(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given g[tex]\equiv g_{ij} = [/tex]

[-1 0;

0 1]

Show that A= [tex]A^{i}_{j}[/tex] =

[1 2

-2 1]

is symmetric wrt innter product g, has complex eigenvalues, but eigenvectros have zero length wrt the complex inner product.

3. The attempt at a solution

Im sure this is just a simple linear algebra problem but im having trouble figuring out how to compute the dot procut with this 1,1 tensor.

My guess would be to break the matrix [tex]A^{i}_{j}[/tex] intro rows and compute { (row1)g(row2)^t} then to show this is symmetric calculate { (row2)g(row1)^t}. But that seems wrong.

I can calculate the eigenvalues (they come out to [tex]\pm i \sqrt{3}[/tex] )

Also i am lost on showing the eigenvecotrs are 0 wrt the this inner product. I would have no idea how to approach this even if i knew how to calculate inner product.

Thanks in advance

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# Homework Help: Inner product with (1,1) tensors: Diff. Geometry/ Lin algebra

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