Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

Transcription

1 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES 2012 Pearso Educatio, Ic.

2 Theorem 8: Let A be a square matrix. The the followig statemets are equivalet. That is, for a give A, the statemets are either all true or all false. a. A is a ivertible matrix. b. A is row equivalet to the idetity matrix. c. A has pivot positios. d. The equatio has oly the trivial solutio. e. The colums of A form a liearly idepedet set Pearso Educatio, Ic. A x = 0 Slide 2.3-2

3 f. The liear trasformatio is oe-tooe. A x = g. The equatio has at least oe solutio for each b i. h. The colums of A spa. i. The liear trasformatio maps oto. j. There is a matrix C such that. k. There is a matrix D such that. l. A T is a ivertible matrix. b x x a Ax a Ax CA = I AD = I 2012 Pearso Educatio, Ic. Slide 2.3-3

4 First, we eed some otatio. If the truth of statemet (a) always implies that statemet (j) is true, we say that (a) implies (j) ad write (a) (j). The proof will establish the circle of implicatios as show i the followig figure. If ay oe of these five statemets is true, the so are the others Pearso Educatio, Ic. Slide 2.3-4

5 Fially, the proof will lik the remaiig statemets of the theorem to the statemets i this circle. Proof: If statemet (a) is true, the i (j), so (a) (j). Next, (j) (d). works for C Also, (d) (c). If A is square ad has pivot positios, the the pivots must lie o the mai diagoal, i which case the reduced echelo form of A is I. Thus (c) (b). Also, (b) (a). A Pearso Educatio, Ic. Slide 2.3-5

6 This completes the circle i the previous figure. Next, (a) (k) because A 1 works for D. Also, (k) (g) ad (g) (a). So (k) ad (g) are liked to the circle. Further, (g), (h), ad (i) are equivalet for ay matrix. Thus, (h) ad (i) are liked through (g) to the circle. Sice (d) is liked to the circle, so are (e) ad (f), because (d), (e), ad (f) are all equivalet for ay matrix A. Fially, (a) (l) ad (l) (a). This completes the proof Pearso Educatio, Ic. Slide 2.3-6

7 Theorem 8 could also be writte as The equatio A x = b has a uique solutio for each b i. This statemet implies (b) ad hece implies that A is ivertible. The followig fact follows from Theorem 8. AB = B = A 1 Let A ad B be square matrices. If, the A ad B are both ivertible, with ad. The Ivertible Matrix Theorem divides the set of all matrices ito two disjoit classes: the ivertible (osigular) matrices, ad the oivertible (sigular) matrices. I A = B Pearso Educatio, Ic. Slide 2.3-7

8 Each statemet i the theorem describes a property of every ivertible matrix. The egatio of a statemet i the theorem describes a property of every sigular matrix. For istace, a sigular matrix is ot row equivalet to I, does ot have pivot positio, ad has liearly depedet colums Pearso Educatio, Ic. Slide 2.3-8

9 Example 1: Use the Ivertible Matrix Theorem to decide if A is ivertible: Solutio: A = A Pearso Educatio, Ic. Slide 2.3-9

10 So A has three pivot positios ad hece is ivertible, by the Ivertible Matrix Theorem, statemet (c). The Ivertible Matrix Theorem applies oly to square matrices. For example, if the colums of a 4 3 matrix are liearly idepedet, we caot use the Ivertible Matrix Theorem to coclude aythig about the existece or oexistece of solutios of equatio of the form A x = b Pearso Educatio, Ic. Slide

11 INVERTIBLE LINEAR TRANSFORMATIONS Matrix multiplicatio correspods to compositio of liear trasformatios. Whe a matrix A is ivertible, the equatio ca be viewed as a statemet about liear trasformatios. See the followig figure. 1 A Ax = x 2012 Pearso Educatio, Ic. Slide

12 INVERTIBLE LINEAR TRANSFORMATIONS T : : A liear trasformatio is said to be ivertible if there exists a fuctio S such that S( T (x)) = x for all x i ----(1) for all x i ----(2) T ( S (x)) = x T : Theorem 9: Let be a liear trasformatio ad let A be the stadard matrix for T. The T is ivertible if ad oly if A is a ivertible matrix. I that case, the liear trasformatio S give 1 by S(x) = A x is the uique fuctio satisfyig equatio (1) ad (2) Pearso Educatio, Ic. Slide

13 INVERTIBLE LINEAR TRANSFORMATIONS Proof: Suppose that T is ivertible. The (2) shows that T is oto, for if b is i ad x = S(b) T (x) = T ( S(b)) = b, the, so each b is i the rage of T. Thus A is ivertible, by the Ivertible Matrix Theorem, statemet (i). Coversely, suppose that A is ivertible, ad let S(x) = 1 A x. The, S is a liear trasformatio, ad S satisfies (1) ad (2). For istace,. Thus, T is ivertible Pearso Educatio, Ic. S T S A A A 1 ( (x)) = ( x) = ( x) = x Slide