What would be a good basis choice to orthonormalize in a a non standard inner product space?

Question

I've encountered the following question:

A non standard inner product on $\mathbb{R}^{3}$ is defined as follows:

$\left\langle v\mid u\right\rangle =v^{t}Au$

$A=\left(\begin{array}{ccc} 2 & -1 & 0\\ -1 & 2 & 0\\ 0 & 0 & 3 \end{array}\right)$

And i'm requested to find an orthonormal basis for V with respect to this inner product.

Now, the standard way to approach this is to start with the standard basis and apply the Gram-Shcmidt procedure. Is there a "smart" choice for a starting basis which would make the Gram-Shcmidt procedure easier?

Answer 1

Yes. Apply Gram-Shmidt to $\{(1,0,0),(0,1,0)\}$. You will get two vectors $u$ and $v$, both of which are of the form $(x,y,0)$. Then your orthonormal basis will be $\bigl(u,v,(0,0,\sqrt{1/3})\bigr)$, because $(0,0,1)$ is orthogonal to any vector of the type $(x,y,z)$.

Answer 2

Two worthwhile tricks:

1. For every $2 \times 2$ matrix $A$ and every real number $c$, the vector $(0, 0, 1)$ is a $c$-eigenvector of the block matrix $$ \left[\begin{array}{cc} A & 0 \\ 0 & c \\ \end{array}\right]. $$ (Similarly, $(1, 0, 0)$ is a $c$-eigenvector if $A$ is in the lower-right corner, and $(0, 1, 0)$ is a $c$-eigenvector if $A$ is "split" among the four corner entries.)

2. If $a$ and $b$ are real, the vectors $(1, 1)$ and $(1, -1)$ are eigenvectors of the $2 \times 2$ matrix $$ A = \left[\begin{array}{cc} a & b \\ b & a \\ \end{array}\right], $$ with respective eigenvalues $a + b$ and $a - b$.

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Combining these allows an orthonormal eigenbasis to be written down immediately, by inspection.