Cayley-Hamilton Theorem - Revision history

Kfagerstrom at 18:20, 10 March 2023

2023-03-10T18:20:44Z

Kfagerstrom at 18:12, 10 March 2023

2023-03-10T18:12:31Z

Kfagerstrom at 18:03, 10 March 2023

2023-03-10T18:03:55Z

Kfagerstrom: Created page with "Let $F$ be a field, and let $V$ be a finite dimensional $F$ -vector space. If $t : V → V$ is a linear transformation, then $m..."$

2023-03-10T18:03:44Z

Created page with "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..."

New page

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>.
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>.

<math> </math>

← Older revision		Revision as of 18:20, 10 March 2023
Line 9:		Line 9:
	By Lemma 4.21, <math>m = g_k</math> and <math>c = g_1 · · · g_k</math>, so <math>m \| c</math>. By definition, we <math>m(A) = 0</math>. Since <math>m\|c</math>, we conclude that <math>c(A) = 0</math>.		By Lemma 4.21, <math>m = g_k</math> and <math>c = g_1 · · · g_k</math>, so <math>m \| c</math>. By definition, we <math>m(A) = 0</math>. Since <math>m\|c</math>, we conclude that <math>c(A) = 0</math>.
	<math> </math>		<math> </math>
		+
		+	===Lemma 4.21===
		+
		+	Let <math>F </math> be a field, let <math> V</math> be a finite dimensional <math> F</math>-vector space, and <math> _V → V</math> be a linear transformation with invariant factors <math>g_1\| · · · \|g_k. </math> Then <math> c_t = g_1 · · · g_k</math> and mt = gk.
		+
		+	===Proof===
		+	The product of the elements on the diagonal of the Smith Normal Form of <math> xI_n − A</math>
		+	is the determinant of <math>xI_n − A</math>. Thus the product of the invariant factors <math>g_1 · · · g_k</math> of <math>V_t</math> is the characteristic polynomial <math>c_t</math> of <math>t</math>. Notice here that we chose our invariant factors <math>g_1, . . . , g_k</math> to be monic, so that <math>g_1 · · · g_k</math> is monic, and thus actually equal to <math>c_t</math> (not just up to multiplication by a unit). By Problem Set 5, <math>\text{ann}_{F[x]}(V_t) = (g_k)</math>, and since <math>g_k</math> is monic we deduce that <math>m_t = g_k</math>.

@@ Line 1: / Line 1: @@
 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>.
 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>

Cayley-Hamilton Theorem - Revision history

Kfagerstrom at 18:20, 10 March 2023

Kfagerstrom at 18:12, 10 March 2023

Kfagerstrom at 18:03, 10 March 2023

Kfagerstrom: Created page with "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..."

Kfagerstrom: Created page with "Let $F$ be a field, and let $V$ be a finite dimensional $F$ -vector space. If $t : V → V$ is a linear transformation, then $m..."$