Difference between revisions of "Cayley-Hamilton Theorem"
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Let <math>F</math> be a field, and let <math>V</math> be a finite dimensional <math>F</math>-vector space. If <math>t : V → V</math> is a linear transformation, then <math>m_t\mid c_t | Let <math>F</math> be a field, and let <math>V</math> be a finite dimensional <math>F</math>-vector space. If <math>t : V → V</math> is a linear transformation, then <math>m_t\mid c_t | ||
</math>, and hence <math>c_t(t) = 0</math>. | </math>, and hence <math>c_t(t) = 0</math>. | ||
| − | Similarly, for any matrix <math>A ∈ M_n(F)</math> over a field <math>F we have <math>m_A|c_A</math> and <math>c_A(A) = 0</math>. | + | Similarly, for any matrix <math>A ∈ M_n(F)</math> over a field <math>F</math> we have <math>m_A|c_A</math> and <math>c_A(A) = 0</math>. |
<math> </math> | <math> </math> | ||
Revision as of 18:03, 10 March 2023
Let \(F\) be a field, and let \(V\) be a finite dimensional \(F\)-vector space. If \(t : V → V\) is a linear transformation, then \(m_t\mid c_t \), and hence \(c_t(t) = 0\). Similarly, for any matrix \(A ∈ M_n(F)\) over a field \(F\) we have \(m_A|c_A\) and \(c_A(A) = 0\).
\( \)
