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Cayley-Hamilton reduces every power of $(\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array})$ to a linear combination of $A$ and $I$

Example

Let $A = (\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array})$ , and define $F_{0} = 0$ , $F_{1} = 1$ , and $F_{m + 1} = F_{m} + F_{m - 1}$ for $m \geq 1$ . Then, over any field and for every $m \geq 1$ ,

$A^{m} = F_{m} A + F_{m - 1} I_{2},$

with the integer coefficients interpreted in the field.

Facts & Assumptions

Given: The displayed matrix and recurrence.

[L1]

Cayley-Hamilton states that a matrix satisfies its characteristic polynomial (Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, $χ_{T} (T) = 0$ ).

Verification

technique · induction on $m$

1.1

Direct calculation gives $χ_{A} (x) = x^{2} - x - 1$ and $A^{2} = (\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}) = A + I_{2}$ , agreeing with [L1]. For $m = 1$ , $A = F_{1} A + F_{0} I_{2}$ ; for $m = 2$ , $A^{2} = F_{2} A + F_{1} I_{2}$ .

baseL1algebra

2.1

Assume $m \geq 2$ and $A^{m} = F_{m} A + F_{m - 1} I_{2}$ . Multiplication by $A$ and $A^{2} = A + I_{2}$ give $A^{m + 1} = F_{m} A^{2} + F_{m - 1} A = (F_{m} + F_{m - 1}) A + F_{m} I_{2} = F_{m + 1} A + F_{m} I_{2}$ .

ihstep 1.1algebra

3.1

The base cases and recurrence prove the formula for every $m \geq 1$ , so every positive power lies in $span {A, I_{2}}$ .

step 1.1step 2.1discharge-induction∎

Depends on

Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, $\chi_T(T)=0$

Used by

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Cayley-Hamilton reduces every power of (1110)(11​10​) to a linear combination of AA and II