Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1994 • Question 1

Question

Math 6410 • ODE Fall 1994 • Question 1

Assume that the eigenvalues of a real \(n\times n\) matrix \(A\) have negative real part strictly less than \(-5\). Prove that there exists a positive constant \(M\) such that for all \(x\in \mathbb {R}^n\) and \(t\ge 0\) \[ \left \lVert e^{At}x\right \rVert \le M\lVert x\rVert e^{-5t}. \]

Raw LaTeX

Assume that the eigenvalues of a real \(n\times n\) matrix \(A\)
have negative real part strictly less than \(-5\). Prove that there exists a positive
constant \(M\) such that for all \(x\in\R^n\) and \(t\ge0\)
\[
\left\lVert e^{At}x\right\rVert\le M\lVert x\rVert e^{-5t}.
\]

Topic Tags

Linear Systems and Matrix Exponentials Spectral Theory of Operators Stability of Linear Systems Variation of Constants Formula