# Hadamard vector product

Given a vector $\vec{a} = a_x\vec{e}_x + a_y\vec{e}_y + a_z\vec{e}_z$, and an other vector $\vec{b}$ we can define a component-wise (Hadamard) product $\vec{c} = \vec{a} \star \vec{b}$, by,

$\vec{c} = \vec{a} \star \vec{b} = a_xb_x\vec{e}_x + a_yb_y\vec{e}_y + a_zb_z\vec{e}_z.$ Indeed, $\vec{e}_i\star\vec{e}_j = \begin{cases} \vec{e}_i, & \text{for } i = j \\ 0, & \text{for } i \neq j \end{cases}$

Euclidian vectors equipped with such a binanary operation would form an Abelian group and should be less controversial than the scalar/dot/inner product or vector/cross product. In order to investigate why it is not popular we look at its geometric properties:

- The “length” of the product is equal to the inner product: $\|\vec{a}\star\vec{b}\| = \vec{a} \cdot \vec{b}$.
- The identity element breaks the symmetry between negative and positive directions …
- … Hence, the product does not retain its orientation under rotation.
- Its direction … ?

Lets see if we can get inspiration from some examples:

The direction of the product is not very intuitive.

# Jargon in linear algebra

During my Bsc. studies, I found that the cryptic names of the various mathematical concepts made grasping them (as a Dutch person) harder than it should have been. As I am now teaching linear algebra, I wish to alleviate this problem by explaining the relation of these names with their associated concepts to the students. Here is a list of what I found.

A womb can be viewed as a container for its contents, similar(?) to how a rectangular array of numbers can be contained in a algebraic object. As such, the Latin and (old) French word for womb is used to denote a

**Matrix**^{1}.When doing row reduction/Gaussian elimination, the row sweeping operation revolves around a non-zero entity that is selected as a reference to guide the row subtractions. An center of an axle that is a central entity around which stuff revolves is also called a

**pivot location**.Some advantages to positioning military units on a battle field in a line or in a frontal formation may combined by placing them in as staggered, so-called

**echelon form**. A similar stair-case like arrangements is made by the pivot locations of a matrix after Gaussian elimination has taken place. The word**echelon**in turn stems from a French word for ladder (“échelle”).The word homogeneous refers to things begin “the same”. The right-most column of the augmented matrix for the

**homogeneous equations**is always the same (a column of zeros) and remains as such, even after elementary row operations have been applied.The central entity of an object may be called a

**kernel**. Since every vector in a vector space is accompanied by a vector in the opposite direction, the “center” of mass of the range of a transformation ($T: \mathbb{R}^n \rightarrow \mathbb{R}^m$) may be said the be the zero vector ($i.e. \mathbf{0} \in \mathbb{R}^m$). As such we say that the**kernel**($\text{Ker}(T) \in \mathbb{R}^n$) is mapped there.One may carry something from one position to another. The displacement entity is then a carrier. This displacement is often denoted with a

**vector**, the Latin word for carrier.An invertible matrix $A$ can be paired with its inverse $A^{-1}$. Note that $A$ is also the inverse of $A^{-1}$. A matrix that is not invertible, cannot be paired an therefore remains alone. We therefore say that this is a

**singular**matrix.When a criminal is caught via the hits he/she left behind on the scene, these hints can be said to have made a trail that led to the perpetrator. Such a trail can also “follow” a matrix $A$, of particular interest is the “similarity” transformation ($C$) via an invertible matrix $B$ defined as $C = B^{-1}AB$. A prime example of such a matrix trail is therefore called the

**trace**of a matrix, which does not change even tough generally $C\neq A$.The length ($\|\vec{a}\|$) of a vector ($\vec{a}$) can be expressed using the

**inner product**($\|\vec{a}\|^2 = \vec{a}\cdot\vec{a}$). H. Grassmann was the first to study this product of a vector with itself before it was applied as a product of two different vectors. He therefore found this product was inward-looking/intovert, and named it as such (albeit in German).In a linear transformation, there may exist characteristic vectors that are only scaled by a certain characteristic value. These values may be found as the root of the

**characteristic polynomial**. Since they are so characteristic, they belong to the transformation. A loose German (and Dutch) translation of something that “belongs” to you is “eigen”, giving rise to the**eigenvectors and eigenvalues**^{2}.The

**singular value**decomposition (SVD) of a matrix ($A$) places**singular values**($\sigma_i$) in a diagonal matrix. These values are such that $A^TA - \sigma_i^2I$ is a singular matrix (c.f. eigen values).