Internal documentation

eigen_safe(x)

Compute the eigendecomposition of x. If x is a StaticMatrix, it is converted to a Matrix before the decomposition.

isnormal(x; kwargs...) -> Bool

Check if the matrix or number x is normal, that is, if it commutes with its adjoint:

\[x x^\mathrm{H} = x^\mathrm{H} x.\]

By default, this is an equality check. Provide kwargs for isapprox to perform an approximate check.

log_safe(x)

Compute the matrix logarithm of x. If x is a StaticMatrix, it is converted to a Matrix before computing the log.

log_safe!(y, x)

Compute the matrix logarithm of x. If the eltype of y is real, then the imaginary part of x is ignored, and a DomainError is raised if real(x) has no real logarithm.

mul!_safe(Y, A, B) -> Y

Call mul! safely, that is, A and/or B are permitted to alias with Y.

nzsign(z[, absz])

Compute a modified sign(z) that is always nonzero, i.e. where

\[\operatorname(nzsign)(z) = \begin{cases} 1 & \text{if } z = 0\\ \frac{z}{|z|} & \text{otherwise} \end{cases}\]

realify(X::AbstractMatrix{T𝔽}, 𝔽::AbstractNumbers) -> Y::AbstractMatrix{<:Real}

Given a matrix $X ∈ 𝔽^{n×n}$, compute $Y ∈ ℝ^{m×m}$, where $m = n \operatorname{dim}_𝔽$, and $\operatorname{dim}_𝔽$ is the real_dimension of the number field $𝔽$, using the map $ϕ \colon X ↦ Y$, that preserves the matrix product, so that for all $C,D ∈ 𝔽^{n×n}$,

\[ϕ(C) ϕ(D) = ϕ(CD).\]

See realify! for an in-place version, and unrealify! to compute the inverse of $ϕ$.

realify!(Y::AbstractMatrix{<:Real}, X::AbstractMatrix{T𝔽}, 𝔽::AbstractNumbers)

In-place version of realify.

realify!(Y::AbstractMatrix{<:Real}, X::AbstractMatrix{<:Complex}, ::typeof(ℂ))

Given a complex matrix $X = A + iB ∈ ℂ^{n×n}$, compute its realified matrix $Y ∈ ℝ^{2n×2n}$, written where

\[Y = \begin{pmatrix}A & -B \\ B & A \end{pmatrix}.\]

symmetrize(X)

Given a square matrix X compute 1/2 .* (X' + X).

symmetrize!(Y, X)

Given a square matrix X compute 1/2 .* (X' + X) in place of Y.

unrealify!(X::AbstractMatrix{T𝔽}, Y::AbstractMatrix{<:Real}, 𝔽::AbstractNumbers[, n])

Given a real matrix $Y ∈ ℝ^{m×m}$, where $m = n \operatorname{dim}_𝔽$, and $\operatorname{dim}_𝔽$ is the real_dimension of the number field $𝔽$, compute in-place its equivalent matrix $X ∈ 𝔽^{n×n}$. Note that this function does not check that $Y$ has a valid structure to be un-realified.

See realify! for the inverse of this function.

usinc(θ::Real)

Unnormalized version of sinc function, i.e. $\operatorname{usinc}(θ) = \frac{\sin(θ)}{θ}$. This is equivalent to sinc(θ/π).

usinc_from_cos(x::Real)

Unnormalized version of sinc function, i.e. $\operatorname{usinc}(θ) = \frac{\sin(θ)}{θ}$, computed from $x = cos(θ)$.

vec2skew!(X, v, k)

Create a skew symmetric matrix in-place in X of size $k×k$ from a vector v, for example for v=[1,2,3] and k=3 this yields

[  0  1  2;
  -1  0  3;
  -2 -3  0
]