document.write( "Question 122231: if i have a parametric matrix and i want to find parametric Eigenvectors and Eigenvalues....what i have to do??? and these Eigenvectors and Eigenvalues are still parametric or i have to equal them????
\n" ); document.write( "thanx \n" ); document.write( "

Algebra.Com's Answer #89749 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
Although it's difficult to give a general answer, here goes.
\n" ); document.write( "You do it the same way you solve an eigenvalue/eigenvector problem for a regular (non-parametric) matrix.
\n" ); document.write( "Let's use an example.
\n" ); document.write( "Let A be your 2x2 parametric matrix,
\n" ); document.write( " $\"A=%28matrix%282%2C2%2C1%2Cs%2C3%2C0%29%29\"$
\n" ); document.write( "where s is your parameter.
\n" ); document.write( "First, set up your eigenvalue problem,
\n" ); document.write( " $\"A-%28sigma%29I=%28matrix%282%2C2%2C1-sigma%2Cs%2C3%2C-sigma%29%29\"$
\n" ); document.write( "I'm using $\"sigma\"$ instead of lambda, the traditional eigenvalue Greek symbol, here.
\n" ); document.write( "Find your characteristic polynomial,
\n" ); document.write( " $\"%281-sigma%29%28-sigma%29-3s=0\"$
\n" ); document.write( " $\"sigma%5E2-sigma-3s=0\"$
\n" ); document.write( " $\"sigma+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A%28-3s%29%29%29%2F%282%29+\"$
\n" ); document.write( " $\"sigma+=+%281+%2B-+sqrt%28+1%2B12s%29%29%2F%282%29+\"$
\n" ); document.write( "Now it gets tricky.
\n" ); document.write( "As you can see from the solution for $\"sigma\"$ , the parameter(s) position(s) in the parametric matrix determines what effect it has on the characteristic polynomial.
\n" ); document.write( "In this example, depending on whether s is positive or negative, could lead to real or complex roots.
\n" ); document.write( "Once you have the eigenvalues, you would go back to your matrix,
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\n" ); document.write( "The solution vectors ( $\"x%5B1%5D\"$ , $\"x%5B2%5D\"$ ) are the eigenvectors tied to each of the specific eigenvalues $\"sigma\"$ .
\n" ); document.write( "The parameter information is included through the characteristic polynomial for the eigenvalue and the matrix for the eigenvector.
\n" ); document.write( "Hope it helps! \n" ); document.write( "

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