Editing Diagonal matrix (section)

== Vector operations ==
Multiplying a vector by a diagonal matrix multiplies each of the terms by the corresponding diagonal entry. Given a diagonal matrix <math>\mathbf{D} = \operatorname{diag}(a_1, \dots, a_n)</math> and a vector <math>\mathbf{v} = \begin{bmatrix} x_1 & \dotsm & x_n \end{bmatrix}^\textsf{T}</math>, the product is:
<math display="block">\mathbf{D}\mathbf{v} = \operatorname{diag}(a_1, \dots, a_n)\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix} =
  \begin{bmatrix}
    a_1 \\
        & \ddots \\
        &        & a_n
  \end{bmatrix}
  \begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix} =
  \begin{bmatrix}a_1 x_1 \\ \vdots \\ a_n x_n\end{bmatrix}.
</math>

This can be expressed more compactly by using a vector instead of a diagonal matrix, <math>\mathbf{d} = \begin{bmatrix} a_1 & \dotsm & a_n \end{bmatrix}^\textsf{T}</math>, and taking the [[Hadamard product (matrices)|Hadamard product]] of the vectors (entrywise product), denoted <math>\mathbf{d} \circ \mathbf{v}</math>:

<math display="block">\mathbf{D}\mathbf{v} = \mathbf{d} \circ \mathbf{v} =
  \begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix} \circ \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} =
  \begin{bmatrix} a_1 x_1 \\ \vdots \\ a_n x_n \end{bmatrix}.
</math>

This is mathematically equivalent, but avoids storing all the zero terms of this [[sparse matrix]]. This product is thus used in [[machine learning]], such as computing products of derivatives in [[backpropagation]] or multiplying IDF weights in [[TF-IDF]],<ref>{{cite book |last=Sahami |first=Mehran |date=2009-06-15 |title=Text Mining: Classification, Clustering, and Applications |url=https://books.google.com/books?id=BnvYaYhMl-MC&pg=PA14 |publisher=CRC Press |page=14 |isbn=9781420059458}}</ref> since some [[BLAS]] frameworks, which multiply matrices efficiently, do not include Hadamard product capability directly.<ref>{{cite web |url=https://stackoverflow.com/questions/7621520/element-wise-vector-vector-multiplication-in-blas |title=Element-wise vector-vector multiplication in BLAS? |author=<!--Not stated--> |date=2011-10-01 |website=stackoverflow.com |access-date=2020-08-30}}</ref>