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The question is, mathematically, what is eigenvalue and eigenvector. Again, its application. For instance, find the eigenvalue and Eigenvectorof the matrix A (2/1/2. 3/4/10. -2/-2/-5).

Remember that a matrix represents a linear transformation, so for example your 3x3 matrix A takes as input a vector X of length 3 and outputs the vector A*X.

Eigenvectors are just 1-dimensional subspaces that are preserved under this action, so you should think of them as invariant lines through the origin, if A*X=cX then the line spanned by X has every vector multiplied by c. The eigenvalue tells you the scalar that the matrix acts on this line by.

So for example imagine a rotation matrix in R^2. You can tell right away it won't have any eigenvectors, since no line through the origin is preserved by a rotation.

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