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normalized eigenvector $u_1=\frac{1}{\sqrt{5}}\begin{bmatrix} 1 \\ 2 \\ \end{bmatrix}$ need to find orthonormal basis of $Span[{u_1}]^ ⊥$ using mathematica. I tried with Orthogonalize command and did not work.

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  • $\begingroup$ You mean Cross[1/Sqrt[5] {1, Sqrt[2]}]? $\endgroup$ – yode Nov 5 '16 at 10:52
  • $\begingroup$ Actually, I was going to solve Schur Decomposition. I could do that using SchurDecomposition[N[{{7, -2}, {12, -3}}]] in Mathematica Software. But I want to do that using different method. $\endgroup$ – Falcon Nov 5 '16 at 11:28
  • $\begingroup$ Extending from a vector to a basis can be done with NullSpace. Then orthogonalize. $\endgroup$ – Daniel Lichtblau Nov 6 '16 at 19:15
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Since the span of $u_1$ is the same as the span of {1,2}, you could simply do this:

  1. Find some vector orthogonal to {1,2}:

        v1 = {1, 2}
        Solve[{x, y} . v1 == 0}]
    (* {{y -> - x/2}} *)
    
  2. Accordingly, take say x -> 2 in that solution and then y -> -1. Thus set:

        v2 = {2, -1} 
    
  3. Thus v2 is in the 1-dimensional space orthogonal to the span of your $u_1$. Now just normalize v2.

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