Determinants and Diagonalization of Square Matrices

Chapter 3 Determinants and Diagonalization

OUR GOAL o How to find the determinant of a square matrix? o How to diagonalize a square matrix?

Determinant of a square matrix Determinant of an nxn matrix A are denoted by det(A) or |A|. For 2 x 2 matrices: det? ? ? Or ? ? ? ? ? = ?? ?? = ?? ??

3 x 3matrices: + - + ? ? ? ? ? ? ? ? ? = det(A) = - b.det? ? ? + c.det? ? +a.det? ? ? ? ? = aei afh (bdi bgf) + cdh cge

Example Find the determinant of the matrix 3 2 2 2 0 1 1 ? = 1 3

The determinant of 3x3 matrix (only) 2 2 1 1 3 col co col a b c a b d e f d e g h i g h + + - - col col l ????det ? , -3 2 2 -2 0 1 3 2 2 2 0 1 1 ? ??? ? = 1 3 - + = + + cdh ceg afh bdi det A aei bfg

Definition a d g b e h c f i e h f i d g f i d g e h ( ) + ( ) ( ) + = = + + d e t . . . . . . A a b c a a a 11 12 13 (1,1) (1 ,2) (1,3) cofactor c ofactor cofactor If A is an mxm matrix then the determinant of A is defined by detA=ai1ci1(A)+ai2ci2(A)+ +aimcim(A) or detA= a1jc1j(A)+a2jc2j(A)+ +amjcmj(A) 1 0 0 0 2 6 7 1 1 5 4 1 0 8 6 4 0 0 = = 17 1 0 8 68 1 2 2

The determinant of triangular matrices Determinant = a11.a22 ann

Examples Find det(A), det(B), det(AB), det(A+B) 2 3 1 2 5 1 2 4 ? = ??? ? = det(A.B) = det(A).det(B) det(A+B) det(A) + det(B)

Examples Find det(A), det(3A), det(A2) if 2 1 3 5 ? = o det(cA) = cndet(A) o det(Ak) = [det(A)]k

Properties For all nxn matrices A, B: o det(A.B) = det(A).det(B) o det(kA) = kndet(A) o det(AT) = det(A) o det(A-1) = 1/det(A) o det(Ak) = [det(A)]k

The determinant of triangular matrices Determinant = a11.a22 ann

Examples o Find the determinants 1 0 0 2 3 0 3 2 |?| = = 1.3. 2 = 6, 2 0 3 2 0 2 3 = 6 // from A, interchange row 1 and row 2 1 0 2 4 3 0 2 0 0 6 2 2 = 12 and // from A, -2.(row 1)

Examples o Find the determinants 1 0 2 2 5 2 1 5 = 5 4 And 1 0 0 2 5 0 2 1 1 = 5 The second matrix is obtained from the first matrix by (2*row1 + row3), they have the same determinants.

Determinants and elementary operators 1. i ch 2 d ng (ho c 2 c t) cho nhau, matrix thu c v matrix ban u c nh th c tr i d u nhau. 2. Nh n m t d ng (ho c c t) v i m t s k matrix thu c c nh th c g p k l n det c a matrix c . //kri 3. N u nh n c v o d ng ri r i c ng v o d ng rj (ho c th c hi n tr n c t) nh th c kh ng thay i. // // ri rj cri + rj

Examples 0 2 3 2 2 2 4 1 9 4 2 2 2 2 4 6 1 1 2 3 1 0 0 0 1 2 2 1 3 9 + r 3 r r r 1 + = 3 6 1 0 0 3 2 2 4 1 9 2 2 0 3 2 2 4 1 9 2 2 = 3 r r 1 2 1 4 = 2 2 1 0 1 0 1 4 8 3 3 3 7 4 9 1 0 0 0 1 2 3 1 0 0 0 1 2 3 1 0 0 0 1 2 3 + + 2 7 r r r r 2 2 = 3 4 1 4 8 1 4 7 24 8 7 4 1 4 1 24 8 1 7 = r r ( ) 2 3 = 2 2 2.7 2 7 1 4 9 9 0 0 0 0 7 4 1 0 0 0 1 2 3 1 4 1 0 8 1 23 + 24 r r ( ) ( ) 3 4 = = 1 .1. 2.7 2.7.1. 23 0 0 1 0 2 6 - - 1 5 4 4 4 3 6 5 Do yourself: Find 1 3 1 2

Next det(A) and existence of A-1 o A is invertible det(A) 0 o A formula for finding A-1

(i,j)-cofactor (i,j)-cofactor of a matrix [aij] is defined by cij = (-1)i+jdet(Aij), where Aij is the matrix obtained from A by deleting row ith and column jth 2 3 1 2 1 0 1 For example, given A = row 2 2 1 column 3 Then, c23 = (-1)2+3det 2 = -1.(-1) = 1 3 2 1 Do yourself. Find all cofactors cij of the matrix A

How to find A-1? An nxn matrix A is invertible if and only if det(A) 0 Furthermore, 1 A-1= det ????(?), where the adjugate matrix adj A is defined as adj(A) = [ i,j cofactors]Transpose

Adjugate matrix The adjugate matrix of A is the matrix ... ... ... ... c c c c c c 11 21 1 n = 12 ... c 2 2 2 n adjA ... c ... c 1 2 n n nn For example, 1 1 2 0 3 0 1 1 2 1 ( ) ( ) 1 1 + 1 2 + = = = = = A 3 1 . We have c 1 3, c 1 3, 11 12 1 1 2 0 1 = = = = = = = = = c 3, c 3, c 6 3 2 1 2 1 5 11 12 13 = c c 2, c 2, c 1, c 1, c 4, adjA 3 6 21 22 23 5 4 31 32 33

Theorem of Adjugate Formula If A is any square matrix, then A(adjA)=(detA)I In particular, ifdetA 0 then A is invertible and 1 = 1 A adjA det A For example, 1 1 2 2 1 3 = = A 0 0 2 0 1 det A 2 and adjA= 0 1 0 1 2 1 0 3/2 2 1 3 1 1/2 10 2 A = = 1 1 1 0 1/2 1/2 0 0 2 0 0 1 Note that [ ]

Diagonal matrices An nxn matrix is called diagonal matrix if all its entries off the main diagonal are zeros 0 ... ... ... ... 0 0 ... 1 0 ... 0 ( ) = = 2 , ,..., D diag 1 2 n ... 0 n For example 3 0 0 0 1 0 0 0 0 0 0 0 2 ( ) 3, 2,1,4 = diag 0 0 4

Diagonalization Diagonalizing a matrix A is to find an invertible matrix P such that P-1AP is a diagonal matrix P-1AP=diag(1, 2, , n) For example, 2 4 1 1 1 o Given a matrix A, A = 1 4 1 o Find a matrix P, P = o Compute P-1AP, 3 0 0 = 1 P AP 2

How to find P? 4 x x + x 2 ( ) x ( ) ( )( ) = + = 2 Find c =det xI A : 2 x 1 4 x x 6 a th c c tr ng A 1 1 (characteristic polynomial) ( ) x = = = c 0 x 3 x 2 C c gi tr ri ng (eigenvalues) c a A + = + A = = x 4 y 0 ( ) x=3: solve the system 3 I A X 0 x 4 y 0 = = N u t 0 th X=(-4t,t) c g i y x t x 4 = = X t l v c t ri ng (eigenvectors) ng v i gi tr ri ng x=3 1 y 4 t + = 4 = x 4 y 0 ( ) = x=-2: solve the system 2 I A X 0 x y 0 = = N u t 0 th X=(t,t) Relationship between eigenvalues and eigenvectors : eigenvalue (a number) X: -eigenvector (remember: vector X 0) ( I-A)X=0 AX= X y t x 1 = = X t c g i l v c t ri ng ng v i x=-2 1 y x t 4 1 1 3 0 0 = 1 Choose P= P AP 1 2 2 0 4 1 4 2 1 4 1 4 1 4 = = = = = 1 P , P , P , P ,...are allowed. In case P= , P AP 1 1 1 2 1 1 1 1 1 1 0 3

Example 1 2 1 2 = Find the eigenvalues ang eigenvectors and then diagonalize the matrix The characteristic polynomial of A is 1 1 ( ) ( 2 2 0,3 are eigenvalues = x A x = = 2) 2 = = = = 1)( ( 3) 0 0 3 c x x x x x x x A x ( ) = 0: Solve the system 0I-A X=0 x 1 1 = P x 1 2 1 0 2 0 1 10 0 0 1 = X t 1 2 0 1 0 0 0 3 ( ) = 3: Solve the system 3I-A X=0 = 1 P AP 2 10 1 0 2 0 10 0 0 1 2 = X t 2

Use the fact: if x1, x2,, xm are eigenvalues of an nxn matrix , then det(A) = x1.x2 xm First, det(A) = 4 We know that det(A) = the product of eigenvalues 4 = 1.2.2 f // x1=1, x2 = x3 = 2.

Use the fact: if X is an eigenvector of a matrix A corresponding an eigenvalue k, then AX = kX

When is A diagonalizable? Theorem A is diagonalizable iff every eigenvalue of multiplicity m yields exactly m basic eigenvectors, that is, iff the general solution of the system ( I-A)X=0 has exactly m parameters. For example, 0 1 = A is not diagonalizable. 1 2 x 1 ( ) x ( ) ( ) ( ) 2 = = = + = + = = = 2 In fact, c det xI A x x 2 1 x 2 x 1 x 1 0 x 1 A 1 x 2 1 0 1 1 1 0 ( ) = = x 1 (multiplicity 2): solve the system 1 I A X 0 1 1 0 0 0 0 1 = X t one parameter not diag onalizable 1

When is A diagonalizable? 0 1 1 0 1 1 is not diagonalizable. = = A 2 0 0 x 1 1 ( ) x ( ) ( ) ( ) 2 = = = + = = = 3 In fact, c det xI A 1 2 x 1 x x 3 x 2 x 1 x 2 0 x 1 x 2 A 0 1 0 1 1 1 1 1 0 ( ) 1 (multiplicity 2): solve the system = = x 1 I A X 0 1 2 1 1 0 1 0 0 0 0 2 0 0 1 0 0 one parameter not diagonalizable

When is A diagonalizable? 0 1 1 0 1 1 is not diagonalizable. = = A 2 0 0 x 1 1 ( ) x ( ) ( ) ( ) 2 = = = + = = = 3 In fact, c det xI A 1 2 x 1 x x 3 x 2 x 1 x 2 0 x 1 x 2 A 0 1 0 1 1 1 1 1 0 ( ) 1 (multiplicity 2): solve the system = = x 1 I A X 0 1 2 1 1 0 1 0 0 0 0 2 0 0 1 0 0 one parameter not diagonalizable

SUMMARY o Determinants of nxn matrices o Properties: o det(AB) = det(A)det(B) o det(cA) = cndet(A) o det(AT) = det(A) o det(A-1) = 1/det(A) o Determinants and elementary operators o Determinants and inverse of a matrix o An nxn matrix has an inverse if and only if det(A) 0 o A-1 = adj(A)/detA o Diagonalization o Characteristic polynomial o Eigenvalues o Eigenvectors

THANKS