E Step # :Replace row with the sum of row and times row. I E Page of 9
Step # :Replace row with the sum of row and times row. I E Step # 5:Replace row with the sum of row and times row. I E Step # 6:Relace row with times row. I E Page of 9
Theorem: If an elementary row operation is performed on an mxn matrix "", the resulting matrix can be written as E ij, where E ij is created by performing the same row operations on I m. Example : Show that this works for each of the six individual steps in Examples 9 & above. Step # :Replace row with the sum of row and times row. E Step # :Replace row with the sum of row and times row. E ( E ) 8 Step # :Replace row with the sum of row and times row. E E E 8 Page of 9
Step # :Replace row with the sum of row and times row. E E E E Step # 5:Replace row with the sum of row and times row. E E E E E E E E E E Step # 6:Relace row with times row. E E E E E E Page of 9
E E E E E E Thus, the ordered product of all of these individual elementary matrices with the original matrix "" produces the Identity Matrix demonstrating that that product is equivalent to performing the corresponding set of elementary row operations. lso, since ( E E E E E E ) I, ( E E E E E E ) and thus E E E E E E Theorem: Each elementary matrix " E ij " is invertible. The inverse matrix " E ij " is the elementary matrix of the same type that transforms "E ij " back into the Identity Matrix. Example : Find the inverse of E of example. E Page 5 of 9
The matrix "E " replaces row by the sum of row and times row. The inverse is constructed by undoing that operation. The matrix " E " replaces row by the sum of row and times row. E E E I E E I Theorem: If "" and "B" are nxn invertible matrices, then so is their product. B B Page 6 of 9
( B ) ( B ) B B B I B ( B ) B B I ( ) B Similarly, ( B ) B ( B ) I. Therefore, Matrix products can be calculated by regarding one of the two factors as a Compound Elementary Matrix. Example : Find the product C matrices and B Bof the assuming that "" is the Compound Elementary Matrix., C B Row ( C) ( ) + ( ) + ( ) Row ( C) ( ) Page 7 of 9
Row ( C) ( ) + ( ) ( ) + ( ) Row ( C) ( 5 ) Row ( C) ( ) + ( ) + ( ) Row ( C) ( 6) C B 5 6 Example : Find the product C matrices and B Bof the assuming that "B" is the Compound Elementary Matrix., C B Page 8 of 9
Col C + 5 Col C + + Col C + 6 C B 5 6 Page 9 of 9