Questions tagged [hadamard-product]
For questions about Hadamard product between two matrices, or it can concern analytic functions.
0
votes
0answers
28 views
Matrix square root of squared correlation matrix
Setup:
Given $y \sim N(0,\Sigma)$, suppose we want to transform $y$ to a new space so entries have zero covariance. We can use the inverse square root and apply to transform $\tilde{y} = \Sigma^{-1/...
1
vote
0answers
15 views
How to simplify repeated convolution and Hadamard multiplication
I’ve determined that the following expression gives me the correct answer in a programming challenge:
$$ (A \circledast M) \times H) \circledast M) \times H) \circledast M) \times H) \circledast M) \...
0
votes
1answer
68 views
Differentials to derivatives involving trace of matrices
Suppose $P$ is a real-valued function of the $p\times m$ (real) matrix $\mathbf{Q}$. After taking its differential, one arrives with the following:
$$
d(P(\mathbf{Q}))
= \operatorname{trace}\...
0
votes
0answers
16 views
Chain and product rule for Hadamard product differentiation
(Asked a similar question before but deleted to add further detail)
Similar to this question and a related to this question, how can I apply the chain and product rule to find the Jacobian of
$$
f_1(...
0
votes
1answer
45 views
Hessian of quadratic form of function using Hadamard and Frobenius notation
Related to this question, I am trying to compute the Hessian of
$$
g(r, \theta) = [r\cos(\theta)]^{\top} A \, [r\cos(\theta)] = f(r, \theta) ^{\top} A \, f(r, \theta) \tag{$*$}
$$
for $r, \theta \in \...
0
votes
1answer
60 views
Matrix Differentiation (involving Hadamard products)
I am trying to differentiate over the following Frobenius Norm: $$\Phi =||A-(B\circ C)D ||^2_F$$
with respect to B, C, D respectively, i.e.:
$$\frac{\partial \Phi}{\partial B}, \frac{\partial \Phi}{\...
1
vote
1answer
30 views
Derivative of $tr(A(C \circ X)BB'(C' \circ X')A')$
Can we differentiate this function: $tr(A(C \circ X)BB'(C' \circ X')A')$ w.r.t $X$?
Also, $tr(A(C \circ X)Y)$ w.r.t $X$.
1
vote
0answers
51 views
Determinant of Hadamard product / sum of matrices (one diagonal)
I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $...
2
votes
0answers
42 views
Hadamard product derivative
If $\circ$ represents the Hadamard product, and $^*$ the conjugate-transpose operation. Given
$$f_{(\mathbf{x})} =(\mathbf{x} \circ \mathbf{x})^*H(\mathbf{x} \circ \mathbf{x}) - (\mathbf{x} \circ \...
1
vote
2answers
28 views
Hadamard product and being unitary
Let $A\in\mathbb{C}^{n\times n}$ and $A=B\circ B$ where $B$ is a unitary matrix and $\circ$ accounts for the Hadamard product. Can we say any thing about $A$ to be unitary or not?
1
vote
0answers
43 views
Diagonalization and the Hadamard product
Let $B \in \mathbb{C}^{n\times n}$ be unitarily diagonalizable such that $B=V\Lambda V^*$. Let $A=B\circ B$ where $\circ$ accounts for the Hadamard product. Then we can say that $A$ is also unitarily ...
0
votes
0answers
21 views
Conditionally (almost) definite matrices and Hadamard product
There seem to be different uses of the terminology. One says that a matrix $M$ is almost definite if $x'Mx=0 \Rightarrow Mx=0$. But here I am referring to a different one, that is there exist some $A$ ...
0
votes
0answers
59 views
Duality of the convolution theorem in Fourier Domain
Convolution in the time domain can be represented as a Hadamard (pointwise) product in the Frequency domain. Using the instructions specified at
https://in.mathworks.com/matlabcentral/answers/38066-...
0
votes
1answer
63 views
Derivation Numerical Method with partial derivatives, vectors, matrices and scalar product
I need help in finding a way to combine the equations
\begin{equation}
\frac{\partial J}{\partial W} \cdot \delta W = \langle Y_M^T (\eta^{'}(Y_M W) \odot (\eta(Y_MW)-C)),\delta W \rangle
\end{...
0
votes
0answers
48 views
Derivative of Hadamard product with respect to matrix
I'm trying to calculate this derivative wrt matrix $F_{i}$ and simplify the whole expression:
$ \frac{ \partial J_{term 1}}{\partial \mathbf{F_{i}}}=
2 (\sum_{j:G(i,j)=1} (\mathbf{W}_{i,j} \...
0
votes
0answers
27 views
Hadamard product and minimality of operator norm
Let $Y \in \mathbf{R}^{n \times n}$ be a given matrix, and let $1 \leq r \leq n$ be an integer.
Suppose that $A \in \mathbf{R}^{n \times n}$ is such that
$$
\|X - Y\| \geq \|A - Y\|, \quad \text{...
1
vote
1answer
46 views
Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$?
Is it true that $\|A\circ \bar{A}\|_2\leq\|AA^{\dagger}\|_2$? Here $\circ$ is the Hardamard product and $\|•\|_2$ is the Frobenius norm.
0
votes
1answer
59 views
Conditions for Hadamard to distribute with matrix multiplication
Let $A$ be an $n \times m$ matrix, and let $\circ$ be the Hadamard product. What are sufficient conditions on $A$ for the following to be true for all $m$-vectors $x$ and $y$?
$$
Ax \, \circ \, Ay = A(...
2
votes
1answer
47 views
Solution to $Mx-x^2=0$ where $x^2$ is the square of the elements of vector $x$
I have been trying to find the solutions for
$$Mx=x\circ x$$
where $\circ$ is the element wise product.
One solution is $x=0$. But there is another solution $x\neq 0$, if $M$ has a real positive ...
1
vote
2answers
195 views
Frobenius Norm of Hadamard Product and Trace
I'm trying to relate the Frobenius Norm of a Hadamard Product to a trace that does not include another Hadamard Product, if possible. In other words, if A and B are (sxr) matrices, with not all ...
0
votes
0answers
46 views
Solve System of Quadratic Equations with Hadamard Product from pde collocation
I have the following system of equations:
$ A x - z \cdot B x \cdot B y + c = 0$
$D y - z \cdot B x \cdot B y + d = 0$
Where I want to solve for $ x,y \in \mathbb{R}^N$. $A$, $B$ and $D$ are all ...
-2
votes
1answer
58 views
matrix gradient of the Hadamard product
What is the matrix gradient for the function $||A(B \circ X) ||_F^2 $ with respect to $X$. Here $A,B \in \mathbb{C}^{n \times n}$ and $X \in \mathbb{R}^{n \times n}$.($\circ$) is the Hadamard ...
0
votes
0answers
20 views
Prove that $\|\mathbf{A}\circ uv'\|_F = 1$ when $\mathbf{A}$ is sign matrix
I want to prove that
$$
\|\mathbf{A}\circ uv'\|_F = \|u\|_2 \, \|v\|_2 = 1
$$ when $\mathbf{A}$ is sign matrix.
My proof is as follows ($:$ denote the Frobenius product
$$
(\mathbf{A}\circ uv'):(\...
1
vote
0answers
106 views
rank inequality for Hadamard product
How I can show that
$$
\operatorname{rank}(A\circ B) \leq \operatorname{rank}(A)\operatorname{rank}(B)
$$
where $A,B$ are rectangular matrices and $\circ$ is the Hadamard product between the two.
...
0
votes
1answer
59 views
Derivative of a Matrix Expression with Hadamard
I have the following expression at hand:
$1_{nx1}^T (X_{nxk} \circ X_{nxk})(Y_{kx1} \circ Y_{kx1})$
$X$ is a matix, $Y$ is a vector, and $1$ is a vector of ones with indicated dimensions. $\circ$ is ...
3
votes
3answers
207 views
Can Hadamard (schur) product $A \circ B$ be positive definite, if $A \succeq 0$ (positive semi-definite) and $B \succ 0$ (positive definite)?
Dear Linear Algebra experts,
According to Schur Product Theorem, if both $A \succ 0$ and $B \succ 0$ are positive definite, then the Hadamard product of $(A \circ B) \succ 0$ is also positive ...
2
votes
1answer
347 views
Matrix notation for element-wise raising to the power of $n$
The Hadamard product $A \odot B$ gives the element-wise multiplication of matricies $A$ and $B$.
How do I denote the raising a matrix to the power $n$, element-wise?
0
votes
1answer
155 views
Derivative of trace involving inverse and Hadamard product
Let $A, B$ be symmetric $(n \times n)$ matrices and let $A$ be invertible. I am looking for the derivative
$$ \frac{\partial}{\partial A} \operatorname{tr}[A^{-1}(A \odot B)], $$
where $\odot$ is the ...
1
vote
1answer
232 views
Derivative of Frobenius norm of Hadamard Product
I am trying to find:
$\frac{\partial}{\partial A} \left||Q\circ C \right||^2_F $
$\quad$ and $\quad$
$\frac{\partial}{\partial B} \left||Q\circ C \right||^2_F $
where
$C= B A^T$
C is (pxq) ...
0
votes
1answer
33 views
Determinant defined as Product of Columns
Let $N$ be a non-singular matrix, $v_i$ be the column $i$ of $N$, and $M$ be a matrix with $e_i$ as columns.
$M$ and $N$ have the same dimensions.
I do not understand how
$|\det(N)|=\prod_i ||v_i|...
0
votes
1answer
25 views
Check Reasoning On Calculation Involving Diagonal Matrix and Matrix and Hadamard Products
I apologize in advance, as I kind of realize this is a dumb question. But I need a little more mathematical rigor to my naive logic.
Suppose I have an $n \times n$ diagonal matrix, $\mathbf{A}$, ...
0
votes
1answer
201 views
solve matrix equation involving Hadamard products
I'm trying to solve the equation
$$
(\Sigma \circ C)^{-1} \circ C = \left[ (\Sigma \circ C)^{-1} S (\Sigma \circ C)^{-1} \right] \circ C
$$
for $\Sigma$ or $\Sigma \circ C$, where $\circ$ denotes the ...
1
vote
0answers
42 views
Relation for the determinant of a special Hadamard product.
Inspired by this very related question.
Let $A$ be a circulant matrix and $X$ an anti-circulant matrix of size $n\times n$. Moreover let the sum of each row in $X$ be zero, $\sum_i x_{ij}=0$. It ...
0
votes
1answer
30 views
inequality on matrix Hadamard Products $\|A \odot X\|_F$
I have two matrix with same size $A$ and $X$, $A$ is a binary matrix. $X$ is nonnegative matrix. Is there any inequality show that
$\|A \odot X\|_F <= f(A) \|X\|_F$, how to find $f(A)$?
my goal is ...
2
votes
0answers
18 views
Bounding the determinant of principal sub-matrices of the Kroneker product
I have a matrix $A$ that is 2 dimensional and has negative determinant. I have a matrix $B$ that is 2 dimensional and has a positive determinant. Both have strictly positive elements. I want to show ...
0
votes
1answer
366 views
Hadamard product of a positive semidefinite matrix with a negative definite matrix
If I have a positive semidefinite matrix $A$ and a negative definite matrix $B$, is it true that their Hadamard product $A\circ B$ is negative semidefinite? Ideally I am looking for a proof / a ...
0
votes
2answers
186 views
what is transposition Hadamard product?
$$F=||Q \circ X||_F^2
$$
where $\circ $ is hadamard product.
How I can convert it to style of general matrix multiplication?
1
vote
0answers
107 views
When does Hadamard product rank upper bound hold?
We know $rank(A\circ B)\leq rank(A)rank(B)$. Under what conditions do we have equality?
0
votes
1answer
231 views
How to deal with hadamard product of matrix with vector
$$Ax=b$$
$A$ is a real square symmetric matrix with all non-zero entries, $x$ and $b$ are vectors.
The hadamard inverse of $A$ is defined
$$A^{\circ-1} \circ A = E$$
where $E$ is a matrix of ones....
1
vote
1answer
114 views
Differentiating trace of matrix product when matrix elements are functions of a vector
According to a well known formula (Eqs. 100-104 here)
$$\frac{\partial}{\partial B} tr(AB)=A^T$$
For square real-valued matrices $A,B$. For simplicity assume these matrices are symmetric.
But...
1)...
0
votes
1answer
139 views
Derivative of trace of Hadamard and dot product
Im struggling solving an equation and I have tried to find solution in the matrix cookbook but did not find a clue. How can I calculate the derivative of the equation which is a combination of ...
3
votes
1answer
118 views
How to solve the linear equation $A\circ (XB) + CX = D$
How to solve the follow equation $X$ is the variable:
$A\circ (XB) + CX = D$
where $ A \circ B$ is element-wise product or Hadamard product.
if the $A = 1_{n \times n}$. the above equation become
$(...
0
votes
1answer
114 views
Second order gradient of a non-linear element-wise function and a Hadamard product
I am applying the mean value theorem to an analysis, but I am running into problems computing the second derivative.
let:
$$ J = d^Tf(Wx) $$
with $f$ a generic continuous and differentiable element-...
1
vote
0answers
46 views
Is sum of inverse of max a kernel?
For $X$ be a nonempty set, a function $f :X\times X\to\mathbb R$ is called a kernel on $X$ if for all $m\in \mathbb N$ and all $x_1,\cdots,x_m \in X$
$$K_f\equiv\begin{bmatrix}
f(x_1,x_1) & \cdots ...
2
votes
1answer
287 views
Algorithm for computing Hadamard product of two rational generating functions
If I have two generating functions $A(x) = \sum_na_nx^n$ and $B(x) = \sum_n b_nx^n$
then the Hadamard product is $(A \star B)(x) = \sum_{n} a_nb_nx^n$.
Now when $A$ and $B$ are both rational ...
1
vote
1answer
261 views
Derivative of Hadamard product with functions
I'm new to matrix derivatives, and I'm having a bit of trouble with this one in particular. I have this equation for the function:
$f(x) = M(g(x) ∘ g(x))$
Where M is a non-square matrix, '$∘$' is ...
0
votes
1answer
71 views
A is positive definite, B is positive semidefinite and all the diagonal element is positive, prove that Schur product their is positive definite
$A$ is positive definite, $B$ is positive semidefinite and all the diagonal element is positive, prove that $A\circ B$ is positive definite
It is easy to prove it with Oppenheim Inequality:
...
1
vote
1answer
99 views
Differentiate hadamard product of square matrix $S \odot VV^T \in R^{n \times n} $ over rectangular matrix $V \in R^{n \times r}$
I want to differentiate $f = \log\det(L)$ over $V$ where $L = S \odot VV^T$.
The thing that I know is $df = L^{-T} : dL = (S \odot VV^T) : (dS \odot VV^T+ S \odot (VdV^T+dVV^T ))$ where $:$is a ...
0
votes
1answer
195 views
Distributing matrix-vector product across a element-wise (hadamard) product
Given this equation, how do you solve for $c_3$ (in terms of $M$, $c_1$ and $c_2$)?
$(Mc_1 + e_1) \odot (Mc_2 + e_2) = Mc_3 + e_3$
I'm not sure how to algebraically distribute across the hadamard ...
-1
votes
1answer
213 views
Can anyone tell me what is the derivative of hadamard product
I want to know the derivative of ${\bf{(}}{{\bf{w}}^{\mathop{\rm H}\nolimits} }{\bf{A}} \odot {\bf{(}}{{\bf{w}}^{\mathop{\rm H}\nolimits} }{\bf{A)^{\bf{*}} - }}{{\bf{g}}^H}{\bf{)(}}{{\bf{A}}^H}{\bf{w}}...
