1. Find the eigen values and print the corresponding eigen vectors for
above matrix .
2. Are the eigen vector (c) normalized .
3. Show that for matrix ‘A’, sum of eigen value =trace of A and
product of eigen value=det of A .
4. Find and prove the nature of the above matrices or matrix (c) .
5. For matrix D show that eigen vectors are mutually orthogonal .
6. Show that for matrix D,inv(p)*D*p will give diagonal matrix with
diagonal element as eigen values .
Apparatus- Scilab software , Laptop .
Algorithm- 1. Input the all given matrix e.g.,A,B,C,D,E,F and
identity matrix .
2. Using the spec command for find the eigen values and
eigen vectors .
3. To normalized the eigen vectors of corresponding matrices
and display the values .
4. Using the sum and trace command for show sum of eigen
value is equal to trace of A.
5. Find the det of eigen value of A and det of A using the det
command .
6. Check the orthogonality of eigen vectors of matrix D.
7.Find the diagonal of the matrix D .
8. Tofind the nature of the matrices as determinant for
harmitian,orthogonal,symmetric and unitry etc.
INPUT-
A=[2 1 1;1 3 2;3 1 4]
B=[1 -%i 3+4*%i;%i 2 4;3-4*%i 4 3]
C=[2 -%i 2*%i;%i 4 3;-2*%i 3 5]
D=(1/3)*[1 2 2;2 1 -2;2 -2 1]
E=[-3 -7 -5;2 4 3;1 2 2]
F=1/sqrt(3)*[1 1+%i;1-%i -1]
I=[1 0 0;0 1 0;0 0 1]
[a,b]=spec(A)
disp(a,"eigen vector of A",b,"eigen values of A")
[c,d]=spec(B)
disp(c,"eigen vector of B",d,"eigen values of B")
[e,f]=spec(C)
disp(e,"eigen vector of C",f,"eigen values of C")
[g,h]=spec(D)
disp(g,"eigen vector of D",h,"eigen values of D")
[i,j]=spec(E)
disp(i,"eigen vector of E",j,"eigen values of E")
[k,l]=spec(F)
disp(k,"eigen vector of F",l,"eigen values of F")
for i=1:3
disp(norm(e(:,i)),"normalised of eigenvalue of C")
end
disp(trace(A),"trace of A")
disp(sum(b),"sum of eigen value of A")
disp(det(b),"det of eigen value of A")
disp(det(A),"det of A")
disp(g'*g,"eigen vector of D are mutually orthogonal")
disp(inv(g)*D*g,"digonal matrix of D")
Y1=B'
disp(B'
,"dagger of matrix B")
disp(B,"matrix of B")
disp("if B dagger is equal to B so it harmitian")
Y2=C'
disp(C'
,"dagger of matrix B")
disp(C,"matrix of B")
disp("if C dagger is equal to C so it harmitian")
Y3=D'
disp(D'
,"dagger of matrix B")
disp(D,"matrix of B")
disp("if D dagger is equal to D and D is real so it symmetric")
Y4=D'*D
disp("if transpose is equal to its inverse so D is also orthogonal
matrix")
disp(Y4)
Y5=F'
disp(F'
,"dagger of matrix F")
disp(F,"matrix of F")
disp("if F dagger is equal F so it harmitian")
Y6=F'*F
disp(Y6)
disp("dagger is equal to its inverse so F is also unitary matrix")
OUTPUT-
eigen values of A
6.095824 0. 0.
0. 1.452088 + 0.4336988i 0.
0. 0. 1.452088 - 0.4336988i
eigen vector of A
0.3243216 0.3899937 - 0.1875413i 0.3899937 + 0.1875413i
0.5849985 0.5379993 + 0.2718959i 0.5379993 - 0.2718959i
0.7433655 -0.6703451 -0.6703451
eigen values of B
-4.746829 0. 0.
0. 2.3968018 0.
0. 0. 8.3500273
eigen vector of B
-0.3404771 - 0.5212558i -0.2734105 + 0.5541666i -0.3192331 - 0.3586139i
-0.4539695 + 0.0504648i -0.3645474 - 0.6890355i -0.4256442 - 0.0502727i
0.6353997 0.1023784 -0.7653665
eigen values of C
-0.3871996 0. 0.
0. 3.6916109 0.
0. 0. 7.6955887
eigen vector of C
0.6662617i -0.7255184i 0.1723904i
0.5184148 0.6168052 0.5922816
-0.5360424 -0.3052448 0.7870731
eigen values of D
-1. 0. 0.
0. 1. 0.
0. 0. 1.
eigen vector of D
-0.5773503 -0.7634414 0.2895237
0.5773503 -0.1309858 0.8059214
0.5773503 -0.6324555 -0.5163978
eigen values of E
1.0000075 + 0.000013i 0. 0.
0. 1.0000075 - 0.000013i 0.
0. 0. 0.9999849
eigen vector of E
-0.9045338 -0.9045338 -0.9045344
0.3015128 + 0.0000026i 0.3015128 - 0.0000026i 0.3015085
0.3015105 - 0.0000013i 0.3015105 + 0.0000013i 0.301513
eigen values of F
-1. 0.
0. 1.
eigen vector of F
0.3250576 + 0.3250576i 0.627963 + 0.627963i
-0.8880738 0.4597008
normalised of eigenvalue of C
1.
normalised of eigenvalue of C
1.
normalised of eigenvalue of C
1.
trace of A
9.
sum of eigen value of A
9.
det of eigen value of A
14. + 4.441D-16i
det of A
14.
eigen vector of D are mutually orthogonal
1. 0. 1.244D-16
0. 1. 0.
1.244D-16 0. 1.
digonal matrix of D
-1. 0. -1.244D-16
0. 1. 0.
1.330D-16 0. 1.
dagger of matrix B
1. -i 3. + 4.i
i 2. 4.
3. - 4.i 4. 3.
matrix of B
1. -i 3. + 4.i
i 2. 4.
3. - 4.i 4. 3.
if B dagger is equal to B so it harmitian
dagger of matrix C
2. -i 2.i
i 4. 3.
-2.i 3. 5.
matrix of C
2. -i 2.i
i 4. 3.
-2.i 3. 5.
if C dagger is equal to C so it harmitian
dagger of matrix D
0.3333333 0.6666667 0.6666667
0.6666667 0.3333333 -0.6666667
0.6666667 -0.6666667 0.3333333
matrix of D
0.3333333 0.6666667 0.6666667
0.6666667 0.3333333 -0.6666667
0.6666667 -0.6666667 0.3333333
if D dagger is equal to D and D is real so it symmetric
if transpose is equal to its inverse so D is also orthogonal matrix
1. 0. 0.
0. 1. 0.
0. 0. 1.
dagger of matrix F
0.5773503 0.5773503 + 0.5773503i
0.5773503 - 0.5773503i -0.5773503
matrix of F
0.5773503 0.5773503 + 0.5773503i
0.5773503 - 0.5773503i -0.5773503
if F dagger is equal F so it harmitian
1. 0.
0. 1.
dagger is equal to its inverse so F is also unitary matrix
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