Articles of hadamard product

Is there a formula for the inverse of Hadamard product?

Say $A$ and $B$ are two square, positive-semidefinite matrices. Is there an expression in terms of matrix product, transpose, and inverse for the Hadamard product $A∘B$? For example, “$(A∘B)^{-1} = A^{-1} ∘ B^{-1}$” (which is not true). Edit: I understand that $A∘B$ may not be invertible, but is there any expression if invertibility is given?

generic rule matrix differentiation (Hadamard Product, element-wise)

I struggle with taking the derivative of the Hadamard-Product? Let us consider $f(x)=x^TAx=x^T(Ax)$. We know $$\frac{\partial}{\partial x} x^TAx = (A+A^T)x.$$ The Matrix-Cookbook claimed $d(XY)=d(X)Y+Xd(Y)$ and $$\frac{\partial}{\partial x} x^Ta = \frac{\partial}{\partial x}a^Tx = a.$$ Setting $X:=x^T$ and $Y:=Ax$ we have \begin{align*} X &= x^TE & d_x(X) = E\\ Y &= Ax & d_x(Y) = A\\ \end{align*} […]

Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices

The component-wise product (Hadamard product) of two positive definite matrices is a positive definite matrix (Schur product theorem). I encountered the following proof of it: $A=(a_{ij})$ and $B=(b_{ij})$ are positive definite. Let $a_{ij} = \displaystyle \sum_{k=1}^N \lambda_k t_{ik} t_{jk}$ where $T=(t_{ij})$ is an orthogonal matrix and $\lambda_k$ are the eigenvalues of $A$. $$\sum_{i,j=1}^N a_{ij} b_{ij} […]

If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease?

Given a positive semidefinite matrix $P$, if we scale down its entries individually, will its operator norm always decrease? Put it another way: Suppose $P\in M_n(\mathbb R)$ is positive semidefinite and $B\in M_n(\mathbb R)$ is a $[0,1]$-matrix, i.e. $B$ has all entries between $0$ and $1$ (note: $B$ is not necessarily symmetric). Let $\|\cdot\|_2$ denotes […]

Element-wise (or pointwise) operations notation?

Is there a notation for element-wise (or pointwise) operations? For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same length z, where $z_i = x_i * y_i$ . In mathematical notation, there doesn’t seem to be a standard for […]