# Find orthonormal basis using mathematica

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.

• You mean Cross[1/Sqrt[5] {1, Sqrt[2]}]? – yode Nov 5 '16 at 10:52
• 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. – Falcon Nov 5 '16 at 11:28
• Extending from a vector to a basis can be done with NullSpace. Then orthogonalize. – Daniel Lichtblau Nov 6 '16 at 19:15

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.