# Proving Euclidian Norm squared is equivalent to transpose times matrix for x in R^n

Apologies if this has been answered already but I can't seem to find an answer that I think answers my question (or at least one I understand).

Anyways the question is,


Prove ‖x‖22=xTx for all x in Rn


When I took a swing at it I got a scalar value for the Euclidian (or Frobenius?) norm and a matrix value (of dimension n x n) for x-transpose times x. I really have no clue what I'm doing but I know it can't be good if I have a scalar and matrix set to be equivalent to one another.

thanks

• Note that $\mathbf{ x}^T$ is ${1}\times n$ and $\mathbf{x}$ is $n\times 1$, so $\mathbf{x}^T\mathbf{x}$ is size $1\times 1$ (essentially a scalar). Mar 20, 2019 at 5:11

If we think of $$x \in \mathbb{R}^d$$ as a column vector, then $$x^tx = \sum_{j=1}^d x_j^2 = \lvert x \rvert^2.$$ Edit: It seems you may be a bit confused on matrix multiplication. Since $$x^t$$ is a $$1 \times d$$ matrix and $$x$$ is a $$d \times 1$$ matrix, $$x^tx$$ will be a $$1\times 1$$ matrix (ie, a scalar): $$x^tx = (x_1, \ldots, x_d) \begin{pmatrix} x_1 \\ \vdots \\ x_d \end{pmatrix} = \sum_{j=1}^d x_j^2.$$