Motivation
Given a square matrix A, we want to find a polynomial whose roots are precisely the eigenvalues of A. For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a1, a2, a3, etc. then the characteristic polynomial will be:
This works because the diagonal entries are also the eigenvalues of this matrix.
For a general matrix A, one can proceed as follows. A scalar is an eigenvalue of A if and only if there is an eigenvector such that
or
(where I is the identity matrix). Since v is non-zero, this means that the matrix I − A is singular (non-invertible), which in turn means that its determinant is 0. Thus the roots of the function det( I − A) are the eigenvalues of A, and it is clear that this determinant is a polynomial in .
Read more about this topic: Characteristic Polynomial
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