"(时任法国数学会Vice-会长) Hadamard 1936年曾受清華大學邀請至中國讲学3个多月。（同時期者尚有美國數學家诺伯特·维纳）。他为偏微分方程创造了适定性问题概念。他也給了其名字予论体积的阿达马不等式，还有阿达马矩阵，是阿达马变换所以建基的。量子计算的阿达马门使用这个矩阵。在阿达马所著的《数学领域的发明心理学》他用內省來描述数学思维过程。他描述他的数学思考大部分是无字的，往往有心象伴随著，浓缩了证明的整体思路。" -- https://zh.wikipedia.org/wiki/雅克·阿达马

https://zh.wikipedia.org/wiki/阿达马矩阵:

阿达马矩阵是一个方阵，每个元素都是 +1 或 −1，每行都是互相正交的。

在阿达马矩阵理论最重要的开放性问题是存在性的问题。即

阿达马猜想: 对于每个4的倍数 n = 4k，k 为自然数，都存在 n 阶的阿达马矩阵。

西尔维斯特构造法给出了阶数为1, 2, 4, 8, 16, 32 等等的阿达马矩阵，之后阿达马本人给出了阶数为12和20的阿达马矩阵。Raymond Paley 随后给出了任何q+1 阶的阿达马矩阵的方法，其中q 是任何模4为3的质数任意次幂。他也给出了形式为2(q+1)的阿达马矩阵的方法，其中q 是任何模4为1的质数任意次幂。Hadi Kharaghani 和 Behruz Tayfeh-Rezaie 2004年6月21日宣布他们构造出了428阶的阿达马矩阵。现在最小的尚未被构造出来的4k阶阿达马矩阵是668阶。

"The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is 92. A Hadamard matrix of this order was found using a computer by Baumert, Golomb, and Hall in 1962 at JPL. They used a construction, due to Williamson, that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.

As of 2008, there are 13 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known. They are: 668, 716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964." -- https://en.wikipedia.org/wiki/Hadamard_matrix

Practical applications:

• Olivia MFSK – an amateur-radio digital protocol designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands.

• Balanced repeated replication (BRR) – a technique used by statisticians to estimate the variance of a statistical estimator.

• Coded aperture spectrometry – an instrument for measuring the spectrum of light. The mask element used in coded aperture spectrometers is often a variant of a Hadamard matrix.

• Feedback delay networks – Digital reverberation devices which use Hadamard matrices to blend sample values

• Plackett–Burman design of experiments for investigating the dependence of some measured quantity on a number of independent variables.

• Robust parameter designs for investigating noise factor impacts on responses

• Compressed sensing for signal processing and undetermined linear systems (inverse problems)

• Quantum Hadamard gate

References:

[1] http://mathworld.wolfram.com/HadamardMatrix.html

[2] https://web.math.sinica.edu.tw/math_media/d184/18408.pdf (Hadamard 矩陣及其應用, 1893-1993), https://web.math.sinica.edu.tw/media/, https://web.math.sinica.edu.tw/bulletin/

[3] https://press.princeton.edu/titles/8342.html (Hadamard Matrices and Their Applications - K. J. Horadam - Chapter 1 Introduction) 

[4] http://blog.sciencenet.cn/blog-420554-1199513.html (Costas array https://en.wikipedia.org/wiki/Costas_array) 

[5] http://www.kepu.net.cn/gb/hlg100/page_1011.htm (著名法国数学家阿达玛（1865-1963）于1896证明了黎曼函数的部分猜想，并证明了极难的素数定理：命表示不超过x的素数个数, )