Cayley–Hamilton theorem
Every square matrix over a commutative ring satisfies its own characteristic equation / From Wikipedia, the free encyclopedia
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In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation.
The characteristic polynomial of an n × n matrix A is defined as[5] , where det is the determinant operation, λ is a variable scalar element of the base ring, and In is the n × n identity matrix. Since each entry of the matrix is either constant or linear in λ, the determinant of is a degree-n monic polynomial in λ, so it can be written as By replacing the scalar variable λ with the matrix A, one can define an analogous matrix polynomial expression, (Here, is the given matrix—not a variable, unlike —so is a constant rather than a function.) The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that ;} that is, the characteristic polynomial is an annihilating polynomial for
One use for the Cayley–Hamilton theorem is that it allows An to be expressed as a linear combination of the lower matrix powers of A: When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.
A special case of the theorem was first proved by Hamilton in 1853[6] in terms of inverses of linear functions of quaternions.[2][3][4] This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. Cayley in 1858 stated the result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.[7][8] As for n × n matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”. The general case was first proved by Ferdinand Frobenius in 1878.[9]