Editing Matrix exponential (section)

=== Diagonalizable case ===

If a matrix is [[diagonal matrix|diagonal]]:
<math display="block">A = \begin{bmatrix}
a_1 & 0 & \cdots & 0 \\
0 & a_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & a_n
\end{bmatrix} ,</math>
then its exponential can be obtained by exponentiating each entry on the main diagonal:
<math display="block">e^A = \begin{bmatrix}
e^{a_1} & 0 & \cdots & 0 \\
0 & e^{a_2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & e^{a_n}
\end{bmatrix} .</math>

This result also allows one to exponentiate [[diagonalizable matrix|diagonalizable matrices]]. If
{{block indent|em=1.2|text={{math|1=''A'' = ''UDU''<sup>−1</sup>}} }}
then
{{block indent|em=1.2|text={{math|1=''e''<sup>''A''</sup> = ''Ue''<sup>''D''</sup>''U''<sup>−1</sup>}},}}
which is especially easy to compute when {{math|''D''}} is diagonal.

Application of [[Sylvester's formula]] yields the same result.  (To see this, note that addition and multiplication, hence also exponentiation, of diagonal matrices is equivalent to element-wise addition and multiplication, and hence exponentiation; in particular, the "one-dimensional" exponentiation is felt element-wise for the diagonal case.)