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document Upper/LowerTriangular
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and clarify that Symmetric/Hermitian are views as well

add references to Upper/LowerTriangular in the manual

Ref #22504
(cherry picked from commit c964958)
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@fredrikekre @ararslan
fredrikekre authored and ararslan committed Sep 15, 2017
1 parent 21dcfec commit cdb5335
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2 changes: 1 addition & 1 deletion base/linalg/cholesky.jl
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Expand Up @@ -160,7 +160,7 @@ end
chol(A) -> U
Compute the Cholesky factorization of a positive definite matrix `A`
and return the UpperTriangular matrix `U` such that `A = U'U`.
and return the [`UpperTriangular`](@ref) matrix `U` such that `A = U'U`.
# Example
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6 changes: 4 additions & 2 deletions base/linalg/symmetric.jl
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Expand Up @@ -8,7 +8,8 @@ end
"""
Symmetric(A, uplo=:U)
Construct a `Symmetric` matrix from the upper (if `uplo = :U`) or lower (if `uplo = :L`) triangle of `A`.
Construct a `Symmetric` view of the upper (if `uplo = :U`) or lower (if `uplo = :L`)
triangle of the matrix `A`.
# Example
Expand Down Expand Up @@ -57,7 +58,8 @@ end
"""
Hermitian(A, uplo=:U)
Construct a `Hermitian` matrix from the upper (if `uplo = :U`) or lower (if `uplo = :L`) triangle of `A`.
Construct a `Hermitian` view of the upper (if `uplo = :U`) or lower (if `uplo = :L`)
triangle of the matrix `A`.
# Example
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45 changes: 45 additions & 0 deletions base/linalg/triangular.jl
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Expand Up @@ -50,6 +50,51 @@ LowerTriangular(U::UpperTriangular) = throw(ArgumentError(
UpperTriangular(U::LowerTriangular) = throw(ArgumentError(
"cannot create an UpperTriangular matrix from a LowerTriangular input"))

"""
LowerTriangular(A::AbstractMatrix)
Construct a `LowerTriangular` view of the the matrix `A`.
# Example
```jldoctest
julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 9.0
julia> LowerTriangular(A)
3×3 LowerTriangular{Float64,Array{Float64,2}}:
1.0 ⋅ ⋅
4.0 5.0 ⋅
7.0 8.0 9.0
```
"""
LowerTriangular
"""
UpperTriangular(A::AbstractMatrix)
Construct an `UpperTriangular` view of the the matrix `A`.
# Example
```jldoctest
julia> A = [1.0 2.0 3.0; 4.0 5.0 6.0; 7.0 8.0 9.0]
3×3 Array{Float64,2}:
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 9.0
julia> UpperTriangular(A)
3×3 UpperTriangular{Float64,Array{Float64,2}}:
1.0 2.0 3.0
⋅ 5.0 6.0
⋅ ⋅ 9.0
```
"""
UpperTriangular

imag(A::UpperTriangular) = UpperTriangular(imag(A.data))
imag(A::LowerTriangular) = LowerTriangular(imag(A.data))
imag(A::UnitLowerTriangular) = LowerTriangular(tril!(imag(A.data),-1))
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58 changes: 29 additions & 29 deletions doc/src/manual/linear-algebra.md
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Expand Up @@ -144,29 +144,29 @@ specialized routines that are specially developed for particular matrix types.
The following tables summarize the types of special matrices that have been implemented in Julia,
as well as whether hooks to various optimized methods for them in LAPACK are available.

| Type | Description |
|:------------------------ |:-------------------------------------------------------------------------------- |
| [`Hermitian`](@ref) | [Hermitian matrix](https://en.wikipedia.org/wiki/Hermitian_matrix) |
| `UpperTriangular` | Upper [triangular matrix](https://en.wikipedia.org/wiki/Triangular_matrix) |
| `LowerTriangular` | Lower [triangular matrix](https://en.wikipedia.org/wiki/Triangular_matrix) |
| [`Tridiagonal`](@ref) | [Tridiagonal matrix](https://en.wikipedia.org/wiki/Tridiagonal_matrix) |
| [`SymTridiagonal`](@ref) | Symmetric tridiagonal matrix |
| [`Bidiagonal`](@ref) | Upper/lower [bidiagonal matrix](https://en.wikipedia.org/wiki/Bidiagonal_matrix) |
| [`Diagonal`](@ref) | [Diagonal matrix](https://en.wikipedia.org/wiki/Diagonal_matrix) |
| `UniformScaling` | [Uniform scaling operator](https://en.wikipedia.org/wiki/Uniform_scaling) |
| Type | Description |
|:------------------------- |:-------------------------------------------------------------------------------- |
| [`Hermitian`](@ref) | [Hermitian matrix](https://en.wikipedia.org/wiki/Hermitian_matrix) |
| [`UpperTriangular`](@ref) | Upper [triangular matrix](https://en.wikipedia.org/wiki/Triangular_matrix) |
| [`LowerTriangular`](@ref) | Lower [triangular matrix](https://en.wikipedia.org/wiki/Triangular_matrix) |
| [`Tridiagonal`](@ref) | [Tridiagonal matrix](https://en.wikipedia.org/wiki/Tridiagonal_matrix) |
| [`SymTridiagonal`](@ref) | Symmetric tridiagonal matrix |
| [`Bidiagonal`](@ref) | Upper/lower [bidiagonal matrix](https://en.wikipedia.org/wiki/Bidiagonal_matrix) |
| [`Diagonal`](@ref) | [Diagonal matrix](https://en.wikipedia.org/wiki/Diagonal_matrix) |
| `UniformScaling` | [Uniform scaling operator](https://en.wikipedia.org/wiki/Uniform_scaling) |

### Elementary operations

| Matrix type | `+` | `-` | `*` | `\` | Other functions with optimized methods |
|:------------------------ |:--- |:--- |:--- |:--- |:------------------------------------------------------------------- |
| [`Hermitian`](@ref) |   |   |   | MV | [`inv()`](@ref), [`sqrtm()`](@ref), [`expm()`](@ref) |
| `UpperTriangular` |   |   | MV | MV | [`inv()`](@ref), [`det()`](@ref) |
| `LowerTriangular` |   |   | MV | MV | [`inv()`](@ref), [`det()`](@ref) |
| [`SymTridiagonal`](@ref) | M | M | MS | MV | [`eigmax()`](@ref), [`eigmin()`](@ref) |
| [`Tridiagonal`](@ref) | M | M | MS | MV |   |
| [`Bidiagonal`](@ref) | M | M | MS | MV |   |
| [`Diagonal`](@ref) | M | M | MV | MV | [`inv()`](@ref), [`det()`](@ref), [`logdet()`](@ref), [`/()`](@ref) |
| `UniformScaling` | M | M | MVS | MVS | [`/()`](@ref) |
| Matrix type | `+` | `-` | `*` | `\` | Other functions with optimized methods |
|:------------------------- |:--- |:--- |:--- |:--- |:------------------------------------------------------------------- |
| [`Hermitian`](@ref) |   |   |   | MV | [`inv()`](@ref), [`sqrtm()`](@ref), [`expm()`](@ref) |
| [`UpperTriangular`](@ref) |   |   | MV | MV | [`inv()`](@ref), [`det()`](@ref) |
| [`LowerTriangular`](@ref) |   |   | MV | MV | [`inv()`](@ref), [`det()`](@ref) |
| [`SymTridiagonal`](@ref) | M | M | MS | MV | [`eigmax()`](@ref), [`eigmin()`](@ref) |
| [`Tridiagonal`](@ref) | M | M | MS | MV |   |
| [`Bidiagonal`](@ref) | M | M | MS | MV |   |
| [`Diagonal`](@ref) | M | M | MV | MV | [`inv()`](@ref), [`det()`](@ref), [`logdet()`](@ref), [`/()`](@ref) |
| `UniformScaling` | M | M | MVS | MVS | [`/()`](@ref) |

Legend:

Expand All @@ -178,15 +178,15 @@ Legend:

### Matrix factorizations

| Matrix type | LAPACK | [`eig()`](@ref) | [`eigvals()`](@ref) | [`eigvecs()`](@ref) | [`svd()`](@ref) | [`svdvals()`](@ref) |
|:------------------------ |:------ |:--------------- |:------------------- |:------------------- |:--------------- |:------------------- |
| [`Hermitian`](@ref) | HE |   | ARI |   |   |   |
| `UpperTriangular` | TR | A | A | A |   |   |
| `LowerTriangular` | TR | A | A | A |   |   |
| [`SymTridiagonal`](@ref) | ST | A | ARI | AV |   |   |
| [`Tridiagonal`](@ref) | GT |   |   |   |   |   |
| [`Bidiagonal`](@ref) | BD |   |   |   | A | A |
| [`Diagonal`](@ref) | DI |   | A |   |   |   |
| Matrix type | LAPACK | [`eig()`](@ref) | [`eigvals()`](@ref) | [`eigvecs()`](@ref) | [`svd()`](@ref) | [`svdvals()`](@ref) |
|:------------------------- |:------ |:--------------- |:------------------- |:------------------- |:--------------- |:------------------- |
| [`Hermitian`](@ref) | HE |   | ARI |   |   |   |
| [`UpperTriangular`](@ref) | TR | A | A | A |   |   |
| [`LowerTriangular`](@ref) | TR | A | A | A |   |   |
| [`SymTridiagonal`](@ref) | ST | A | ARI | AV |   |   |
| [`Tridiagonal`](@ref) | GT |   |   |   |   |   |
| [`Bidiagonal`](@ref) | BD |   |   |   | A | A |
| [`Diagonal`](@ref) | DI |   | A |   |   |   |

Legend:

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2 changes: 2 additions & 0 deletions doc/src/stdlib/linalg.md
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Expand Up @@ -18,6 +18,8 @@ Base.LinAlg.SymTridiagonal
Base.LinAlg.Tridiagonal
Base.LinAlg.Symmetric
Base.LinAlg.Hermitian
Base.LinAlg.LowerTriangular
Base.LinAlg.UpperTriangular
Base.LinAlg.lu
Base.LinAlg.lufact
Base.LinAlg.lufact!
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