Editing Matrix exponential (section)

== Illustrations ==

Suppose that we want to compute the exponential of
<math display="block">B = \begin{bmatrix}
  21 & 17 &  6 \\
  -5 & -1 & -6 \\
   4 &  4 & 16
\end{bmatrix}.</math>

Its [[Jordan normal form|Jordan form]] is
<math display="block">J = P^{-1}BP = \begin{bmatrix}
  4 &  0 &  0 \\
  0 & 16 &  1 \\
  0 &  0 & 16
\end{bmatrix},</math>
where the matrix ''P'' is given by
<math display="block">P = \begin{bmatrix}
  -\frac14 &  2 &  \frac54 \\
   \frac14 & -2 & -\frac14 \\
         0 &  4 &        0
\end{bmatrix}.</math>

Let us first calculate exp(''J''). We have
<math display="block">J = J_1(4) \oplus J_2(16) </math>

The exponential of a {{math|1 × 1}} matrix is just the exponential of the one entry of the matrix, so {{math|1=exp(''J''<sub>1</sub>(4)) = [''e''<sup>4</sup>]}}. The exponential of ''J''<sub>2</sub>(16) can be calculated by the formula {{math|1=''e''<sup>(λ''I'' + ''N'')</sup> = ''e''<sup>''λ''</sup> ''e''<sup>N</sup>}} mentioned above; this yields<ref>This can be generalized; in general, the exponential of {{math|''J''<sub>''n''</sub>(''a'')}} is an upper triangular matrix with {{math|''e''<sup>''a''</sup>/0!}} on the main diagonal, {{math|''e''<sup>''a''</sup>/1!}} on the one above, {{math|''e''<sup>''a''</sup>/2!}} on the next one, and so on.</ref>

<math display="block">\begin{align}
        &\exp \left( \begin{bmatrix} 16 & 1 \\ 0 & 16 \end{bmatrix} \right)
    = e^{16} \exp \left( \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \right) = \\[6pt]
  {}={} &e^{16} \left(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + {1 \over 2!}\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \cdots {} \right)
    = \begin{bmatrix} e^{16} & e^{16} \\ 0 & e^{16} \end{bmatrix}.
\end{align}
</math>

Therefore, the exponential of the original matrix {{mvar|B}} is
<math display="block">\begin{align}
  \exp(B)
  &= P \exp(J) P^{-1}
   = P \begin{bmatrix} e^4 & 0 & 0 \\ 0 & e^{16} & e^{16} \\ 0 & 0 & e^{16} \end{bmatrix} P^{-1} \\[6pt]
  &= {1 \over 4} \begin{bmatrix}
    13e^{16} - e^4 & 13e^{16} - 5e^4 &  2e^{16} - 2e^4 \\
    -9e^{16} + e^4 & -9e^{16} + 5e^4 & -2e^{16} + 2e^4 \\
    16e^{16}       & 16e^{16}        &  4e^{16}
  \end{bmatrix}.
\end{align}</math>