Eigenvalue Problem

[Steven Spear]

Hi, could someone show me how to use Mathematica to sucessfully answer the
following problem:

Show that the Eigenvalues of A =

[a][b]
[c][d]

must be real numbers. Thanks, Steve

[Ken Levasseur]

Steve:

I can show that they don't have to be real:

In[4]:=
Eigenvalues[{{1, 1}, {-1, 1}}]
Out[4]=
{1 - \[ImaginaryI], 1 + \[ImaginaryI]}

Ken Levasseur
Math. Sci.
UMass Lowell

[Jens-Peer Kuska]

Hi,

for what do you need Mathematica ?
For a real matrix the matrix must symmetric,
for a complex matrix it must be hermitian.

Regards
Jens

[BobH...@aol.com]

In a message dated 12/16/01 4:01:32 AM, smi...@mindspring.com writes:

>Hi, could someone show me how to use Mathematica to sucessfully answer
>the
>following problem:
>
>Show that the Eigenvalues of A =
>
>[a][b]
>[c][d]
>
>must be real numbers. Thanks, Steve
>

A = {{a,b},{c,d}};

{\[Lambda], \[Mu]} = FullSimplify[Eigenvalues[A]]

{(1/2)*(a + d - Sqrt[(a - d)^2 +
4*b*c]), (1/2)*(a + d +
Sqrt[(a - d)^2 + 4*b*c])}

{u,v} = FullSimplify[Eigenvectors[A]]

{{-((-a + d + Sqrt[(a - d)^2 +
4*b*c])/(2*c)), 1},
{(a - d + Sqrt[(a - d)^2 +
4*b*c])/(2*c), 1}}

Simplify[{A.u-\[Lambda]*u,A.v-\[Mu]*v}]

{{0, 0}, {0, 0}}

For the elements of the eigensystem to be real requires

(a - d)^2 + 4*b*c >= 0

Bob Hanlon
Chantilly, VA USA