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Hadamard

Hadamard is a surname most commonly associated with Jacques Salomon Hadamard (1865–1963), a French mathematician whose work spanned analysis, number theory, and differential equations. The name is also attached to several mathematical concepts named in his honor.

Jacques Hadamard made foundational contributions to the theory of partial differential equations and complex analysis. He

Hadamard matrix: a square matrix with entries ±1 whose rows are mutually orthogonal. Such matrices have order

Hadamard transform: the discrete transform defined by a Hadamard matrix, used in digital signal processing and

Hadamard product: the elementwise product of two matrices, denoted A ∘ B. It is widely used in matrix

Hadamard inequality: for an n×n matrix A with row vectors a_i, the absolute value of the determinant

Hadamard space: a complete metric space with nonpositive curvature in the sense of Alexandrov; it appears in

introduced
the
notion
of
a
well-posed
problem,
developed
maximum
principles
and
estimates,
and
proved
results
on
the
propagation
of
singularities
for
hyperbolic
equations.
He
also
formulated
the
Hadamard
three-lines
theorem
in
complex
analysis.
n
equal
to
1,
2,
or
a
multiple
of
4.
The
Hadamard
conjecture
posits
that
a
Hadamard
matrix
exists
for
every
order
n
that
is
a
multiple
of
4.
data
compression.
The
fast
Hadamard
transform
exploits
structure
to
run
in
O(n
log
n)
time
for
sizes
that
are
powers
of
two.
analysis
and
preserves
certain
positivity
properties
under
appropriate
conditions.
satisfies
|det(A)|
≤
∏
||a_i||,
where
||a_i||
is
the
Euclidean
norm
of
the
i-th
row.
This
provides
a
bound
on
determinants
based
on
row
lengths.
geometry
and
geometric
group
theory.