Definition of minimal polynomial of a linear transformation
Created almost 9 years ago, updated 5 months ago
ID24045
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).
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
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).
**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).
**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
**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 \(A - 2I = 0\).