08 - Polynomials on Linear Maps

Minimal Polynomial of $T$ Let $V$ be a finite dimensional vector space Let $T:V\to V$ be a linear transformation

Minimal polynomial of $T$ is defined as

$$m_T(x)=m_A(x)$$

where $A={}_{\mathcal{B}}[T{]}_{\mathcal{B}}$ for some basis $\mathcal{B}$ of $V$

As we have showed previously that $m_A(x)=m_{P^{-1}AP}(x)$ then $m_T(x)$ is independent of choice basis

Characteristic Polynomial of $T$

Let $V$ be a finite dimensional vector space
Let $T:V\to V$ be a linear transformation

Characteristic polynomial of $T$ is defined as

$${\chi }_T(x)={\chi }_A(x)$$

where $A={}_{\mathcal{B}}[T{]}_{\mathcal{B}}$ for some basis $\mathcal{B}$ of $V$

As we have showed previously that ${\chi }_A(x)={\chi }_{P^{-1}AP}(x)$ then ${\chi }_T(x)$ is independent of choice basis

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