34
$\begingroup$

Let $f:V \to V$ be a linear map such that $(f\circ f)(v) = -v$. Prove that if $V$ is a finite dimensional vector space over $\mathbb R$, $V$ is even dimensional.

From what I can figure out for myself, if $V$ is finite dimensional, then every basis of $V$ is finite, i.e. a linearly independently subset of $V$ has a finite number of vectors.

And I figure that if $V$ is even dimensional, then every basis of $V$ is even, i.e. a linearly independently subset of $V$ has an even number of vectors.

But I'm not sure how to connect these two points.

$\endgroup$
  • 2
    $\begingroup$ Just a note: you say "then the basis", but this doesn't make sense. There isn't a unique basis. $\endgroup$ – user223391 Jun 17 '16 at 3:59
  • $\begingroup$ Oh, right - thanks for pointing that out :) Would it be correct if I just changed it to "a basis" rather than "the basis"? $\endgroup$ – maths123 Jun 17 '16 at 4:00
  • 1
    $\begingroup$ I would say "then every basis is finite" (not finite dimensional, a basis doesn't have a dimension, a vector space does). $\endgroup$ – user223391 Jun 17 '16 at 4:04
  • 5
    $\begingroup$ (1) I'm not sure there's much to connect just yet. So far we've ignored $f$. (2) What field are you working over? I think that matters here. $\endgroup$ – Hoot Jun 17 '16 at 4:15
  • 4
    $\begingroup$ @Hoot is right, the field is important. If we're working over $\mathbb F_5$ as a 1-dimensional vector space, $v \mapsto 2v$ has the desired properties but our space has odd dimension. $\endgroup$ – amcerbu Jun 17 '16 at 4:35
61
$\begingroup$

I heard this story from David Lieberman:

Once this question was included in the Qual (qualifying exam) for Harvard graduate students. As it turned out this one question perfectly predicted all students' performance, so the exam's other 17 questions were not necessary! Indeed:

  • Every student who did not solve this question flunked their Qual.

  • Every student who solved this problem by fiddling with Jordan canonical forms and the like got a "conditional pass".

  • Every student who solved this problem using the determinant passed.

$(\det f)^2 = \det (f \circ f) = \det -I = (-1)^n$ where $n = \dim V$, so $(-1)^n \geq 0$ and $n$ is even.

[Note: This assumes that the matrix has real coefficients. As noted in the comments, the result would be wrong over the complex numbers (e.g., let $n=1$ and $f=i$) and some other fields.]

$\endgroup$
  • $\begingroup$ Great, that makes sense! Thank you :) $\endgroup$ – maths123 Jun 17 '16 at 5:05
  • $\begingroup$ @NoamD.Elkies I just got it and now I'm kicking myself.I'm surprised they didn't have to put those graduate students on suicide watch after that. $\endgroup$ – Mathemagician1234 Jun 17 '16 at 5:10
  • 1
    $\begingroup$ That must have been in the quaint old days, if out of 17 graduate students none observed that f(v) is the data of a complex structure on V. $\endgroup$ – zyx Jun 18 '16 at 3:30
  • 3
    $\begingroup$ This story goes back 30+ years. Indeed nowadays graduate students and even the top undergraduates come with much stronger backgrounds than we could expect a generation ago. The Internet probably deserves a good part of the credit for this... $\endgroup$ – Noam D. Elkies Jun 18 '16 at 18:48
  • 2
    $\begingroup$ @NoamD.Elkies Can confirm, just graduated and view the internet (and this website!) as being highly responsible for my development of mathematics both as a skill and as an interest. $\endgroup$ – Stella Biderman Apr 4 '17 at 17:31
40
$\begingroup$

Here is another solution. We can put the structure of a complex vector space on $V$ by defining $$ (a+bi)\cdot v=av+bf(v) $$ for all $v\in V$ and for all $a,b\in\mathbf R$. This works exactly since $f^2=-1$. Then $V$ is of course a finite dimensional vector space over $\mathbf C$, and therefore of even dimension over $\mathbf R$.

$\endgroup$
  • 12
    $\begingroup$ Sheldon "use $\det$ only as a last resort" Axler would be proud. +1 $\endgroup$ – Noam D. Elkies Jun 17 '16 at 18:25
22
$\begingroup$

This is kind of similar to Noam's answer, but...

You could also argue that there are no eigenvectors with real eigenvalues, since if $f(v)=\lambda v$, then $-v=f(f(v))=\lambda^2 v$, so $\lambda^2 = -1$. However, if $V$ is odd dimensional, then the characteristic polynomial is odd degree, and therefore has a real root, and therefore $f$ has a real eigenvalue.

$\endgroup$
8
$\begingroup$

Just for fun...

As in Callus's answer, observe that if $f(f(x))=-x$ for all $x\in\mathbb R^n$ then there is no $x\ne 0$ such that $f(x)=\lambda x$ for some $\lambda\in\mathbb R$ (since then we would have $\lambda^2x=-x$, which is impossible). We may define a map $F\colon\mathbb R^n\to\mathbb R^n$ by $$ F(x) = \left<x,x\right>f(x) - \left<x,f(x)\right>x $$ which is a scalar multiple of the usual projection map of $f(x)$ on to the plane tangent to $x$. This map has no zeroes other than $x=0$; indeed, if $x\ne 0$ and $F(x)=0$, it means that $$ f(x) = \frac{\left<x,f(x)\right>}{\left<x,x\right>}x $$ which we already know to be impossible unless $x=0$. Observing that $\left<x,F(x)\right>=0$ for all $x\in\mathbb R^n$, we see that $F$ restricts to a smooth tangent field on $S^{n-1}$ with no zeroes; by the hairy ball theorem, it follows that $n-1$ is odd, and so $n$ is even.

$\endgroup$
5
$\begingroup$

Use induction on the dimension. For non zero $v$, $\{v, f(v)\}$ is linear independent and spans a 2d subspace closed under f. Then f is well defined on the quotient $V/\{v,f(v)\}$, which has dimension 2 less. (tyro1's answer is I think along these lines but a bit garbled).

$\endgroup$
4
$\begingroup$

Since $f^2 = -I$, we see that the minimal polynomial of $f$ over $\mathbb{R}$ is $x^2 + 1 \in \mathbb{R}[x]$. The minimal polynomial and characteristic polynomial share their root sets, so the characteristic polynomial of $f$ must be $(x^2 + 1)^k$ for some $k\ge 1$. But the degree $2k$ of the characteristic polynomial is the dimension of $V$, hence $V$ is even-dimensional.

$\endgroup$

protected by user223391 Jun 19 '16 at 4:56

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.