# What does the notation $(x, y)_2$ mean?

My professor for a numerical methods course has been occasionally using the notation $$(x, y)_2$$ where $$x$$ and $$y$$ are vectors or matrices, but she hasn't explained what it means. Here are some examples of it appearing in the notes:

A (symmetric) matrix $$A \in \mathbb{R}^{nxn}$$ is called positive definite if $$(Ax, x)_2 = \sum_{i,j=1}^{n} x_i a_{i,j} x_j > 0$$ for all $$x \in \mathbb{R}^n, x \neq 0$$.

Relating to QR matrix decomposition:

Convert the column vectors of $$A$$ $$(a_1, \dots, a_n)$$ into orthonormal vectors $$q_1, \dots, q_n$$. This means $$\|q_i\|_2 = 1$$ and $$(q_i, q_j)_2 = 0$$ if $$i \neq j$$.

And regarding singular value decomposition:

Let $$A = U\Sigma V^T$$ be the singular value decomposition of a matrix $$A \in \mathbb{R}^{mxn}$$ and let $$r = rank(A)$$. Then $$\overline{x} = \sum_{i=1}^{r} \frac{(u_i, b)_2}{\sigma_i} v_i$$ is the unique solution of the normal equation with minimal Euclidean norm. Furthermore, the residual is $$\| b - A\overline{x} \|_{2}^{2} = \sum_{i=r+1}^{m} |(u_i, b)_2 |^2.$$

Based on these and some other examples, my best guess is that it's some sort of matrix/vector norm, but I'm not sure what. I haven't been able to find the notation online, I haven't seen it in my other linear algebra experience, and unfortunately I'm not able to make it to my professor's office hours as they're during another of my classes. Hopefully someone here can help! Thanks in advance!

• If $x=(x_1,...,x_n)$ and $y=(y_1,...,y_n)$, then they are denoting by $(x,y)_2=x_1y_2+...+x_ny_n$.
– user752802
Mar 4, 2020 at 21:34
• It's the usual inner product of vectors: $(x,y)_2 = \sum_{i=1}^n x_i y_i$. Mar 4, 2020 at 21:34
• For more details, look up p-norm en.wikipedia.org/wiki/Norm_(mathematics)#p-norm Mar 4, 2020 at 21:35
• The above comments taken together constitute a full answer to the question. Mar 4, 2020 at 21:38

The definition of $$(Ax,x)_2$$ is right in the text you quoted... In general, if $$x,y \in \mathbb{R}^n$$, $$(x,y)_2$$ stands for the usual inner product, i.e. $$(x,y)_2 = \sum_{i=1}^n x_i y_i.$$
From this definition you can deduce the other formulas, for instance, $$(Ax , x)_2 = \sum_{i=1}^n (Ax)_i x_i = \sum_{i=1}^n\sum_{j=1}^n A_{ij} x_j x_i = x^T A x.$$
Generally, you'll see the inner product written as $$\langle u_i,u_j \rangle _2$$ for example.
This is not solely used for $$n$$-dimensional vector spaces either. In functional analysis it is common to discuss the inner product of 2 functions, say $$f,g$$, such that $$\langle f, g \rangle_2 = \int_a^b f(x)g(x) dx$$ where $$x\in (a,b)$$, for example.
Often you see the vector 2 norm defined written as $$\langle u_i,u_i \rangle_2 = \Vert u_i \Vert_2^2$$