Assume you know eigenvectors and eigenvalues for a matrix, find the matrix.
a1,a2,a3 are eigenvalues    b1,b2,b3 are eigenvectors

assume matrix ttt is |a11  a12  a13|
                     |a21  a22  a23|
                     |a31  a32  a33|

a1*b1--ttt*a1=jp1
a2*b2--ttt*a2=jp2
a3*b3--ttt*a3=jp3
Solve jp1=0  jp2=0   jp3=0  for amn
Here is TI-89 solution

a=| -0.5  -3.0   8.0 |
  | -1.0   3.0  -5.0 |
  | -0.1  -0.2  -0.1 |

matrix to be determined

========
TI-89 program

ff()
Prgm
ClrIO
SetMode("Vector Format", "Rectangular")
SetMode("Exact/Approximate","Approximate")

[[-0.5,-3.0,8.0][-1.0,3.0,-5.0],[-0.1 -0.2 -0.1]]->a

                Matrix to be solved for

eigVc(a)->ea
subMat(ea,1,1,3,1)->a1
subMat(ea,1,2,3,2)->a2
subMat(ea,1,3,3,3)->a3

eigVl(a)->h
product(mid(h,1,1))->b1
product(mid(h,2,1))->b2
product(mid(h,3,1))->b3

DelVar a11_,a12_,a13_
DelVar a21_,a22_,a23_
DelVar a31_,a32_,a33_,x

[[a11_,a12_,a13_][a21_,a22_,a23_][a31_,a32_,a33_]]->ttt

ttt*a1-a1*b1->jp1_
ttt*a1-a1*b1->jp1_
ttt*a1-a1*b1->jp1_

                First

subMat(jp1_,1,1,1,1)->j1_
subMat(jp2_,1,1,1,1)->j2_
subMat(jp3_,1,1,1,1)->j3_

det(jp1_)->v1_
det(jp2_)->v2_
det(jp3_)->v3_

csolve(v1_=0 and v2_=0 and v3_=0,{a11_,a12_,a13_})->nn

part(nn,1)->zz0
part(nn,2)->zz1
part(zz1,1)->zz2
part(zz1,2)->zz3
part(zz0,2)->aa11
part(zz2,2)->aa12
part(zz3,2)->aa13
               
   Solve for second set

subMat(jp1_,2,1,2,1)->j1_
subMat(jp2_,2,1,2,1)->j2_
subMat(jp3_,2,1,2,1)->j3_

det(j1_)->v1_
det(j2_)->v2_
det(j3_)->v3_

csolve(v1_=0 and v2_=0 and v3_=0,{a21_,a22_,a23_})->nn

part(nn,1)->zz0
part(nn,2)->zz1
part(zz1,1)->zz2
part(zz1,2)->zz3
part(zz0,2)->aa21
part(zz2,2)->aa22
part(zz3,2)->aa23

     Solve for third set
subMat(jp1_,3,1,3,1)->j1_
subMat(jp2_,3,1,3,1)->j2_
subMat(jp3_,3,1,3,1)->j3_

det(j1_)->v1_
det(j2_)->v2_
det(j3_)->v3_

csolve(v1_=0 and v2_=0 and v3_=0,{a31_,a32_,a33_})->nn

part(nn,1)->zz0
part(nn,2)->zz1
part(zz1,1)->zz2
part(zz1,2)->zz3
part(zz0,2)->aa31
part(zz2,2)->aa32
part(zz3,2)->aa33

         Answer

[[aa11,aa12,aa13][aa21,aa22,aa23][aa31,aa32,aa33]]->b
Disp b

EndPrgm


Results b on main screen

| -.5     -3.+3.E-14*i    8.+2.4E-13*i |
|  -1.    -3.            -5.-2.5E-13*i |
| -.1     -.2-4.E-15*i  -.1            |

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