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Home / Linear Algebra / Cayley- Hamilton Theorem: Every Matrix A is a root of its characterstic polynomial ∆(t).

Cayley- Hamilton Theorem: Every Matrix A is a root of its characterstic polynomial ∆(t).

Cayley- Hamilton Theorem: Every Matrix A is a root of its characterstic polynomial ∆(t).

Proof: Let A be an arbitrary n- square matrix and let ∆(t) be its characterstic polynomial, say,

∆(t) = |tI-A| = tn +an-1tn-1+ …+a1t+a0

Now let B(t) denote the classical adjoint of the matrix tI-A. The elements of B(t) are cofactors of the matrix tI-A, and hence are polynomials in t of degree not exceeding n-1. Thus

B(t) = Bt-1tn-1 + …+ Btt+B0

Where the Bi are n-square matrices over K which are independent of t. By the fundamental property of the clasical adjoint (tI-A) B(t) = |tI-A|I, or, we have

(tI-A)(Bn-1tn-1+ …+B1t + B0 = (tn+an-1tn-1+…+ a1t+a0)I

Removing the parenthesis and equating the corresponding powers of t yields

Bn-1 = I, Bn-2-ABn-1 = an-1I, …, B0-AB1=a1I, -AB0 = a0I

Multiplying the above the equations by An, An-1, …, A, I, respectively, yields

AnBn-1 = AnI, An-1Bn-2-AnBn-1 = an-1An-1,…, AB0 – A2B1 = aiA, -AB0 = a0I

Adding the above matrix equations yields 0 on the left hand side and ∆(A) on the right hand side, that is,

0 = An + an-1An-1+…+ a1A+a0I

Therefore, ∆(A) = 0, which is the Cayley- Hamilton theorem. (Proved)

 

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