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# Questions tagged [hadamard-product]

For questions about Hadamard product between two matrices, or it can concern analytic functions.

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### Hadamard's theorem proof related steps

$$\int_0^{2 \pi} \frac{\rho e^{i \theta} d \theta}{\left(\rho e^{i \theta}-z\right)^{h+2}}=\frac{-i}{h+1}\left[\frac{1}{\left(\rho e^{i \theta}-z\right)^{h+1}}\right]_0^{2 \pi}=0$$ $\therefore$ We ...
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### Definition of Hadamard product between matrix and vector

I have read here that: $$M \circ \vec{v} = \operatorname{Diag}(\vec{v}) \, M$$ That is, the Hadamard product between an $n\times m$ matrix and an $n \times 1$ vector is equivalent to the dot product ...
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### Does matrix multiplication distribute over the Hadamard product?

I know that the Hadamard product is distributive over addition. But suppose that $A, B \in \mathbb{R}^{n \times n}$ and $v \in \mathbb{R}^n$. Then can we say $$v^T (A \circ B) v = v^TAv \cdot v^TBv$$ ...
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### Hadamard (element-wise multiplication) product rank

I am having some problems on understanding an inequality regarding the rank of the Hadamard product (element-wise product). I have $B=A\circ A$ where $A$ is a $n\times r$ matrix, and $\circ$ is the ...
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### An invertible matrix is orthogonal if and only if the inverse is equal to the transpose on nonzero elements

Let $A$ be an invertible real matrix, and suppose that $(A^{-1})_{i,j} = (A^{T})_{i,j}$ whenever $(A^{T})_{i,j}\ne 0$. Is it true that $A$ is orthogonal? I found this statement in a paper without ...
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