Semisimple operator

<h2 id="notes">Notes</h2>

<ul><li>Hoffman, Kenneth; <a href="/facts/Ray_Kunze/qe72t0wF">Kunze, Ray</a> (1971). "Semi-Simple operators". <i>Linear algebra</i> (2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0276251">0276251</a>.</li>
<li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (1979). <i>Lie algebras</i>. New York. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-63832-4. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/6499793">6499793</a>.{{cite book}}:  CS1 maint: location missing publisher (link)</li>
<li>Lam, Tsit-Yuen (2001). <i>A first course in noncommutative rings</i>. Graduate texts in mathematics. Vol. 131 (2 ed.). Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95183-0.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lam (2001), p. 39 <a href="https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA39&dq=%22linear+operator%22" target="_blank">https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA39&dq=%22linear+operator%22</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Jacobson 1979, A paragraph before Ch. II, § 5, Theorem 11. - Jacobson, Nathan (1979). Lie algebras. New York. ISBN 0-486-63832-4. OCLC 6499793. <a href="https://search.worldcat.org/oclc/6499793" target="_blank">https://search.worldcat.org/oclc/6499793</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Lam (2001), p. 39 <a href="https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA39&dq=%22linear+operator%22" target="_blank">https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA39&dq=%22linear+operator%22</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>This is trivial by the definition in terms of a minimal polynomial but can be seen more directly as follows. Such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any basis for this space can be extended to an eigenbasis. <a href="/wiki/Hyperplane" target="_blank">/wiki/Hyperplane</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Semisimple operator open-in-new

Semisimple operator