Characteristic Polynomials Characteristic Polynomials In linear algebra, every square matrix is associated with a characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace. The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs with the same characteristic polynomial have five nodes. 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.
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(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 . Formal Definition :- We start with a field K (such as the real or complex numbers) and an n×n matrix A over K. The characteristic polynomial of A, denoted by pA(t), is the polynomial defined by pA(t) = det(t I − A) where I denotes the n-by-n identity matrix and the determinant is being taken in K[t], the ring of polynomials in t over K. (Some authors define the characteristic polynomial to be det(A − t I). That polynomial differs from the one defined here by a sign (−1)n, so it makes no difference for properties like having as roots the eigenvalues of A; however the current definition always gives a monic polynomial, whereas the alternative definition always has constant term det(A).) Properties :- The polynomial pA(t) is monic (its leading coefficient is 1) and its degree is n. The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of A are precisely the roots of pA(t) (this also holds for the minimal polynomial of A, but its degree may be less than n). The coefficients of the characteristic polynomial are all polynomial expressions in the entries of the matrix. In particular its constant coefficient is equal to (−1)ndet(A), and the coefficient of tn − 1 is equal to (−1)n − 1 tr(A), the matrix trace of A. For a 2×2 matrix A, the characteristic polynomial is therefore given by t 2 − tr(A)t + det(A).
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The Cayley–Hamilton theorem states that replacing t by A in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term c as c times the identity matrix) yields the zero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the minimal polynomial of A divides the characteristic polynomial of A. Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar. The matrix A and its transpose have the same characteristic polynomial. A is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over K (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case A is similar to a matrix in Jordan normal form.
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