Question

Find all matrices $A$ such that $e^{t A}=0$.

    Find all matrices $A$ such that $e^{t A}=0$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 10, Problem 8

Instant Answer

Solved by Expert Nick Johnson

01/22/2023

Step 1

The exponential of a matrix $A$, denoted $e^{tA}$, is defined by the power series: \[ e^{tA} = \sum_{n=0}^\infty \frac{(tA)^n}{n!}. \] This series converges for all $t$ and for any square matrix $A$. Show more…

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Key Concepts

Matrix Exponential

The matrix exponential is defined via a power series expansion: e^(tA) = I + tA + (tA)^2/2! + (tA)^3/3! + ... for any square matrix A. This definition guarantees that the series converges for all t and any matrix A, making the matrix exponential a fundamental concept in solving systems of linear differential equations and understanding continuous transformations.

Invertibility of the Matrix Exponential

A crucial property of the matrix exponential is that it is always invertible. The inverse of e^(tA) is given by e^(-tA), which follows directly from its series definition and the properties of exponentiation. This inherent invertibility means that e^(tA) can never be the zero matrix, because a zero matrix does not have an inverse. This concept is key to understanding why no matrix A can satisfy e^(tA) = 0.

Spectral Mapping Theorem

The spectral mapping theorem connects the eigenvalues of a matrix A to those of its exponential e^(tA). Specifically, if ? is an eigenvalue of A, then e^(t?) is an eigenvalue of e^(tA). Since e^(t?) is never zero for any finite ? and t, this theorem reinforces the fact that e^(tA) cannot be the zero matrix, regardless of the choice of A.

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