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From Axler's Linear Algebra,

If $T\in L(U)$ is an operator on a finite dimensional vector space $U$, then $U$ has a basis consisting of eigenvectors if and only if there is an inner product that makes $T$ self adjoint.

I have no problem proving the statement, however I want to see if I understand the idea correctly.

A sketch of the proof in one direction:

If $(v_1,\dots ,v_m)$ is the basis consisting of eigenvectors, the point is to define an inner product make them orthogonal to each other and have unit length, specifically, define:

$$\langle a_1 v_1+\dots +a_m v_m, b_1 v_1 +\dots + b_m v_m \rangle = a_1b_1 +\dots + a_m b_m$$

With this definition, it is clear that any two eigenvectors are orthogonal, and they have length $1$ if we define norm as in the usual way.

Inspired by the problem, now my question is, is that we can define any basis to be an orthonormal basis by choosing the right inner product? In the usual way of defining dot product on $\Bbb R^n$, $\langle (a_1,\dots ,a_n), (b_1,\dots , b_n) \rangle =a_1b_1 +\dots +a_nb_n$, it is defined using the standard basis, which we chose to be orthonormal because they matches our usual idea of what perpendicular means.

lEm
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1 Answers1

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The answer to your question:

is that we can define any basis to be an orthonormal basis by choosing the right inner product?

is Yes. two vectors are defined orthogonal if its inner product is null, so the notion of orthogonality is ''inner product dependent''. And any basis is orthogonal with respect a suitable inner product.

About the same problem you can see the answers to my question: What really is ''orthogonality''?

Community
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Emilio Novati
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  • I Think that the question of why we represents ''orthogonal'' vectors as in a typical Cartesian coordinate system (with $90°$ angles$) is more subtle, and has a physical justification. See: http://math.stackexchange.com/questions/1710515/a-cartesian-coordinate-system-is-a-mathematical-or-physical-thing – Emilio Novati Sep 07 '16 at 15:13
  • Thank you for the answer, this is exactly what I was looking for. Certainly the concept of orthogonality comes from the physical sense of perpendicular, but further generalising it as having inner product equals zero brings up richer concepts. And I must agree with you that coordinate is sometimes confusing, because a coordinate system "seemingly defines some geometry" for spaces like R^n, but it is actually nothing more than a way of labelling vectors/points (which unfortunately and unavoidable involving choosing a particular basis if we are talking about vector spaces). – lEm Sep 07 '16 at 17:20