The minimal polynomial of a linear transformation exists and is unique.
Created almost 9 years ago, updated 5 months ago
ID24047
Theorem: For every linear transformation (T: V o V) on a finite-dimensional nontrivial vector space, the minimal polynomial exists and is unique.
Existence: The space (\mathcal{L}(V,V)) of linear operators on (V) has dimension (n^2) (where (n = \dim V)). Therefore the (n^2 + 1) operators (I, T, T^2, \ldots, T^{n^2}) are linearly dependent, so some nonzero polynomial (p) satisfies (p(T) = 0). Among all such polynomials, there is one of least degree.
Uniqueness: If two monic polynomials (m_1) and (m_2) both have minimal degree and annihilate (T), then (m_1 - m_2) also annihilates (T) but has lower degree, which is only possible if (m_1 - m_2 = 0), i.e., (m_1 = m_2).
Making the polynomial monic (leading coefficient = 1) ensures uniqueness.
**Theorem:** For every linear transformation \(T: V o V\) on a finite-dimensional nontrivial vector space, the minimal polynomial exists and is unique.
**Existence:** The space \(\mathcal{L}(V,V)\) of linear operators on \(V\) has dimension \(n^2\) (where \(n = \dim V\)). Therefore the \(n^2 + 1\) operators \(I, T, T^2, \ldots, T^{n^2}\) are linearly dependent, so some nonzero polynomial \(p\) satisfies \(p(T) = 0\). Among all such polynomials, there is one of least degree.
**Uniqueness:** If two monic polynomials \(m_1\) and \(m_2\) both have minimal degree and annihilate \(T\), then \(m_1 - m_2\) also annihilates \(T\) but has lower degree, which is only possible if \(m_1 - m_2 = 0\), i.e., \(m_1 = m_2\).
Making the polynomial monic (leading coefficient = 1) ensures uniqueness.