Property of the identity matrix

Question

Is the identity the only matrix $A \in \mathbb R^{n \times n}$ with real positive eigenvalues that is equal to its inverse?

Thanks.

No. Any matrix $;A;$ s.t. $;A^2=I;$ will fulfill that. – DonAntonio Jun 19 '13 at 17:59 — DonAntonio, Jun 19 '13 at 17:59
Can you give an example with all eigenvalues equal to one? – roger Jun 19 '13 at 18:04 — roger, Jun 19 '13 at 18:04

score 5 · Answer 1 · answered Jun 19 '13 at 18:09

5

Yes. If a matrix squares to the identity all of its eigenvalues are $\pm 1$, so in your case, all eigenvalues are $+1$. Now consider the Jordan normal form of $A$; if any Jordan block is different from the identity, then $A^2 \not= I$, so the Jordan normal form of $A$ is the identity matrix, and therefore $A$ itself is the identity.

answered Jun 19 '13 at 18:09

fuglede

6,716

That's perfect thanks! – roger Jun 19 '13 at 18:10

score 1 · Answer 2 · answered Jun 19 '13 at 21:55

1

If $A^2-I=0$, the minimal polynomial $m_A(x)$ of $A$ divides $x^2-1$. Yet $x+1$ is not a factor of $m_A(x)$ because $-1$ is not an eigenvalue of $A$. Hence $m_A(x)=x-1$, i.e. $A=I$.

answered Jun 19 '13 at 21:55

user1551

139,064

Property of the identity matrix

2 Answers2