Can we say scaling matrix is necessarily diagonal?

Question

Can we say scaling matrix is necessarily diagonal?

According to wikipedia, yes

According to this video, no

$S$ is scaling along orthogonal directions according to this

So, how to put them both together?

Answer 1

As pointed out by Nij, the Wikipedia article you quoted states that:

Non-uniform scaling is accomplished by multiplication with any symmetric matrix.

A symmetric matrix does not necessarily have to be diagonal. But there is a theorem that a symmetric matrix is diagonalizable.

Answer 2

A matrix multiplication is a linear transformation $$A :\mathbb{R}^n \rightarrow \mathbb{R}^{m} \\ x \mapsto A x$$ that maps $x \in \mathbb{R}^n$ into the column space of $A \in \mathbb{R}^{m\times n}$, i.e $$A = \begin{pmatrix} \mathbf{a}_1 \ | \ \mathbf{a}_2 \ | \cdots \ |\ \mathbf{a}_n \end{pmatrix} \begin{pmatrix}x_1 \\ \vdots \\ x_n \end{pmatrix} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \cdots + x_n \mathbf{a}_n$$ where $\mathbf{a}_i \in \mathbb{R}^m$. So if $A$ is diagonal, the $\mathbf{a}_i$'s are an orthogonal basis, and the product correponds to an scaling along these orthogonal directions.