General how to find the eigenvectors/values

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To find the eigenvalues and eigenvectors of a 3x3 matrix, start by determining the roots of the characteristic equation, which is a cubic polynomial derived from the matrix. This involves solving for values of c such that the determinant of (T - cI) equals zero, indicating that the matrix is singular. Once the eigenvalues are found, substitute them back into the equation (T - cI)v = 0 to solve for the corresponding eigenvectors. Understanding the geometric properties of the transformation can also provide insights into the potential eigenvalues. This process is fundamentally similar to that for 2x2 matrices but is inherently more complex due to the cubic nature of the characteristic equation.
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this is not an assignment question...but i am just wondering in general how to find the eigenvectors/values for a 3X3 matrix...can someone show me the step by steps with an example?? don't worry, soon i will give back to this web site.:biggrin:
 
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Find the roots of the characteristic equation, solve the linear algbebra problem.
 
Pretty much the way you find eigenvalues for 2 by 2 matrices- except that it is harder! The characteristic equation is a cubic polynomial equation. Solve that for the eigenvalues.
 
an eigenvalue for T is a number c such that Tv = cv for some non zero vector v.

that means (T-c)v = 0, i.e. that T-c is singular, hence has determinant zero.

so look for those c such that det(T-c) = 0. this is a cubic equation in the coefficients of T.

I.e. if you choose a basis for the space, T becomes a matrix and you can calculate this equation and hopefully find its roots.Or you may know that T is length preserving, say it camer from geometry as a reflection or a rotation, or something like that, and then that imposes restrictions on the possible eigenvalues.
 
ah, fantastico
 

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