# Questions tagged [hadamard-matrices]

In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either $+1$ or $−1$ and whose rows are mutually orthogonal.

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### Apply the Hadamard transform to different state vectors

I am in the process of understanding a proof. First, the following is said there: $$H\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix}=\frac{1}{\sqrt{N}}\begin{pmatrix}1\\1\\1\\1\end{pmatrix}$$ This is ...
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### Hadamard transforms seen as rotations in higher dimensions

The Hadamard transform – i.e. the multiplication of a vector $x \in \mathbb{R}^{N}$ (with $N = 2^n$) by the Hadamard matrix $H_n$ – yields another vector $k = H_n x$ in $\mathbb{R}^{N}$. ...
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### Inductive proof of $D_N=-H_n\cdot R_N \cdot H_n$ (Grover Iterator)

I am currently working on an inductive proof $D_N=-H_n\cdot R_N \cdot H_n$ (H is the Hadamard matrix for n Bits), the induction assumption (base case) and the induction condition have been done by me,...
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### Constructing Symmetric Hadamard Matrices

I'm trying to understand the exact degrees of freedom involved in constructing Hadamard matrices with elements in $\{1, -1\}$, and if possible, reduce them in such a way that they can be ...
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### Is there any proof that there doesn't exist a Hadamard matrix of size $6 \times 6$?

A matrix $H \in {\pm 1}^n$ is Hadamard matrix if $HH^T=nI_n$, where $I$ is $n\times n$ identity matrix. Hadamard's conjecture said that there exists Hadamard matrix of order 1,2 or $4n$, for every ...
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### Matrices with nearly balanced matches and mismatches between columns

I am interested in binary matrices with a near balanced number of matched and mismatched entries between columns. For example, a Hadamard matrix has a perfect balance between matched and mismatched ...
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### Fast calculation of the Hadamard transform of a 4X4 matrix

I'm reading a book envolves Hadamard transformations, and in one of the examples they are trying to reconstruct Y from RX+Z where all are 4X4 matrixes. R is given by: R= 1/4 * \begin{array}{l}5&3&...
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### Efficient matrix-vector multiplication for “partial” Hadamard matrices

I've recently been working on an algorithm for bilinear systems in the form $y = (Lw) \odot (Rx)$, where $\odot$ denotes the elementwise product between two matrices of compatible sizes. In the above, ...
Definition: Let a field $\mathbb{F}$. Consider an $2^n \times 2^n$ matrix $\bf H$ over $\mathbb{F}$. $\bf H$ is called Hadamard over $\mathbb{F}$ if and only if  {\bf H}=\left( \begin{array}{cc}...