4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns


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1 L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows 41
2 Left and right inverse AB BA in general, so we have to distinguish two types of inverses Left inverse: X is a left inverse of A if XA = I A is left invertible if it has at least one left inverse Right inverse: X is a right inverse of A if AX = I A is right invertible if it has at least one right inverse Matrix inverses 42
3 Examples A = , B = [ ] A is left invertible; the following matrices are left inverses: 1 9 [ ], [ 0 1/ /2 2 ] B is right invertible; the following matrices are right inverses: , , Matrix inverses 43
4 Some immediate properties Dimensions a left or right inverse of an m n matrix must have size n m Left and right inverse of (conjugate) transpose X is a left inverse of A if and only if X T is a right inverse of A T A T X T = (XA) T = I X is a left inverse of A if and only if X H is a right inverse of A H A H X H = (XA) H = I Matrix inverses 44
5 Inverse if A has a left and a right inverse, then they are equal and unique: XA = I, AY = I = X = X(AY ) = (XA)Y = Y we call X = Y the inverse of A (notation: A 1 ) A is invertible if its inverse exists Example A = , A 1 = Matrix inverses 45
6 Linear equations set of m linear equations in n variables A 11 x 1 + A 12 x A 1n x n = b 1 A 21 x 1 + A 22 x A 2n x n = b 2.. A m1 x 1 + A m2 x A mn x n = b m in matrix form: Ax = b may have no solution, a unique solution, infinitely many solutions Matrix inverses 46
7 Linear equations and matrix inverse Left invertible matrix: if X is a left inverse of A, then Ax = b = x = XAx = Xb there is at most one solution (if there is a solution, it must be equal to Xb) Right invertible matrix: if X is a right inverse of A, then x = Xb = Ax = AXb = b there is at least one solution (namely, x = Xb) Invertible matrix: if A is invertible, then Ax = b x = A 1 b there is a unique solution Matrix inverses 47
8 Outline left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
9 Linear combination a linear combination of vectors a 1,..., a n is a sum of scalar products x 1 a 1 + x 2 a x n a n the scalars x i are the coefficients of the linear combination can be written as a matrixvector product x 1 a 1 + x 2 a x n a n = [ ] a 1 a 2 a n x 1 x 2.. x n the trivial linear combination has coefficients x 1 = = x n = 0 (same definition holds for real and complex vectors/scalars) Matrix inverses 48
10 Linear dependence a set of vectors a 1, a 2,..., a n is linearly dependent if x 1 a 1 + x 2 a x n a n = 0 for some scalars x 1,..., x n, not all zero vector 0 can be written as a nontrivial linear combination of a 1,..., a n equivalently, at least one vector a i is a linear combination of the rest: if x i 0 a i = x 1 x i a 1 x i 1 x i a i 1 x i+1 x i a i+1 x n x i a n Matrix inverses 49
11 Example the vectors a 1 = , a 2 = , a 3 = are linearly dependent 0 can be expressed as a nontrivial linear combination of a 1, a 2, a 3 : 0 = a 1 + 2a 2 3a 3 a 1 can be expressed as a linear combination of a 2, a 3 : (and similarly a 2 and a 3 ) a 1 = 2a 1 + 3a 3 Matrix inverses 410
12 Linear independence a 1,..., a n are linearly independent if they are not linearly dependent the zero vector cannot be written as a nontrivial linear combination: x 1 a 1 + x 2 a x n a n = 0 = x 1 = x 2 = = x n = 0 none of the vectors a i is a linear combination of the other vectors Matrix with linearly independent columns A = [ a 1 a 2 a n ] has linearly independent columns if Ax = 0 only for x = 0 Matrix inverses 411
13 Example the vectors a 1 = 1 2 0, a 2 = 1 0 1, a 3 = are linearly independent: only if x 1 = x 2 = x 3 = 0 x 1 a 1 + x 2 a 2 + x 3 a 3 = x 1 x 2 2x 1 + x 3 x 2 + x 3 = 0 Matrix inverses 412
14 Dimension inequality if n vectors a 1, a 2,..., a n of length m are linearly independent, then n m (proof is in textbook) if an m n matrix has linearly independent columns then m n if an m n matrix has linearly independent rows then m n Matrix inverses 413
15 Outline left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
16 Nonsingular matrix for a square matrix A the following four properties are equivalent 1. A is left invertible 2. the columns of A are linearly independent 3. A is right invertible 4. the rows of A are linearly independent a square matrix with these properties is called nonsingular Nonsingular = invertible if properties 1 and 3 hold, then A is invertible (page 45) if A is invertible, properties 1 and 3 hold (by definition of invertibility) Matrix inverses 414
17 Proof left invertible (b ) (a) linearly independent columns (b) linearly independent rows (a ) right invertible we show that (a) holds in general we show that (b) holds for square matrices (a ) and (b ) follow from (a) and (b) applied to A T Matrix inverses 415
18 Part a: suppose A is left invertible if B is a left inverse of A (satisfies BA = I), then Ax = 0 = BAx = 0 = x = 0 this means that the columns of A are linearly independent: if A = [ a 1 a 2 a n ] then x 1 a 1 + x 2 a x n a n = 0 holds only for the trivial linear combination x 1 = x 2 = = x n = 0 Matrix inverses 416
19 Part b: suppose A is square with linearly independent columns a 1,..., a n for every nvector b the vectors a 1,..., a n, b are linearly dependent (from inequality on page 413) hence for every b there exists a nontrivial linear combination x 1 a 1 + x 2 a x n a n + x n+1 b = 0 we must have x n+1 0 because a 1,..., a n are linearly independent hence every b can be written as a linear combination of a 1,..., a n in particular, there exist nvectors c 1,..., c n such that Ac 1 = e 1, Ac 2 = e 2,..., Ac n = e n, the matrix C = [ ] c 1 c 2 c n is a right inverse of A: A [ ] [ ] c 1 c 2 c n = e1 e 2 e n = I Matrix inverses 417
20 Examples A = , B = A is nonsingular because its columns are linearly independent: x 1 x 2 + x 3 = 0, x 1 + x 2 + x 3 = 0, x 1 + x 2 x 3 = 0 is only possible if x 1 = x 2 = x 3 = 0 B is singular because its columns are linearly dependent: Bx = 0 for x = (1, 1, 1, 1) Matrix inverses 418
21 Example: Vandermonde matrix A = 1 t 1 t 2 1 t n t 2 t 2 2 t n t n t 2 n t n 1 n with t i t j for i j we show that A is nonsingular by showing that Ax = 0 only if x = 0 Ax = 0 means p(t 1 ) = p(t 2 ) = = p(t n ) = 0 where p(t) = x 1 + x 2 t + x 3 t x n t n 1 p(t) is a polynomial of degree n 1 or less if x 0, then p(t) can not have more than n 1 distinct real roots therefore p(t 1 ) = = p(t n ) = 0 is only possible if x = 0 Matrix inverses 419
22 Inverse of transpose and product Transpose and conjugate transpose if A is nonsingular, then A T and A H are nonsingular and (A T ) 1 = (A 1 ) T, (A H ) 1 = (A 1 ) H, we write these as A T and A H Product if A and B are nonsingular and of equal size, then AB is nonsingular with (AB) 1 = B 1 A 1 Matrix inverses 420
23 Outline left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
24 Gram matrix A = [ a 1 a 2 a n ] the Gram matrix associated with A is the matrix of column inner products for real matrices: A T A = a T 1 a 1 a T 1 a 2 a T 1 a n a T 2 a 1 a T 2 a 2 a T 2 a n... a T na 1 a T na 2 a T na n for complex matrices A H A = a H 1 a 1 a H 1 a 2 a H 1 a n a H 2 a 1 a H 2 a 2 a H 2 a n... a H n a 1 a H n a 2 a H n a n Matrix inverses 421
25 Nonsingular Gram matrix Gram matrix is nonsingular if only if A has linearly independent columns suppose A R m n has linearly independent columns: A T Ax = 0 = x T A T Ax = (Ax) T (Ax) = Ax 2 = 0 = Ax = 0 = x = 0 therefore A T A is nonsingular suppose the columns of A R m n are linearly dependent x 0, Ax = 0 = x 0, A T Ax = 0 therefore A T A is singular (for A C m n, replace A T with A H and x T with x H ) Matrix inverses 422
26 Pseudoinverse suppose A R m n has linearly independent columns this implies that A is tall or square (m n); see page 413 the pseudoinverse of A is defined as A = (A T A) 1 A T this matrix exists, because the Gram matrix A T A is nonsingular A is a left inverse of A: A A = (A T A) 1 (A T A) = I (for complex A with linearly independent columns, A = (A H A) 1 A H ) Matrix inverses 423
27 Summary the following three properties are equivalent 1. A is left invertible 2. the columns of A are linearly independent 3. A T A is nonsingular 1 2 was already proved on page : we have seen that the pseudoinverse is a left inverse 2 3: proved on page 422 a matrix with these properties must be tall or square Matrix inverses 424
28 Outline left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
29 Pseudoinverse suppose A R m n has linearly independent rows this implies that A is wide or square (m n); see page 413 the pseudoinverse of A is defined as A = A T (AA T ) 1 A T has linearly independent columns hence its Gram matrix AA T is nonsingular, so A exists A is a right inverse of A: AA = (AA T )(AA T ) 1 = I (for complex A with linearly independent rows, A = A H (AA H ) 1 ) Matrix inverses 425
30 Summary the following three properties are equivalent 1. A is right invertible 2. the rows of A are linearly independent 3. AA T is nonsingular 1 2 and 2 3: by transposing result on page : we have seen that the pseudoinverse is a right inverse a matrix with these properties must be wide or square Matrix inverses 426
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