# Is the matrix in the given dot product orthogonal

A dot product is given in the space $\mathbb{R}^2$ with the following formula:

$$\left\langle\begin{bmatrix}x_1\\x_2\end{bmatrix}, \begin{bmatrix}y_1\\y_2\end{bmatrix}\right\rangle = 2x_1y_1 + x_2y_2$$

Is the matrix $\begin{bmatrix}1 & 1\\2 & 1\end{bmatrix}$ in this dot product orthogonal?

I'm not sure if I know what this exercise is asking from me. I know we can prove that a matrix is orthogonal if: $AA^T=I$ holds or by showing that each vector in the matrix is a unit vector (it's length is 1) and the dot product of each pair of vectors in the matrix is zero, but I think this isn't the case here.

• You can find some good starting points on how to format mathematics on the site here and here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. – Zev Chonoles Feb 5 '13 at 17:32
• @Zev: Why not the MathJax basic tutorial and quick reference on our meta? I believe it was intended to be the optimal one-stop intro and reference for beginners. – Rahul Feb 5 '13 at 17:35

Let the columns of the matrix $\mathbf A$ be $\mathbf a_1$ and $\mathbf a_2$. Consider what the condition $\mathbf A^T\mathbf A=\mathbf I$ is really telling you: $$\mathbf A^T\mathbf A=\begin{bmatrix}\mathbf a_1&\mathbf a_2\end{bmatrix}^T\begin{bmatrix}\mathbf a_1&\mathbf a_2\end{bmatrix}=\begin{bmatrix}\mathbf a_1^T\\\mathbf a_2^T\end{bmatrix}\begin{bmatrix}\mathbf a_1&\mathbf a_2\end{bmatrix}=\begin{bmatrix}\mathbf a_1^T\mathbf a_1&\mathbf a_1^T\mathbf a_2\\\mathbf a_2^T\mathbf a_1&\mathbf a_2^T\mathbf a_2\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}.$$ Compare the entries of the last two matrices: $\mathbf a_1^T\mathbf a_1=1$, $\mathbf a_1^T\mathbf a_2=0$, $\mathbf a_2^T\mathbf a_2=1$. This means precisely that the columns of the matrix are orthonormal with respect to the usual inner product $\langle\mathbf x,\mathbf y\rangle=\mathbf x^T\mathbf y=x_1y_1+x_2y_2$. It does not tell you anything about whether they are orthonormal with respect to the different inner product $\langle\mathbf x,\mathbf y\rangle=2x_1y_1+x_2y_2$ that you've been asked to use! So what you need to do is check that orthonormality holds with respect to this inner product instead: $\langle\mathbf a_1,\mathbf a_1\rangle=1$, $\langle\mathbf a_1,\mathbf a_2\rangle=0$, $\langle\mathbf a_1,\mathbf a_2\rangle=1$.