I am interested in the explicit computation of the following norm
$$ |x| := \sup_{t \geq 0} \left\Vert \mathrm {e}^{-At}x \right\Vert_2,$$
$x = (x_1, x_2)^{\top} \in \mathbb{R}^n, A \in \mathbb{R}^{n \times n}$ and $\Vert \cdot \Vert_2$ denotes the standard Euclidean norm. Concretely for the matrix
$$ {A} := \begin{pmatrix} \frac{19}{20} & -\frac{3}{10} \\ \frac{3}{10} & -\frac{1}{20} \end{pmatrix}. $$
This matrix is obviously non-symmetric but positive stable, the Eigenvalues are $\lambda_1 = 1/20$ and $\lambda_2 = 17/20.$ For symmetric and anti-symmetric matrices this norm would coincide with the Euclidean norm but not for this matrix. When computing this problem, the expressions quickly become very intricate. Whereby I chose the approach by explicitly calculating the matrix exponential and subsequent maximization by differentiation, which unfortunately leads to an irrepressible expression. So is there an intelligent way of calculating this norm that leads to a pleasant expression?
Would be very grateful for any help!
EDIT: In this case $$ \mathrm{e}^{-At} = \frac{1}{8} \left(\begin{array}{rr} -\mathrm{e}^{-\frac{1}{20} t} + 9\mathrm{e}^{-\frac{17}{20} t} & 3\mathrm{e}^{-\frac{1}{20} t} - 3\mathrm{e}^{-\frac{17}{20} t} \\ -3\mathrm{e}^{-\frac{1}{20} t} +3 \mathrm{e}^{-\frac{17}{20} t} & 9\mathrm{e}^{-\frac{1}{20} t} - \mathrm{e}^{-\frac{17}{20} t} \\ \end{array} \right). $$ And $ \left \Vert \mathrm{e}^{-At} \left(\begin{array}{rr} x_1 \\ x_2 \end{array} \right) = \right \Vert_2 $ $$ =\frac{1}{8} \sqrt{\left((-\mathrm{e}^{-\frac{1}{20} t} + 9\mathrm{e}^{-\frac{17}{20} t})x_1 + (3\mathrm{e}^{-\frac{1}{20} t} - 3\mathrm{e}^{-\frac{17}{20} t})x_2 \right)^2 + \left((-3\mathrm{e}^{-\frac{1}{20} t} + 3\mathrm{e}^{-\frac{17}{20} t})x_1 + (9\mathrm{e}^{-\frac{1}{20} t} - \mathrm{e}^{-\frac{17}{20} t})x_2 \right)^2} $$ becomes rather cumbersome to work with.