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link Hadamard.jl

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Steven G. Johnson 2023-02-21 07:39:02 -05:00
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@ -69,7 +69,7 @@ Some important examples of unitary matrices:
* [2×2 rotation matrices](https://en.wikipedia.org/wiki/Rotation_matrix)
* the identity matrix I
* any [permutation matrix](https://en.wikipedia.org/wiki/Permutation_matrix) P which re-orders a vector, and is simply a re-ordering of the rows/cols of I
* [Hadamard matrices](https://en.wikipedia.org/wiki/Hadamard_matrix): unitary matrices Hₙ/√n where Hₙ has entries of ±1 only. For n=2ᵏ they are easy to construct recursively, and are known as [Walsh–Hadamard transforms](https://en.wikipedia.org/wiki/Hadamard_transform).
* [Hadamard matrices](https://en.wikipedia.org/wiki/Hadamard_matrix): unitary matrices Hₙ/√n where Hₙ has entries of ±1 only. For n=2ᵏ they are easy to construct recursively, and are known as [Walsh–Hadamard transforms](https://en.wikipedia.org/wiki/Hadamard_transform). (See also the Julia [Hadamard package](https://github.com/JuliaMath/Hadamard.jl).)
* discrete [Haar wavelets](https://en.wikipedia.org/wiki/Haar_wavelet), which are unitary after a diagonal scaling and consist of entries ±1 and 0. They are a form of ["time-frequency analysis"](https://en.wikipedia.org/wiki/Time%E2%80%93frequency_analysis) because they reveal information about *both* how oscillatory a vector is ("frequency domain") and *where* the oscillations occur ("time domain").
* orthonormal eigenvectors can be found for any real-symmetric ("Hermitian") matrix A=Aᵀ: A=QΛQᵀ
* the [SVD](https://en.wikipedia.org/wiki/Singular_value_decomposition) A=UΣVᵀ of *any* matrix A gives (arguably) the "best" orthonormal basis U for C(A) and the "best" orthonormal basis V for C(Aᵀ), which reveal a lot about A.