# Definition of minimal polynomial of a linear transformation

Put content here**Definition:** The *minimal polynomial* of a linear transformation \(T: V 	o V\) is the unique monic polynomial \(m_T(x)\) of least degree such that \(m_T(T) = 0\) (the zero transformation).
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**Properties:**
- The minimal polynomial divides every polynomial \(p(x)\) for which \(p(T) = 0\)
- The minimal polynomial divides the characteristic polynomial
- The minimal polynomial and characteristic polynomial have the same irreducible factors (possibly with different multiplicities)
- The minimal polynomial is unique
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**Example:** For \(T\) represented by \(A = egin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix}\), the characteristic polynomial is \((\lambda-2)^2\) but the minimal polynomial is \((\lambda-2)\) since \(A - 2I = 0\).

# Parents

* Eigenvalues and eigenvectors