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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How to find a particular complex Hadamard matrix?

In a book whose title I don’t remember I read the following problem from S.W. Drury. $U^*$ denotes the conjugate transpose of $U$. Show that every $5 \times 5$ matrix $U$ with complex entries $u_{j,k}$...
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How do you express a function in terms of the Hadamard basis, and what is a Hadamard basis?

I solved problem B2 from the 2014 Putnam competition past paper, however looking at the solution, a reformulation of the question stated that an integrable function could be expressed in terms of a ...
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Small question about symmetric Hadamard matrices

I am trying to understand Hadamard matrix. My question is simply about whether Hadamard matrix can always be made equivalent (by multiplying row/column by -1 or swapping rows/columns) to a symmetric ...
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Proof that Hadamard matrices of order $4k+2$ don't exist

It's known that Hadamard matrices can only exist for orders $1$, $2$ and $4k$. It's easy to show that there are no Hadamard matrices of order $2k+1$. But what is the proof that there are no Hadamard ...
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Another Hadamard matrix of order 4?

Wikipedia states that there is, up to equivalence, a unique Hadamard matrix of order 4, namely $$ \def\p{\phantom+} \begin{pmatrix} \p1&\p1&\p1&\p1 \\ \p1&-1&\p1&-1 \\ \p1&...
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Sylvester matrix column orthogonality

I wanted to prove by induction that all sylvester Matrices defined below have orthogonal columns, which seems trivial to see but I wanted a rigorous proof. $H_0$ = [1] , $H_{i+1}$ = \begin{matrix} ...
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Can the Kronecker product for the Hadamard matrix with itself be written as a matrix multiplication?

The Hadamard is a two by two matrix: $\begin{equation} \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix} \end{equation} $ The Kronecker product of the ...
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Small question regarding Hadamard matrix of order $4m$

I am curious about something after reading stuff about Hadamard matrix conjecture. What I understand is that all of the $4m$ rows are orthogonal. Is it possible that for some $m$ such that the ...
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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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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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Are regular Hadamard matrices symmetric?

I am trying to show that a regular Hadamard matrix must have order $m^2$ for some integer $m$. So far I have found that if $H$ is an $r$-regular Hadamard matrix of order $n$, then $HJ = rJ$ and $HH^...
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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, ...
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Hadamard Form of a Circulant Matrix

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}...
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How to prove the following complex number is not $0$?

Let $n$ be a natural number and $n\geq2$. Denote $\sigma=\exp\left(\dfrac{2\pi i}{n+1}\right)$ and $\omega=\exp\left(\dfrac{2\pi i}{n}\right)$ and $\{\phi_1,\cdots,\phi_n, \psi_1,\cdots,\psi_n\}$ be $...
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Determinant of a Hadamard Matrix as a function of n?

A Hadamard matrix $H$ is a matrix with entries $\pm1$ and orthogonal columns. Given that the matrix is nxn, I got that the determinant is $2^n\times4$. However, this is clearly not correct since the ...
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How many unit vectors can be “close to mutually opposite” in a high dimensional space?

This is actually a follow-up of this previous question: How many vectors can be "close to mutual orthogonal like 80 degrees" in a high dimensional space? To be specific: "almost opposite" ...
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Is there any proof that there doesn't exist a circulant Hadamard matrix of size $8 \times 8$?

Is there any proof that there doesn't exist an $8 \times 8$ circulant Hadamard matrix? A matrix $H \in \{\pm 1\}^{n \times n}$ is Hadamard if $H H^T = n I$, where $I$ is the $n \times n$ identity ...
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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 ...
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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}, \...
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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 ...
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How to get the value of Hadamard matrix given its column and row index?

Please refer to https://en.wikipedia.org/wiki/Hadamard_matrix for the Sylvester's construction of Hadamard matrix. Given a Hadamard matrix $H\in R^{n\times n}$, then how to get the value of $H_{ij}$ ...
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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 ...
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Existence of permutation matrix for a given Hadamard matrix and a permutation matrix. [closed]

For a given Hadamard matrix H and a permutation matrix P, is there a permutation matrix P' such that PHP'=H? H may be any order(2,4,8,12,16,20,24,..ect) where it exists. If it exists list all such P' ...
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Known classes of Hadamard matrices

In the book Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices by Wallis et al., Appendix A of the chapter on Hadamard matrices gives a list of known classes of Hadamard matrices. However, ...
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spectrum of the Hadamard matrices

A (±1)-matrix is a matrix whose entries are 1 and −1. An $n \times n$ (±1)-matrix is called an Hadamard matrix if the rows are orthogonal. Equivalently, An $n \times n$ (±1)-matrix $H$ is Hadamard ⇔ $...
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Kronecher product of two Hadamard matrices is Hadamard matrix

An $n\times n$ matrix $A$ is called a Hadamard matrix if all of its entries are $\pm 1$ and $AA^T=nI$. Show that the Kronecker product $A\otimes B$ of two $n \times n$ Hadamard matrices $A$ and $B$ is ...
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Can (complex) Hadamard matrices of order $n!$ be related to symmetric groups?

For quite a time I wrap my mind about the following: Can (complex) Hadamard matrices of order $n!$ be related to symmetric groups? What I tried so far is some numerics and a lot of unfruitful ...
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Construction of Hadamard Matrices of Order $n!$

I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power. Since $$ n!-1 \bmod 4 = ...