what is hamilton theorem
Answers
⭐In linear algebra, the Cayley–Hamilton theorem (termed after the mathematicians Arthur Cayley and William Rowan Hamilton) says that every square matrix over a commutative ring (for instance the real or complex field) satisfies its own typical equation. If A is a provided as n×n matrix and In is the n×n identity matrix, then the distinctive polynomial of A is articulated as:
p(x) = det(xIn – A)
Where the determinant operation is det and for the scalar element of the base ring, the variable is taken as x. As the entries of the matrix are (linear or constant) polynomials in x, the determinant is also an n-th order monic polynomial in x.⭐
The Cayley–Hamilton theorem says that substituting the matrix A for x in polynomial, p(x) = det(xIn – A), results in the zero matrices, such as:
p(A) = 0
It state that a n x n matrix A is demolished by its characteristic polynomial det(tI – A), which is monic of degree n. The powers of A, found by substitution from powers of x, are defined by recurrent matrix multiplication; the constant term of p(x) provides a multiple of the power A0, which power is described as the identity matrix. The theorem allows An to be articulated as a linear combination of the lower matrix powers of A. If the ring is a field, the Cayley–Hamilton theorem is equal to the declaration that the smallest polynomial of a square matrix divided by its characteristic polynomial
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