Question

linear algebra

Find the characteristic equation of A, the eigenvalues of A, and a basis for the eigenspace corresponding to each eigenvalue. A= - -8 1 3 0 1 1 0 0 4 (a) the characteristic equation of A (b) the eigenvalues of A (Enter your answers from smallest to largest.) (11, 12, 13) = ( ]) (c) a basis for the eigenspace corresponding to each eigenvalue basis for the eigenspace of 11 basis for the eigenspace of 12 basis for the eigenspace of 13

Answer 1

-8 1 3 Given matrix, A=0 1 1 0 0 4. 1 1 a). The eigen values of A are given by the roots of the characterristic equation of A. |-8-2 3 that is |A-A11= 0 = 0 1-2 0 0 4-21 =(-8-)(1-2)(4- 2) - 0 +0 = 0 =(8+2)(1-2)(4-2) = 0. = 7+322 - 362+32 = 0), is the characterristic equation of A B. Consider (8+2)(1-2)(4 - 2) = 0 = A = -8,1,4 are the eigen values of A. By definition, For an eigenvalue 1, its algebraic multiplicityis the mulitplicity of las a root of the characteristic polynomial. Its geometric multiplicity is the maxim al number of linearly independent eigenvectors corresponding to it. The algebraic multiplicity of each 1 = -8,1,4 is 1. c). Case(i). Eigen vectors of A corresponding to 1 =-8 are given by the non-zero solutions of -8 1 3 (A-21)X = 0, where 1=-8. A=0 1 1 0 0 4 TO 1 3 09 1 y = 0;R, R-9R 0 0 12 || 2 0 1 3 x To 12 - 0 0 -26 || Y = 0 R+R+R 26 0 0 12 || 2 0 0 3 0 -100 -26 | 0 0 0 0 Z =y+32 = 0) (1) =-262 = 0 +(2) y = 0, z = 0. A+87 is in echelon form Rank(A+87)=r= No. of non-zer o rows in the echelon form of (A+87) =2 <n= 3(no of variables). Assign arbitrary variables to suitable (n-r)=3-1= 2vari ables so that the system is consistent. Let x=mER, x] [m Now X=y=0 (where n70) are the eigen vectors of A corresponding to A=-8. 1 The geometricic multiplicity of 1 =-8 is 1. i.e. algebraic multiplicity = geometricic multiplicity, in the case of 1=-8. Basis for the eigen space of 2 = -8 is | 0

Case(11). Eigen vectors of A corresponding to 1=1 are given by the non-zero solutions of [-8 1 3 (A-21)X = 0, where 1=1, A=0 1 1 0 4 0 x 0 -9 1 3 = 0 0 1 || y 0;R;R-3R 0 0 0 3 Z x -9 1 3 0 0 1 0 0 0 31-0 3-9x+y +32 = 0) (1) =z=0) +(2) A-/is in echelon form. Rank(A-1)= r = No. of non-zer o rows in the echelon form of (A-1) = 2 <n=3(no of variables). Assign arbitrary variables to suitable (n-r)=3-1= 2variables so that the system is consistent. Let x=mER (1)-9x+y+32 = 0 y=9m Now X = y 10 9m = m 9, (where n = 0) are the eigen vectors of A corresponding to A=1. The geometricic multiplicity of 1 =1 is 1. i.e. algebraic multiplicity = geometricic multiplicity, in the case of 1 =1. 1 Basis for the eigen space of 1, =lis 9

Case(iii). Eigen vectors of A corresponding to 1 = 4 are given by the non-zero solutions of -8 1 3 (A-11)X = 0, where 1 = 4. A=0 1 0 0 4 1 1 -12 0 3|| x -3 1 1-0 0 0 =-12x+y +32 = 0) (1) =-3y +z = 0 +(2) A-47 is in echelon form Rank(A-41)=r = No. of non-zer o rows in the echelon form of (A-41) = 2 <n=3(no of variables). Assign arbitrary variables to suitable (n-r)=3-1= 2variables so that the system is consistent. Let y=mER, (2)=-3y+z= 0 z = 3m (1) -12x+y+32 = 0 5 3X=- *= Now X = (where n70) are the eigen vectors of A corresponding to 1 = 4. 3m 18 The geometricic multiplicity of 1 = 4 is 1. i.e. algebraic multiplicity = geometricic multiplicity, in the case of g = 4. 5 Basis for the eigen space of 13 = 4 is 18 Ans> a. 2+322 - 362+32 = 0, is the characterristic equation of A b. -8,1,4 are the eigen values of A. Basis for the eigen space of 4 =-8 is 0 0 1 Basis for the eigen space of 1 =1 is 9 0 5 Basis for the eigen space of 1, = 4 is 6 18