a 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12

Transcription

1 24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2 matrices = 3 8 = 5, 0 3 = 0 ( 3) = 3, 2 2 = ( 4) = 5 Determinant of 3 3 matrix A typical 3 3 matrix is given by a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 The determinant is defined as a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 = a a 22 a 23 a 32 a 33 a 2 a 2 a 23 a 3 a 33 +a 3 a 2 a 22 a 3 a 32 (8) Example = = = = = = 2 Definition 83 If we exchange rows and columns of the following matrices [ ] a a a a 2 a 3 2, a a 2 a 2 a 22 a a 3 a 32 a 33

2 8 MATRICES 25 to get [ a a 2 a 2 a 22 ], a a 2 a 3 a 2 a 22 a 32 a 3 a 23 a 33 then resulting matrices are called the transpose Determinant for General square matrices Elementary Products The product containing exactly one entry from each row and each column is called an elementary product The signed elementary product is ±a j a 2j2 a njn (82) where the sign is + if the permutation {j,,j n } is even and if it is odd Using these, we define determinant: Definition 84 The determinant of a of square matrix A, denoted by det(a) is the sum of all signed elementary product of A det(a) = ±a j a 2j2 a njn (83) Minors and Cofactors Definition 85 If A is a square matrix, then the minor M ij of the entry a ij is determinant of the submatrix obtained by deleting i-th row and j-th column from A The number C ij = ( ) i+j M ij is called the cofactor of the entry a ij Then the determinant can be expanded wrt any row; for example, wrt first row, we have det(a) = a C +a 2 C 2 + +a n C n (84) In general expanding wrt i-th row, we have det(a) = a i C i +a i2 C i2 + +a in C in (85) or expanding wrt j-th column, we have deta = a j C j +a 2j C 2j + +a nj C nj (86)

3 26 a a 2 a n a 22 a 23 a 2n a 2 a 22 a 2n a 32 a 33 a 3n = a + a n a n2 a nn a n2 a n3 a nn a 2 a 2(i ) a 2(i+) a 2n +( ) +i a 3 a 3(i ) a 3(i+) a 3n a i + a n a n(i ) a n(i+) a nn a 2 a 22 a 2(n ) +( ) +n a 3 a 32 a 3(n ) a n a n a n2 a n(n ) The i-th term on the right is ( ) +i a i times the determinant of (n ) (n ) obtained by deleting first row and i-column 82 Properties of determinant Theorem 82 () Determinant of transposed matrix is the same the determinant of original matrix, ie, det(a T ) = det(a) (2) If we exchange any two rows(columns), then determinant changes signs (3) If any two rows or two columns of A are the same, the determinant os zero (4) det(αa) = α n det(a) (5) Adding a scalar multiple of row (column) to another row (column) does not change determinant Theorem 822 () Determinant of a matrix one of whose row is zero is zero (2) If any two rows (columns) are equal, the determinant is zero Expansion with respect to any row In the RHS of (8) we used expansion wrt the first row By () and (2) of the Theorem 82, we can expand it wrt any row or column, as long as we

4 83 INVERSE OF A MATRIX - CRAMER S RULE 27 multiply the correct sign ( ) i+j If we expand wrt 2nd row the determinant becomes a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 = a 2 a 2 a 3 a 32 a 33 +a 22 a a 3 a 3 a 33 a 23 a a 2 a 3 a 32 We may also expand wrt 3rd column a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 = a 3 a 2 a 22 a 3 a 32 a 23 a a 2 a 3 a 32 +a 33 a a 2 a 2 a 22 Example 823 The followings are expanded wrt 2nd, 3rd row = = = = = = 48 Example 824 The following is expanded wrt 3rd col det(a) = = 0C 3 +( 3)C 23 +0C 33 = ( 3)( ) = 3[6 4 5( 2)] = Inverse of a matrix - Cramer s Rule Recall the determinant expansion formula: Expanding wrt i-th row, we know det(a) = a i C i +a i2 C i2 + +a in C in (87) Now we ask ourselves what will happen (by mistake) if we consider the product of i-th row with the cofactors of another row, say, j i? det(a) = a C 3 +a 2 C a in C 3n (88)

5 28 As an example, consider a a 2 a 3 A = a 2 a 22 a 23 a 3 a 32 a 33 Its determinant is a C +a 2 C 2 +a 3 C 3 and a a 2 a 3 A = a 2 a 22 a 23 a a 2 a 3 Its determinant is clearly zero but expand anyway, to see a C 3 +a 2 C 32 +a 3 C 33 = 0 Thus we arrive at the following conclusion Theorem 83 If the entries of any row (column) of a square matrix are multiplied by the cofactors of corresponding entries of another row(column), then the sum of the product is zero Definition 832 Let the minor M ij of the entry a ij be determinant of the submatrix obtained by deleting i-th row and j-th column from A Let C ij = ( ) i+j M ij be the cofactor of a ij of a square matrix A, then the matrix C C 2 C n C 2 C 22 C 2n C = C n C n2 C nn is called the matrix of cofactors Its transpose is called the adjoint of A, denoted by adj(a) So adj(a) = C T Theorem 833 (Cramer s Rule-inverse) A = adj(a) (89) det(a)

6 83 INVERSE OF A MATRIX - CRAMER S RULE 29 Proof Observe a a 2 a n a 2 a 22 a 2n C C 2 C j C n A adj(a) = C 2 C 22 C j2 a n2 a i a i2 a in C n C 2n C jn C nn a n a n2 a nn Consider the ij-th entry of A adj(a) which is obtained by multiplying i-th row of A with j-th column of adj(a) Thus (A adj(a)) ij = a i C j +a i2 C j2 + +a in C jn This is zero(above theorem) unless i = j, in which case the value becomes det(a) Hence we have with D = det(a) D D 0 A adj(a) = = det(a)i n D Example 834 Find the adjoint and the inverse of 3 2 A = Sol First det(a) = 64 Now the cofactors are C = 2 C 2 =6 C 3 = 6 C 2 = 4 C 22 =2 C 23 = 6 C 3 = 2 C 32 = 0 C 33 = 6 Thus A = 64 adj(a) =

7 30 Theorem 835 (Cramer s Rule) When det(a) 0, the solution of the system Ax = b, ie, a x +a 2 x 2 + +a n x n = b a 2 x +a 22 x 2 + +a 2n x n = b 2 (80) a n x +a n2 x 2 + +a nn x n = b n is given by x = det(a ) det(a), x 2 = det(a 2) det(a),, x n = det(a n) det(a), where A j,(j =,2,,n) is the determinant of the matrix obtained by replacing j-th column by b in A Example 836 Solve 3x + 2y + z = y + z = 0 x + y = 3 84 Eigenvalues and eigenvectors Let A be an n n matrixif there is a vector x 0 and a scalar λ such that Ax = λx (8) holds, then λ is called an eigenvalue, x is called eigenvector Geometrically Ax = λx means A does not change the direction of x Example 84 Find eigenvalues and eigenvectors [ ] 2 A = 2 Sol Since the system has a nonzero solution we must have (A λi)x = 0 det(a λi) = 0 Solving 2 λ 2 λ = (λ 2)2 = 0, λ =,3

8 84 EIGENVALUES AND EIGENVECTORS 3 If λ = we have If λ = 3 The eigenvector is x +x 2 = 0 x +x 2 = 0 ( ) x = k, k is any real x +x 2 = 0 x x 2 = 0 ( ) x = k 2, k 2 any real Example 842 Find eigenvalues and eigenvectors 2 A = Solve λ 2 6 λ 0 2 λ = 0 λ(λ+4)(λ 3) = 0 we obtain λ = 0,λ 2 = 4,λ 3 = 3 For λ = 0, we have 3 R R 2 +R 6R +R 2 R +R (82) (83) Thus k = 3 k 3 and k 2 = 6 3 k 3 Choose k 3 = 3, we get x = k 6, k is any real 3

9 32 If λ 2 = 4 6R +R 2 5R +R 3 2R 2 +R R 2 +R R R R 3 9 R 2 8 R (84) (85) (86) thus k = k 3 and k 2 = 2k 3 Choose k 3 =, we get x 2 = k 2, k is any real Finally, we set λ 3 = 3 2 x 3 = k 3, k is any real 2 Eigenvalues of Powers of a Matrix If p(x) = x n +c x n + +c n x+c n is a polynomial, and if λ is an eigenvalue of A then p(λ) is an eigenvalue of p(a) = A n +c A n + +c n A+c n I Eigenvalues and Eigenvectors of Symmetric Matrices Theorem 843 Let A be a n n symmetric matrix with real entries, then () All the eigenvalues of A are real (2) The eigenspaces of A are all of R n (3) The eigenspaces corresponding to distinct eigenvalues are orthogonal Exercise 844 () Find the image of the unit circle under the following map

10 85 SIMILARITY AND DIAGONALIZATION 33 (a) (b) ( ) 2 2 ( ) (c) (d) ( ) cosθ sinθ sinθ cosθ ( ) 2 2 (2) Diagonalzie (a),(b),(d) 85 Similarity and Diagonalization Similar Matrices In the study of change of bases, we have seen a matrix equation of the form C = P AP Here C and A represent a linear operator wrt different bases Definition 85 If A and C are square matrices with the same size, then we say they are similar if there exists an invertible matrix P such that C = P AP Eigenvalues and Eigenvectors of Similar Matrices Example 852 Find the algebraic multiplicities and geometric multiplicities of the eigenvalues of A = p(λ) = (λ 2)(λ 3) 2 Both of the eigenvalues have geometric multiplicity 2 Example 853 Find the algebraic multiplicities and geometric multiplicities of eigenvalues of A = p(λ) = (λ )(λ 2) 2 λ = have geometric multiplicity λ = 2have geometric multiplicity 2 Theorem 854 Similar matrices have the same eigenvalues and those eigenvalues have the same algebraic multiplicities and geometric multiplicities for both matrices

11 34 Diagonalization Definition 855 A square matrix A is given If there exists an invertible matrix P such that P AP is diagonal, then A is said to be diagonalizable Theorem 856 An n n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors Proof Let {p,p 2,,p n } be the n linearly independent eigenvectors corresponding to the eigenvalues λ,,λ n Let P = [ p p 2 p n ] Then we see PD is p p 2 p n λ 0 0 λ p λ 2 p 2 λ n p n p 2 p 22 p 2n 0 λ 2 0 = λ p 2 λ 2 p 22 λ n p 2n p n p n2 p nn 0 0 λ n λ p n λ 2 p n2 λ n p nn (87) AP = [ ] [ ] Ap Ap 2 Ap n = λ p λ 2 p 2 λ n p n = PD, (88) where D = diag(λ,,λ n ) is the diagonal matrix whose diagonal entries are λ,,λ n Thus A is diagonalizable Conversely, assume there exists an invertible matrix P such that λ 0 0 P 0 λ 2 0 AP = D = λ n is a diagonal matrix Writing this as AP = PD and going reversing the early procedure, we can show that each column p i,i =,2,,n is the eigenvector corresponding to the eigenvalue λ i Method of Diagonalization Example 857 We have seen the eigenvalues of A = 2 0 3

12 85 SIMILARITY AND DIAGONALIZATION 35 are λ = and λ = 2 and the eigenvectors are 2 0 p =, and 0, 0 Thus One can check that P AP = D 2 0 P = 0 0 Linearly Independence of Eigenvectors Theorem 858 If v,v 2,,v k are eigenvectors of an n n matrix A corresponding to the distinct eigenvalues λ,,λ k, then the set is {v,v 2,,v k } linearly independent Proof Assume the vectors v,v 2,,v k are linearly dependent Then one of these vectors are linear combinations of the other ones We may assume the first such vector is v r+ Thus, v r+ = c v +c 2 v 2 + +c r v r, (89) where the set {v,v 2,,v r } is linearly independent Multiplying this equation by A we get λ r+ v r+ = c λ v +c 2 λ 2 v 2 + +c r λ r v r (820) Multiplying (85) by λ r+ and subtracting from (86), we obtain 0 = c (λ λ r+ )v +c 2 (λ 2 λ r+ )v 2 + +c r (λ r λ r+ )v r (82) Since the set {v,v 2,,v r } is linearly independent and all λ i are distinct, we must have c = c 2 = = c r = 0 This means v r+ = 0, a contradiction Thus the vectors v,v 2,,v k must be linearly independent Theorem 859 An n n matrix A with n distinct real eigenvalues is diagonalizable Theorem 850 An n n matrix A is diagonalizable if and only if the sum of geometric multiplicities of its with n eigenvalues is n

13 36 Relationship Between Algebraic Multiplicity Geometric Multiplicity Theorem 85 A is an n n matrix, then () The geometric multiplicity of an eigenvalue of A is less than or equal to its algebraic multiplicity (2) A is diagonalizable if and only the geometric multiplicity of an eigenvalue of A is the same as its algebraic multiplicity A Unifying Theorem Theorem 852 A is an n n matrix, then the followings are equivalent () A is diagonalizable (2) The geometric multiplicity of an eigenvalue of A is less than or equal to its algebraic multiplicity (3) A has n linearly independent eigenvectors (4) R n has a basis consisting of eigenvectors of A (5) The sum of geometric multiplicity of an eigenvalue of A is n (6) The geometric multiplicity of each eigenvalue of A is the same as its algebraic multiplicity 86 Orthogonal Diagonalizability; Functions of a Matrix Orthogonal Similarity Definition 86 If A and C are square matrices with the same size, then we say they are orthogonally similar if there exists an invertible matrix P such that C = P T AP Theorem 862 Two square matrices are orthogonally similar if and only if there exist orthonormal bases wrt which the matrices represent the same linear operator Theorem 863 An n n matrix A is orthogonally diagonalizable if and only if A has an orthonormal set of n eigenvectors

14 86 ORTHOGONAL DIAGONALIZABILITY; FUNCTIONS OF A MATRIX37 Theorem 864 () A matrix A is orthogonally diagonalizable if and only if it is symmetric (2) If A is symmetric, then the eigenvector belonging to different eigenvalues are orthogonal Proof (2) Let Av i = λ i v i be the eigenvector eigenvalues relations Then λ i v T j v i = v T j Av i = (v T j Av i) T = v T i A T v j = v T i Av j (by symmetry) = λ j v T i v j Subtracting, we get 0 = (λ i λ j )v T i v j Since λ i λ j 0, we have v T i v j = 0 Example 865 We have seen the eigenvalues of A = are λ = 2 and λ = 8 and the eigenvectors are v =, 0, and 0 Gram Schmidt orthonormalizing u =, and u 2 =,u 3 = One can check that P T AP = D 2 6 P =

15 38 Spectral Decomposition If A is symmetric and eigenpairs are λ i,u i, then we form the orthogonal matrix P = [u u 2 u n ] Then λ 0 0 u T A = PDP T = [ ] 0 λ 2 0 u T 2 u u 2 u n λ n u T u T 2 = [ ] λ u λ 2 u 2 λ n u n u T n u T n Multiplying out, we see by the orthogonality of u i A = λ u u T +λ 2 u 2 u T 2 + +λ n u n u T n (822) This is called the spectral decomposition of A Example 866 (Geometric Interpretation) A = [ ] are λ = 3 and λ = 2 and the eigenvectors are v = Gram Schmidt orthonormalizing [ ], and 2 [ ] 2 u = [ ], and u 5 2 = [ ] 2 5 A = λ u u T +λ 2 u 2 u T 2 = ( 3)[ 5 2 ] 4 5 +(2)[ We seethat theimage of any vector Ax can bedecomposedalong theeigenspaces ]

16 86 ORTHOGONAL DIAGONALIZABILITY; FUNCTIONS OF A MATRIX39 Powers of Diagonalizable Matrices (P AP) 2 = P APP AP = P A 2 P (P AP) k = P A k P (823) Cayley-Hamilton Theorem Theorem 867 Every square matrix satisfies its characteristic equation, ie, if A is n n matrix with its characteristic polynomial λ n +c λ n + +c n = 0 then A n +c A n + +c n I = 0 (824) Example 868 The characteristic polynomial of is So So p(λ) = (λ 2)(λ 3) 3 = λ 3 8λ 2 +2λ 8 A 3 8A 2 +2A 8I = 0 (825) A 4 = A(A 3 ) = = 43A 2 50A+44I Also the inverse may be found similarly A(A 2 8A+2I) = 8I from which A = 8 (A2 8A+2I) =

17 40 Exponential of a Matrix If then the matrix p(a) is defined as p(x) = a 0 +a x+ +a m x m, p(a) = a 0 I +a A+ +a m A m More generally a function of matrix can be defined by power series(as long as it converges); if f has a convergent power series f(x) = f(0)+f (0)x+ f (0) 2! then we may define f(a) = f(0)i +f (0)A+ f (0) 2! x fm (0) x m + + (826) m! A fm (0) A m + + (827) m! Theorem 869 If A is an n n diagonalizable matrix diagonalized by P and λ,,λ n are the corresponding eigenvalues and if f has a convergent power series on some interval, then f(λ ) f(λ 2 ) 0 f(a) = P P (828) 0 0 f(λ n ) Example 860 Find the exponential e ta of A = Sol 0 0 P AP = = So applying the above theorem to f(a) e t 0 0 e ta = P 0 e 2t 0 P = 0 0 e 2t

18 86 ORTHOGONAL DIAGONALIZABILITY; FUNCTIONS OF A MATRIX49 In general when A is symmetric and has a spectral decomposition A = λ u u T +λ 2 u 2 u T 2 + +λ n u n u T n, The matrix diagonalize A as It can be written as (exer) P = [ u u 2 u n ] f(a) = Pf(D)P T f(a) = f(λ )u u T +f(λ 2 )u 2 u T 2 + +f(λ n )u n u T n, (829) whose eigenvalue can be obtained easily from those of A, as f(λ i ),i =,2,,n