Questions tagged [hadamard-matrices]

Question 1

I'm trying to prove the relationship $QQ^\top=qI-J$ where $Q$ is a Jacobsthal matrix, in order to understand the Paley constructions of Hadamard matrices. Even when I assume for that $q$ is prime ...

Question 2

I am a network optimization engineer working on space-time block code diversity problems. In this field, Hadamard matrices serve as a fundamental building block for creating space-time diversity ...

Question 3

Background While studying two-level full factorial designs and Hadamard matrices, I've noticed several commonalities between these mathematical structures. In particular, both involve matrices with ...

Question 4

Background Let $\{a_n\}_{n=1}^\infty$ be a strictly-increasing sequence of positive integers. We define the density of this sequence in the set of natural numbers as follows: $$ \mu=\lim_{n\to\infty} \...

Question 5

I have read the Wikipedia article about Hadamard matrices that says: Let $H$ be a Hadamard matrix of order $n$, the following is true: $H H^\textsf{T} = n I_n$, where $I_n$ is the identity $n×n$ ...

Question 6

Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2 ,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...

Question 7

The Paley 1 construction method allows to construct an Hadamard matrix (square matrices with orthogonal columns and entries equal to $1$ or $-1$ ) of order N = q+1 where q = 3 (mod 4) is a prime power:...

Question 8

Let $H = (h_{ij})$ be a square matrix of order $n$ such that $|h_{ij}| \leq 1$. Then, by the Hadamard Determinant inequality we know that $$|\det(H)| \leq n^{\frac{n}{2}}$$ I read in this paper that, ...

Question 9

A complex Hadamard matrix of order $N$ is a square matrix $A = [a_{ij}]$ of size $N$ with entries $|a_{ij}| = 1$ for all $i,j$; and further that $A^\dagger A= N I$ (or $A/\sqrt{N} $ is a unitary ...

Question 10

We consider a complex Hadamard matrix $H$ with entries in $\{\pm 1, \pm i\}$. I have seen some readings that often write $H = A + iB$ for matrices $A$ and $B$ with entries in $\{0,\pm 1\}$ where $A \...

Question 11

Can I get some useful links or insight for reading up on Hadamard's Maximum Determinant Problem but with complex entries? What this means is that the maximal determinant is the absolute of the complex ...

Question 12

Hadamard matrices are matrices such $H * H^T = nE$, and all columns are pairwise orthogonal(and all raws are pairwise orthogonal) The Hadamard matrices are equivalent(~) if it is possible to obtain ...

Question 13

A Hadamard matrix of order $n$ is an $n \times n$ matrix with entries $\pm1$ such that any two rows are mutually orthogonal. Any Hadamard matrix must necessarily have order equal to $1$, $2$, or a ...

Question 14

I asked a question about tall semi-orthogonal matrices with zero sum over individual columns and how to arrive at them. This question follows up to deal with binary matrices of the same form and ...

Question 15

More generally, is there a simple condition for which $n$ there are symmetric Hadamard matrices of order $n$? This set of $n$ is closed under multiplication via the Kronecker product.

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Questions tagged [hadamard-matrices]

Jacobsthal matrix used to prove Paley construction

What is the most efficient way to verify whether a given square matrix is a Hadamard matrix?

What is the relationship between two-level full factorial designs and Hadamard matrices?

Non-zero-set construction of Hadamard matrices

Proof about Hadamard matrices that $H H^\textsf{T} = n I_n$ [closed]

Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices

Hadamard matrix of order 28 via Paley 1 construction [closed]

Showing only if for the Hadamard Matrix Equality case with |a| < 1

Spectrum of complex Hadamard matrix

Unique construction of Hadamard matrices

Request for links: Hadamard's Maximum Determinant Problem

Are all Hadamard matrices equivalent to their transposes?

Can there be more than $\log_2(n)$ mutually orthogonal $(\pm1)$-vectors $x$ in $\mathbb{R}^n$ such that $x_{2k-1} \neq x_{2k}$ for all $k$?

Semi-orthogonal tall binary matrices with zero column sum

Is there a 12x12 symmetric Hadamard matrix? [closed]

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