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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Is There a Unitary Transformation Which Maps Non-Orthogonal Vectors into 'Uniform' Vectors?

I would like to find a unitary matrix $U$ which maps normalised vectors $\mathbf{v}^{(i)}$ from the set: \begin{equation} \mathcal{V} = \Bigg\{ \begin{pmatrix} 1 \\ 0\\ 0 \\ \vdots \end{pmatrix}, \...
300 views

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, ...
45 views

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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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 ...
69 views

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&...
40 views

Differentiation of the Hadamard Matrix Product

Thank you for taking the time to read my question. I've had a look at other posts regarding differentiation of Hadamard Products of matrices but none of the examples are the same as the one I am ...
26 views

Using Hadamard's maximal determinant what are the the radii $R$, $r$ for an ellipsoid given a linear program?

Let $Ax \leq b$ be the $m$ inequalities of a linear program in $n$ variables and all entries of $A,b$ be from -1, 0,1. What values of $r, R$ can be supplied to the Ellipsoid algorithm so that when it ...
56 views

Suggestions regarding Hadamard Matrices

I am a grade 12 student, and for this semester, I am interested in doing a math research paper instead of a science one. I've looked up the internet regarding the different topics I can tackle on, but ...
57 views

Legendre symbol for constructing Hadamard matrix

If we have to construct Hadamard matrix of order n, and n is power of 2, we can use Kronecker mutiplication of matrices. I have heard, that in case of arbitrary n (divisible by 4, of course) we can ...
36 views

Prove $\det (A)\le \left(\max_{i,j}|A_{ij}|\right)^n n^{n/2}$

Prove $\det (A)\le \left(\max_{i,j}|A_{ij}|\right)^n n^{n/2}$ How do I prove this as an extension of the Hadamard Inequality? This expression is given as a corollary of the Hadamrd inequality in ...
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 ...
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} \...