Check whether $\exp(pA)-I$ is positive definite?

Question

For every 4×4 real symmetric non-singular matrix A,there exist a positive integer p such that $\exp(pA)-I$ is positive definite.Is it correct?

I know one result which states that if u is eigenvalue of any transformation T and $P(x)\in K[x]$ then p(u) is eigenvalue of p(T).We know that eigenvalue of $exp(pA)-I$ is $\exp(pu)-1$, if u is eigenvalue of A.Here u Can be positive or negative.If it is negative then for positive p $\exp(pu)-1 \lt 0$.Hence $\exp(pA)-I$ is not positive definite.Is my approach correct?

score 3 · Answer 1 · answered Apr 22 '18 at 07:36

3

Exactly, if $A$ is not positive definite, it has an eigenvalue $\lambda$ with $\lambda\leq 0$. Then, for any $p>0$, $e^{p\lambda}-1\leq0$, so $e^{pA}-I$ is not positive definite.

answered Apr 22 '18 at 07:36

Martin Argerami

217,281

Eigenvalues value of A can not be zero as A is non singular. – ASHWINI SANKHE Apr 22 '18 at 09:20
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That's irrelevant, as my answer and your reasoning only talk about «not positive definite». – Martin Argerami Apr 22 '18 at 12:50
@MartinArgerami does this is appicable only for polynomials or any continuous function ? – TheStudent Aug 15 '20 at 02:22
It makes no difference. Given any function $f$, there exists a polynomial $p$ such that $f(A)=p(A)$. – Martin Argerami Aug 15 '20 at 04:54
@MartinArgerami if we suppose greatest integer function then what will be $p$? – TheStudent Aug 15 '20 at 06:30
Any polynomial that agrees with it on the eigenvalues of $A$. – Martin Argerami Aug 15 '20 at 07:09
@MartinArgerami thankyou for your response, but still it is not clear to to ,like here $e^x-1$ how can we write a finite polynomial such that $p(A)=e^{A}-I$ (where $e^x$ can be written in infinite polynomial). can you please elaborate. – TheStudent Aug 15 '20 at 14:37
If the matrix is diagonalizable and $f(\lambda)=g(\lambda)$ for all eigenvalues of $A$, then $f(A)=g(A)$. – Martin Argerami Aug 15 '20 at 15:29
@TheStudent https://math.stackexchange.com/a/1536468/1272145 – Sandeep Tiwari May 10 '24 at 06:37

Check whether $\exp(pA)-I$ is positive definite?

1 Answers1