Content deleted Content added
→Power series: The norm of B was missing a double bar on the left side. Tags: Mobile edit Mobile app edit Android app edit |
Citation bot (talk | contribs) Altered author-link. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Matrix theory | #UCB_Category 110/117 |
||
| (2 intermediate revisions by 2 users not shown) | |||
Line 14:
===Diagonalizable matrices===
A square matrix {{mvar|A}} is [[diagonalizable matrix|diagonalizable]], if there is an [[invertible matrix]] {{mvar|P}} such that <math>D = P^{-1}\,A\,P</math> is a [[diagonal matrix]], that is, {{
<math display="block">D=\begin{bmatrix}
d_1 & \cdots & 0 \\
Line 154:
where <math>\lambda_\pm</math> are the eigenvalues of its characteristic equation, {{math|1={{!}}''A'' − ''λI''{{!}} = 0}}, and are given by
<math display="block">\lambda_\pm = \frac{tr(A)}{2} \pm \sqrt{\left (\frac{tr(A)}{2}\right )^2 - |A|} .</math>
However, if there is degeneracy, the following formula is used, where f' is the derivative of f.
<math display="block">f(A) = f \left( \frac{tr(A)}{2} \right) I + \mathrm{adj} \left( \frac{tr(A)}{2}I - A \right ) f' \left( \frac{tr(A)}{2} \right) .</math>
=== Examples ===
Line 192 ⟶ 194:
==References==
* {{cite book|last1=Higham|first1=Nicholas J.|title=Functions of matrices theory and computation|date=2008|publisher=Society for Industrial and Applied Mathematics|___location=Philadelphia|author-link=
{{Authority control}}
| |||