Elementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
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1 Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015
2 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary matrices to find the inverse of a matrix (if the inverse exists), properties of invertible matrices.
3 Elementary Matrices Definition An n n matrix is an elementary matrix if it can be obtained from I n by a single elementary row operation.
4 Elementary Matrices Definition An n n matrix is an elementary matrix if it can be obtained from I n by a single elementary row operation. Example E is a 2 2 elementary matrix formed by swapping the two rows of I 2. [ ] 0 1 E = 1 0 Note the effect it has upon multiplying an arbitrary matrix. [ ] [ ] [ ] 0 1 a11 a 12 a 13 a21 a = 22 a a 21 a 22 a 23 a 11 a 12 a 13
5 Left Multiplication by E Theorem If E is an elementary matrix obtained from I m by performing a certain elementary row operation and if A is an m n matrix then EA is the matrix that results from performing the same elementary row operation on A.
6 Left Multiplication by E Theorem If E is an elementary matrix obtained from I m by performing a certain elementary row operation and if A is an m n matrix then EA is the matrix that results from performing the same elementary row operation on A. Example Let A = E 2 = , E 1 =, E 3 = ,, and calculate E 1 A, E 2 A, and E 3 A.
7 Inverse Operations Every elementary row operation has an inverse elementary row operation. Operation Inverse Multiply row i by c 0 Multiply row i by 1/c Swap rows i and j Swap rows i and j Add c times row i to row j Add c times row i to row j
8 Inverse Operations Every elementary row operation has an inverse elementary row operation. Operation Inverse Multiply row i by c 0 Multiply row i by 1/c Swap rows i and j Swap rows i and j Add c times row i to row j Add c times row i to row j Example Let E 1 = , E 2 = , E 3 = and find the corresponding inverse operations ,
9 Invertibility Theorem Every elementary matrix is invertible and the inverse is also an elementary matrix.
10 Invertibility Theorem Every elementary matrix is invertible and the inverse is also an elementary matrix. Proof. Let E be an elementary matrix which results from performing an elementary row operation on I. Let E 0 be the matrix that results from performing the inverse elementary row operation on I. By Thm. 3, E 0 E = I.
11 Invertibility Theorem Every elementary matrix is invertible and the inverse is also an elementary matrix. Proof. Let E be an elementary matrix which results from performing an elementary row operation on I. Let E 0 be the matrix that results from performing the inverse elementary row operation on I. By Thm. 3, E 0 E = I. By Thm. 3, EE 0 = I.
12 Invertibility Theorem Every elementary matrix is invertible and the inverse is also an elementary matrix. Proof. Let E be an elementary matrix which results from performing an elementary row operation on I. Let E 0 be the matrix that results from performing the inverse elementary row operation on I. By Thm. 3, E 0 E = I. By Thm. 3, EE 0 = I. Thus, E 0 = E 1.
13 Equivalence Result Theorem If A is an n n matrix, then the following statements are equivalent: 1. A is invertible. 2. Ax = 0 has only the trivial solution. 3. The reduced row echelon form of A is I n. 4. A is expressible as the product of elementary matrices.
14 Proof (1 = 2) Proof. Suppose A is invertible and let x 0 be any solution of Ax = 0. Ax 0 = 0
15 Proof (1 = 2) Proof. Suppose A is invertible and let x 0 be any solution of Ax = 0. Ax 0 = 0 A 1 Ax 0 = A 1 0 (A 1 A)x 0 = 0 Ix 0 = 0 x 0 = 0
16 Proof (1 = 2) Proof. Suppose A is invertible and let x 0 be any solution of Ax = 0. Ax 0 = 0 A 1 Ax 0 = A 1 0 (A 1 A)x 0 = 0 Ix 0 = 0 x 0 = 0 Thus Ax = 0 has only the trivial solution.
17 Proof (2 = 3) Proof. Suppose Ax = 0 has only the trivial solution, x = 0.
18 Proof (2 = 3) Proof. Suppose Ax = 0 has only the trivial solution, x = 0. The augmented matrix a 11 a 12 a 1n 0 a 21 a 22 a 2n a n1 a n2 a nn 0 can be row reduced by elementary row operations to
19 Proof (2 = 3) Proof. Suppose Ax = 0 has only the trivial solution, x = 0. The augmented matrix a 11 a 12 a 1n 0 a 21 a 22 a 2n a n1 a n2 a nn 0 can be row reduced by elementary row operations to Ignoring last column implies A is row reduced to I n.
20 Proof (3 = 4) Proof. Suppose the row reduced form of A is I n.
21 Proof (3 = 4) Proof. Suppose the row reduced form of A is I n. There is a finite sequence of elementary row operations which reduce A to I n.
22 Proof (3 = 4) Proof. Suppose the row reduced form of A is I n. There is a finite sequence of elementary row operations which reduce A to I n. To each elementary row operation corresponds an elementary matrix.
23 Proof (3 = 4) Proof. Suppose the row reduced form of A is I n. There is a finite sequence of elementary row operations which reduce A to I n. To each elementary row operation corresponds an elementary matrix. There exists a finite set of elementary matrices E 1, E 2,..., E k such that E k E k 1 E 2 E 1 A = I n.
24 Proof (3 = 4) Proof. Suppose the row reduced form of A is I n. There is a finite sequence of elementary row operations which reduce A to I n. To each elementary row operation corresponds an elementary matrix. There exists a finite set of elementary matrices E 1, E 2,..., E k such that E k E k 1 E 2 E 1 A = I n. Each elementary matrix has an inverse (which is also an elementary matrix) and thus A = E 1 1 E 1 2 E 1 k 1 E 1 k I n = E 1 1 E 1 2 E 1 k 1 E 1 k.
25 Proof (4 = 1) Proof. Assume A is a product of elementary matrices.
26 Proof (4 = 1) Proof. Assume A is a product of elementary matrices. Since elementary matrices are invertible, then A is a product of invertible matrices, and thus A is invertible.
27 Row Equivalence Definition If matrix B can be obtained from matrix A by a finite sequence of elementary row operations then A and B are said to be row equivalent.
28 Row Equivalence Definition If matrix B can be obtained from matrix A by a finite sequence of elementary row operations then A and B are said to be row equivalent. Remark: An n n matrix A is invertible if and only if A is row equivalent to I n.
29 Matrix Inversion Algorithm Algorithm: to find the inverse of an invertible matrix A, find the set of elementary row operations which reduces A to I and then perform this same sequence of operations on I to produce A 1.
30 Matrix Inversion Algorithm Algorithm: to find the inverse of an invertible matrix A, find the set of elementary row operations which reduces A to I and then perform this same sequence of operations on I to produce A 1. Example Find the inverse of A =
31 Example Example Find the inverse of A =
32 Homework Read Section 1.5 Work exercises 1 6, 8, 9, 11, 15.
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