While studying Linear Algebra, I encountered the following exercise:
Let
$$A = \begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$$
Write $A$ as a sum $$\lambda_{1} u_{1}{u_{1}}^T + \lambda_{2} u_{2}{u_{2}}^T$$ where $\lambda_1$ and $\lambda_2$ are eigenvalues and $u_1$ and $u_2$ are orthonormal eigenvectors.
So what I did is, I computed the eigenvalues: $1$ and $-1$. Then I computed the eigenvectors:
\begin{bmatrix}
1\\1
\end{bmatrix}
and \begin{bmatrix}
-1\\1
\end{bmatrix}.
But, these vectors are not an orthonormal set, thus I used Gram-Schmidt to obtain an orthonormal set of eigenvectors. This gave me:
$u_1 = 1/\sqrt{2}\left( \begin{smallmatrix} 1\\ 1 \end{smallmatrix} \right)$ and $u_2 = 1/\sqrt{2}\left( \begin{smallmatrix} -1\\ 1 \end{smallmatrix} \right)$.
This outcome seems okay, however, I am wondering whether there is a faster method to do this?
Best Answer
The vectors $(1,1)^T$ and $(-1,1)^T$ are orthogonal, so you just had to normalize them (divide them by their norm) to get an orthonormal set.