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Finding the eigenvectors for a self-adjoint matrix when algebraic multiplicity 2
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I have the following 3x3 matrix that I am trying to compute the eigenvectors for

https://preview.redd.it/kyob69zf9x8d1.png?width=1022&format=png&auto=webp&s=ac42913a53ad11c5a93ee96c129098ab5193252b

For eigvalue = -9, the method is straightforward, and I get the following.

https://preview.redd.it/vh5s8ell9x8d1.png?width=1060&format=png&auto=webp&s=4ff5d0cf1ebabec8688ac03c626de0c6ddb6ffe5

However, for the repeated eigval = 3, I am not able to get the eigvector. My thought process is:

  1. find an eigenvector that satisfies the condition -x 2y z = 0

  2. normalize it

  3. find another vector orthonormal to it, using Gram Schmidt.

However, I cannot get a second vector that is orthogonal (inner product=0) to my first vector in 1.

What is wrong with my logic?

https://preview.redd.it/6y6041zp9x8d1.png?width=930&format=png&auto=webp&s=0ddb4aac553846fcdd1e3f8d65131bb8478f6348

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2 months ago