Suppose is a matrix (over a field ). Then the characteristic polynomial of is defined as $P_A(t) = \Det(tI - A)$ (Error compiling LaTeX. ! Undefined control sequence.), which is a th degree polynomial in .
An eigenvector is a non-zero vector that satisfies the relation , for some scalar . In other words, applying a linear operator to an eigenvector causes the eigenvector to dilate. The associated number is called the eigenvalue.
There are at most distinct eigenvalues, whose values are exactly the roots of the characteristic polynomial of the square matrix.
By the Hamilton-Cayley Theorem, the character polynomial of a square matrix applied to the square matrix itself is zero.
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