# Types and methods

## Types and their constructors

None of the types below need to be constructed directly; they arise from performing operations between `LinearMap`

objects or by calling the `LinearMap`

constructor described next.

`LinearMap`

Abstract supertype

`LinearMaps.LinearMap`

— Type```
LinearMap(A::LinearMap; kwargs...)::WrappedMap
LinearMap(A::AbstractVecOrMatOrQ; kwargs...)::WrappedMap
LinearMap(J::UniformScaling, M::Int)::UniformScalingMap
LinearMap{T=Float64}(f, [fc,], M::Int, N::Int = M; kwargs...)::FunctionMap
LinearMap(A::MapOrVecOrMat, dims::Dims{2}, index::NTuple{2, AbstractVector{Int}})::EmbeddedMap
LinearMap(A::MapOrVecOrMat, dims::Dims{2}; offset::Dims{2})::EmbeddedMap
```

Construct a linear map object, either

- from an existing
`LinearMap`

or`AbstractVecOrMat`

/`AbstractQ`

`A`

, with the purpose of

redefining its properties via the keyword arguments `kwargs`

, see below;

- a
`UniformScaling`

object`J`

with specified (square) dimension`M`

; - from a function or callable object
`f`

; - from an existing
`LinearMap`

or`AbstractVecOrMat`

/`AbstractQ`

`A`

, embedded in a larger zero map.

In the case of item 3, one also needs to specify the size of the equivalent matrix representation `(M, N)`

, i.e., for functions `f`

acting on length `N`

vectors and producing length `M`

vectors (with default value `N=M`

). Preferably, also the `eltype`

`T`

of the corresponding matrix representation needs to be specified, i.e., whether the action of `f`

on a vector will be similar to, e.g., multiplying by numbers of type `T`

. If not specified, the devault value `T=Float64`

will be assumed. Optionally, a corresponding function `fc`

can be specified that implements the adjoint (or transpose in the real case) of `f`

.

The keyword arguments and their default values are:

`issymmetric::Bool = false`

: whether`A`

or`f`

act as a symmetric matrix`ishermitian::Bool = issymmetric & T<:Real`

: whether`A`

or`f`

act as a Hermitian matrix`isposdef::Bool = false`

: whether`A`

or`f`

act as a positive definite matrix.

For existing linear maps or matrices `A`

, the default values will be taken by calling internal functions `_issymmetric`

, `_ishermitian`

and `_isposdef`

on the existing object `A`

. These in turn dispatch to (overloads of) `LinearAlgebra`

's `issymmetric`

, `ishermitian`

, and `isposdef`

methods whenever these checks are expected to be computationally cheap or even known at compile time as for certain structured matrices, but return `false`

for generic `AbstractMatrix`

types.

For the function-based constructor, there is one more keyword argument:

`ismutating::Bool`

: flags whether the function acts as a mutating matrix multiplication`f(y,x)`

where the result vector`y`

is the first argument (in case of`true`

), or as a normal matrix multiplication that is called as`y=f(x)`

(in case of`false`

). The default value is guessed by looking at the number of arguments of the first occurrence of`f`

in the method table.

For the `EmbeddedMap`

constructors, `dims`

specifies the total dimensions of the map. The `index`

argument specifies two collections of indices `inds1`

and `inds2`

, such that for the big zero map `L`

(thought of as a matrix), one has `L[inds1,inds2] == A`

. In other words, `inds1`

specifies the output indices, `inds2`

specifies the input indices. Alternatively, `A`

may be shifted by `offset`

, such that (thinking in terms of matrices again) `L[offset[1] .+ axes(A, 1), offset[2] .+ axes(A, 2)] == A`

.

`FunctionMap`

Type for wrapping an arbitrary function that is supposed to implement the matrix-vector product as a `LinearMap`

; see above.

`LinearMaps.FunctionMap`

— Type`FunctionMap{T,iip}(f, [fc,], M, N = M; kwargs...)`

Construct a `FunctionMap`

object from a function or callable object `f`

that represents a linear map of size `(M, N)`

, where `N`

can be omitted for square operators of size `(M,M)`

. Furthermore, the `eltype`

`T`

of the corresponding matrix representation needs to be specified, i.e., whether the action of `f`

on a vector will be similar to, e.g., multiplying by numbers of type `T`

. Optionally, a second function `fc`

can be specified that implements the adjoint (or transpose in the real case) of `f`

.

Accepted keyword arguments and their default values are as in the `LinearMap`

constructor.

**Examples**

```jldoctest julia> F = FunctionMap{Int64,false}(cumsum, 2) 2×2 FunctionMap{Int64,false}(cumsum; issymmetric=false, ishermitian=false, isposdef=false)

julia> F * ones(Int64, 2) 2-element Vector{Int64}: 1 2

julia> Matrix(F) 2×2 Matrix{Int64}: 1 0 1 1

`WrappedMap`

Type for wrapping an `AbstractMatrix`

or `LinearMap`

and to possible redefine the properties `isreal`

, `issymmetric`

, `ishermitian`

and `isposdef`

. An `AbstractMatrix`

will automatically be converted to a `WrappedMap`

when it is combined with other `LinearMap`

objects via linear combination or composition (multiplication). Note that `WrappedMap(mat1)*WrappedMap(mat2)`

will never evaluate `mat1*mat2`

, since this is more costly than evaluating `mat1*(mat2*x)`

and the latter is the only operation that needs to be performed by `LinearMap`

objects anyway. While the cost of matrix addition is comparable to matrix-vector multiplication, this too is not performed explicitly since this would require new storage of the same amount as of the original matrices.

`ScaledMap`

Type for representing a scalar multiple of any `LinearMap`

type. A `ScaledMap`

will be automatically constructed if real or complex `LinearMap`

objects are multiplied by real or complex scalars from the left or from the right.

`LinearMaps.ScaledMap`

— TypeLazy representation of a scaled map `λ * A = A * λ`

with real or complex map `A <: LinearMap{RealOrComplex}`

and real or complex scaling factor `λ <: RealOrComplex`

.

`UniformScalingMap`

Type for representing a scalar multiple of the identity map (a.k.a. uniform scaling) of a certain size `M=N`

, obtained simply as `LinearMap(λI, M)`

, where `I`

is the `LinearAlgebra.UniformScaling`

object. The type `T`

of the resulting `LinearMap`

object is inferred from the type of `λ`

. A `UniformScalingMap`

of the correct size will be automatically constructed if `LinearMap`

objects are multiplied by scalars from the left or from the right (respecting the order of multiplication), if the scalar `λ`

is either real or complex.

`LinearCombination`

, `CompositeMap`

, `TransposeMap`

and `AdjointMap`

Used to add/multiply/transpose/adjoint `LinearMap`

objects lazily, don't need to be constructed explicitly.

`Base.:+`

— Method`+(A::LinearMap, B::LinearMap)::LinearCombination`

Construct a (lazy) representation of the sum/linear combination of the two operators. Sums of `LinearMap`

/`LinearCombination`

objects and `LinearMap`

/`LinearCombination`

objects are reduced to a single `LinearCombination`

. In sums of `LinearMap`

s and `AbstractMatrix`

/`UniformScaling`

objects, the latter get promoted to `LinearMap`

s automatically.

**Examples**

```
julia> CS = LinearMap{Int}(cumsum, 3)::LinearMaps.FunctionMap;
julia> LinearMap(ones(Int, 3, 3)) + CS + I + rand(3, 3);
```

`Base.:*`

— Method`*(A::LinearMap, B::LinearMap)::CompositeMap`

Construct a (lazy) representation of the product of the two operators. Products of `LinearMap`

/`CompositeMap`

objects and `LinearMap`

/`CompositeMap`

objects are reduced to a single `CompositeMap`

. In products of `LinearMap`

s and `AbstractMatrix`

/`UniformScaling`

objects, the latter get promoted to `LinearMap`

s automatically.

**Examples**

```
julia> CS = LinearMap{Int}(cumsum, 3)::LinearMaps.FunctionMap;
julia> LinearMap(ones(Int, 3, 3)) * CS * I * rand(3, 3);
```

`Base.transpose`

— Method`transpose(A::LinearMap)`

Construct a lazy representation of the transpose of `A`

. This can be either a `TransposeMap`

wrapper of `A`

, or a suitably redefined instance of the same type as `A`

. For instance, for a linear combination of linear maps $A + B$, the transpose is given by $A^⊤ + B^⊤$, i.e., another linear combination of linear maps.

`Base.adjoint`

— Method`adjoint(A::LinearMap)`

Construct a lazy representation of the adjoint of `A`

. This can be either a `AdjointMap`

wrapper of `A`

, or a suitably redefined instance of the same type as `A`

. For instance, for a linear combination of linear maps $A + B$, the adjoint is given by $A^* + B^*$, i.e., another linear combination of linear maps.

`KroneckerMap`

and `KroneckerSumMap`

Types for representing Kronecker products and Kronecker sums, resp., lazily.

`Base.kron`

— Method```
kron(A::LinearMap, B::LinearMap)::KroneckerMap
kron(A, B, Cs...)::KroneckerMap
```

Construct a (lazy) representation of the Kronecker product `A⊗B`

. One of the two factors can be an `AbstractMatrix`

, which is then promoted to a `LinearMap`

automatically.

To avoid fallback to the generic `Base.kron`

in the multi-map case, there must be a `LinearMap`

object among the first 8 arguments in usage like `kron(A, B, Cs...)`

.

For convenience, one can also use `A ⊗ B`

or `⊗(A, B, Cs...)`

(typed as `\otimes+TAB`

) to construct the `KroneckerMap`

, even when all arguments are of `AbstractMatrix`

type.

If `A`

, `B`

, `C`

and `D`

are linear maps of such size that one can form the matrix products `A*C`

and `B*D`

, then the mixed-product property `(A⊗B)*(C⊗D) = (A*C)⊗(B*D)`

holds. Upon vector multiplication, this rule is checked for applicability.

**Examples**

```
julia> J = LinearMap(I, 2) # 2×2 identity map
2×2 LinearMaps.UniformScalingMap{Bool} with scaling factor: true
julia> E = spdiagm(-1 => trues(1)); D = E + E' - 2I;
julia> Δ = kron(D, J) + kron(J, D); # discrete 2D-Laplace operator
julia> Matrix(Δ)
4×4 Matrix{Int64}:
-4 1 1 0
1 -4 0 1
1 0 -4 1
0 1 1 -4
```

`LinearMaps.:⊗`

— Function`⊗(k::Integer)`

Construct a lazy representation of the `k`

-th Kronecker power `A^⊗(k) = A ⊗ A ⊗ ... ⊗ A`

, where `A`

can be an `AbstractMatrix`

or a `LinearMap`

.

`LinearMaps.kronsum`

— Function```
kronsum(A, B)::KroneckerSumMap
kronsum(A, B, Cs...)::KroneckerSumMap
```

Construct a (lazy) representation of the Kronecker sum `A⊕B = A ⊗ Ib + Ia ⊗ B`

of two square linear maps of type `LinearMap`

or `AbstractMatrix`

. Here, `Ia`

and `Ib`

are identity operators of the size of `A`

and `B`

, respectively. Arguments of type `AbstractMatrix`

are automatically promoted to `LinearMap`

.

For convenience, one can also use `A ⊕ B`

or `⊕(A, B, Cs...)`

(typed as `\oplus+TAB`

) to construct the `KroneckerSumMap`

.

**Examples**

```
julia> J = LinearMap(I, 2) # 2×2 identity map
2×2 LinearMaps.UniformScalingMap{Bool} with scaling factor: true
julia> E = spdiagm(-1 => trues(1)); D = LinearMap(E + E' - 2I);
julia> Δ₁ = kron(D, J) + kron(J, D); # discrete 2D-Laplace operator, Kronecker sum
julia> Δ₂ = kronsum(D, D);
julia> Δ₃ = D^⊕(2);
julia> Matrix(Δ₁) == Matrix(Δ₂) == Matrix(Δ₃)
true
```

`LinearMaps.:⊕`

— Function`⊕(k::Integer)`

Construct a lazy representation of the `k`

-th Kronecker sum power `A^⊕(k) = A ⊕ A ⊕ ... ⊕ A`

, where `A`

can be a square `AbstractMatrix`

or a `LinearMap`

. This calls `sumkronsum`

on the `k`

-tuple `(A, ..., A)`

for `k ≥ 3`

.

**Example**

```jldoctest julia> Matrix([1 0; 0 1]^⊕(2)) 4×4 Matrix{Int64}: 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2

The common use case of rank-1 matrices/operators with image spanning vector `u`

and projection vector `v`

is readily available as `u ⊗ v'`

or `u ⊗ transpose(v)`

.

There exist alternative constructors of Kronecker products and sums for square factors and summands, respectively. These are designed for cases of 3 or more arguments, and benchmarking intended use cases for comparison with `KroneckerMap`

and `KroneckerSumMap`

is recommended.

`LinearMaps.squarekron`

— Function`squarekron(A₁::MapOrMatrix, A₂::MapOrMatrix, A₃::MapOrMatrix, Aᵢ::MapOrMatrix...)::CompositeMap`

Construct a (lazy) representation of the Kronecker product `⨂ᵢ₌₁ⁿ Aᵢ`

of at least 3 *square* Kronecker factors. In contrast to `kron`

, this function assumes that all Kronecker factors are square, and makes use of the following identity^{[1]}:

\[\bigotimes_{i=1}^n A_i = \prod_{i=1}^n I_1 \otimes \ldots \otimes I_{i-1} \otimes A_i \otimes I_{i+1} \otimes \ldots \otimes I_n\]

where $I_k$ is an identity matrix of the size of $A_k$. By associativity, the Kronecker product of the identity operators may be combined to larger identity operators $I_{1:i-1}$ and $I_{i+1:n}$, which yields

\[\bigotimes_{i=1}^n A_i = \prod_{i=1}^n I_{1:i-1} \otimes A_i \otimes I_{i+1:n}\]

i.e., a `CompositeMap`

where each factor is a Kronecker product consisting of three maps: outer `UniformScalingMap`

s and the respective Kronecker factor. This representation is expected to yield significantly faster multiplication (and reduce memory allocation) compared to `kron`

, but benchmarking intended use cases is highly recommended.

`LinearMaps.sumkronsum`

— Function```
sumkronsum(A, B)::LinearCombination
sumkronsum(A, B, Cs...)::LinearCombination
```

Construct a (lazy) representation of the Kronecker sum `A⊕B`

of two or more square objects of type `LinearMap`

or `AbstractMatrix`

. This function makes use of the following representation of Kronecker sums^{[1]}:

\[\bigoplus_{i=1}^n A_i = \sum_{i=1}^n I_1 \otimes \ldots \otimes I_{i-1} \otimes A_i \otimes I_{i+1} \otimes \ldots \otimes I_n\]

where $I_k$ is the identity operator of the size of $A_k$. By associativity, the Kronecker product of the identity operators may be combined to larger identity operators $I_{1:i-1}$ and $I_{i+1:n}$, which yields

\[\bigoplus_{i=1}^n A_i = \sum_{i=1}^n I_{1:i-1} \otimes A_i \otimes I_{i+1:n},\]

i.e., a `LinearCombination`

where each summand is a Kronecker product consisting of three maps: outer `UniformScalingMap`

s and the respective Kronecker factor. This representation is expected to yield significantly faster multiplication (and reduce memory allocation) compared to `kronsum`

, especially for 3 or more Kronecker summands, but benchmarking intended use cases is highly recommended.

**Examples**

```
julia> J = LinearMap(I, 2) # 2×2 identity map
2×2 LinearMaps.UniformScalingMap{Bool} with scaling factor: true
julia> E = spdiagm(-1 => trues(1)); D = LinearMap(E + E' - 2I);
julia> Δ₁ = kron(D, J) + kron(J, D); # discrete 2D-Laplace operator, Kronecker sum
julia> Δ₂ = sumkronsum(D, D);
julia> Δ₃ = D^⊕(2);
julia> Matrix(Δ₁) == Matrix(Δ₂) == Matrix(Δ₃)
true
```

`BlockMap`

and `BlockDiagonalMap`

Types for representing block (diagonal) maps lazily.

`Base.hcat`

— Function`hcat(As::Union{LinearMap,UniformScaling,AbstractVecOrMatOrQ}...)::BlockMap`

Construct a (lazy) representation of the horizontal concatenation of the arguments. All arguments are promoted to `LinearMap`

s automatically.

**Examples**

```
julia> CS = LinearMap{Int}(cumsum, 3)::LinearMaps.FunctionMap;
julia> L = [CS LinearMap(ones(Int, 3, 3))]::LinearMaps.BlockMap;
julia> L * ones(Int, 6)
3-element Vector{Int64}:
4
5
6
```

`Base.vcat`

— Function`vcat(As::Union{LinearMap,UniformScaling,AbstractVecOrMatOrQ}...)::BlockMap`

Construct a (lazy) representation of the vertical concatenation of the arguments. All arguments are promoted to `LinearMap`

s automatically.

**Examples**

```
julia> CS = LinearMap{Int}(cumsum, 3)::LinearMaps.FunctionMap;
julia> L = [CS; LinearMap(ones(Int, 3, 3))]::LinearMaps.BlockMap;
julia> L * ones(Int, 3)
6-element Vector{Int64}:
1
2
3
3
3
3
```

`Base.hvcat`

— Function`hvcat(rows::Tuple{Vararg{Int}}, As::Union{LinearMap,UniformScaling,AbstractVecOrMatOrQ}...)::BlockMap`

Construct a (lazy) representation of the horizontal-vertical concatenation of the arguments. The first argument specifies the number of arguments to concatenate in each block row. All arguments are promoted to `LinearMap`

s automatically.

**Examples**

```
julia> CS = LinearMap{Int}(cumsum, 3)::LinearMaps.FunctionMap;
julia> L = [CS CS; CS CS]::LinearMaps.BlockMap;
julia> L.rows
(2, 2)
julia> L * ones(Int, 6)
6-element Vector{Int64}:
2
4
6
2
4
6
```

`Base.cat`

— Function`cat(As::Union{LinearMap,AbstractVecOrMatOrQ}...; dims=(1,2))::BlockDiagonalMap`

Construct a (lazy) representation of the diagonal concatenation of the arguments. To avoid fallback to the generic `Base.cat`

, there must be a `LinearMap`

object among the first 8 arguments.

With `using SparseArrays`

, yet another method is available for creating lazy block-diagonal maps.

`SparseArrays.blockdiag`

— Function`blockdiag(As::Union{LinearMap,AbstractVecOrMatOrQ}...)::BlockDiagonalMap`

Construct a (lazy) representation of the diagonal concatenation of the arguments. To avoid fallback to the generic `blockdiag`

, there must be a `LinearMap`

object among the first 8 arguments.

`FillMap`

Type for lazily representing constantly filled matrices.

`LinearMaps.FillMap`

— Type```
FillMap(λ, (m, n))::FillMap
FillMap(λ, m, n)::FillMap
```

Construct a (lazy) representation of an operator whose matrix representation would be an m×n-matrix filled constantly with the value `λ`

.

`EmbeddedMap`

Type for representing linear maps that are embedded in larger zero maps.

`InverseMap`

Type for lazy inverse of another linear map.

`LinearMaps.InverseMap`

— Type`InverseMap(A; solver = ldiv!)`

Lazy inverse of `A`

such that `InverseMap(A) * x`

is the same as `A \ x`

. Letting an `InverseMap`

act on a vector thus requires solving a linear system.

A solver function can be passed with the `solver`

keyword argument. The solver should be of the form `f(y, A, x)`

where `A`

is the wrapped map, `x`

the right hand side, and `y`

a preallocated output vector in which the result should be stored. The default solver is `LinearAlgebra.ldiv!`

.

Note that `A`

must be compatible with the solver function. `A`

can, for example, be a factorization of a matrix, or another `LinearMap`

(in combination with an iterative solver such as conjugate gradient).

**Examples**

```
julia> using LinearMaps, LinearAlgebra
julia> A = rand(2, 2); b = rand(2);
julia> InverseMap(lu(A)) * b
2-element Vector{Float64}:
1.0531895201271027
-0.4718540250893251
julia> A \ b
2-element Vector{Float64}:
1.0531895201271027
-0.4718540250893251
```

`KhatriRaoMap`

and `FaceSplittingMap`

Types for lazy column-wise and row-wise Kronecker product, respectively, also referrerd to as Khatri-Rao and transposed Khatri-Rao (or face-splitting) product.

`LinearMaps.khatrirao`

— Function`khatrirao(A::MapOrVecOrMat, B::MapOrVecOrMat) -> KhatriRaoMap`

Construct a lazy representation of the Khatri-Rao (or column-wise Kronecker) product of two maps or arrays `A`

and `B`

. For the application to vectors, the tranpose action of `A`

on vectors needs to be defined.

`LinearMaps.facesplitting`

— Function`facesplitting(A::AbstractMatrix, B::AbstractMatrix) -> FaceSplittingMap`

Construct a lazy representation of the face-splitting (or row-wise Kronecker) product of two matrices `A`

and `B`

.

## Methods

### Multiplication methods

`Base.:*`

— Method`*(A::LinearMap, x::AbstractVector)::AbstractVector`

Compute the action of the linear map `A`

on the vector `x`

.

**Examples**

```
julia> A=LinearMap([1.0 2.0; 3.0 4.0]); x=[1.0, 1.0];
julia> A*x
2-element Vector{Float64}:
3.0
7.0
julia> A(x)
2-element Vector{Float64}:
3.0
7.0
```

`Base.:*`

— Method`*(A::LinearMap, X::Union{AbstractMatrix,AbstractQ})::CompositeMap`

Return the `CompositeMap`

`A*LinearMap(X)`

, interpreting the matrix `X`

as a linear operator, rather than a collection of column vectors. To compute the action of `A`

on each column of `X`

, call `Matrix(A*X)`

or use the in-place multiplication `mul!(Y, A, X[, α, β])`

with an appropriately sized, preallocated matrix `Y`

.

**Examples**

```
julia> A=LinearMap([1.0 2.0; 3.0 4.0]); X=[1.0 1.0; 1.0 1.0];
julia> A*X isa LinearMaps.CompositeMap
true
```

`Base.:*`

— Method`*(X::Union{AbstractMatrix,AbstractQ}, A::LinearMap)::CompositeMap`

Return the `CompositeMap`

`LinearMap(X)*A`

, interpreting the matrix `X`

as a linear operator. To compute the right-action of `A`

on each row of `X`

, call `Matrix(X*A)`

or `mul!(Y, X, A)`

for the in-place version.

**Examples**

```
julia> X=[1.0 1.0; 1.0 1.0]; A=LinearMap([1.0 2.0; 3.0 4.0]);
julia> X*A isa LinearMaps.CompositeMap
true
```

`LinearAlgebra.mul!`

— Method```
mul!(Y::AbstractVecOrMat, A::LinearMap, B::AbstractVector) -> Y
mul!(Y::AbstractMatrix, A::LinearMap, B::AbstractMatrix) -> Y
```

Calculates the action of the linear map `A`

on the vector or matrix `B`

and stores the result in `Y`

, overwriting the existing value of `Y`

. Note that `Y`

must not be aliased with either `A`

or `B`

.

**Examples**

```
julia> A = LinearMap([1.0 2.0; 3.0 4.0]); B = ones(2); Y = similar(B); mul!(Y, A, B)
2-element Vector{Float64}:
3.0
7.0
julia> A = LinearMap([1.0 2.0; 3.0 4.0]); B = ones(2,2); Y = similar(B); mul!(Y, A, B)
2×2 Matrix{Float64}:
3.0 3.0
7.0 7.0
```

`LinearAlgebra.mul!`

— Method```
mul!(C::AbstractVecOrMat, A::LinearMap, B::AbstractVector, α, β) -> C
mul!(C::AbstractMatrix, A::LinearMap, B::AbstractMatrix, α, β) -> C
```

Combined inplace multiply-add $A B α + C β$. The result is stored in `C`

by overwriting it. Note that `C`

must not be aliased with either `A`

or `B`

.

**Examples**

```
julia> A=LinearMap([1.0 2.0; 3.0 4.0]); B=[1.0, 1.0]; C=[1.0, 3.0];
julia> mul!(C, A, B, 100.0, 10.0) === C
true
julia> C
2-element Vector{Float64}:
310.0
730.0
julia> A=LinearMap([1.0 2.0; 3.0 4.0]); B=[1.0 1.0; 1.0 1.0]; C=[1.0 2.0; 3.0 4.0];
julia> mul!(C, A, B, 100.0, 10.0) === C
true
julia> C
2×2 Matrix{Float64}:
310.0 320.0
730.0 740.0
```

`LinearAlgebra.mul!`

— Method`mul!(C::AbstractMatrix, A::AbstractMatrix, B::LinearMap) -> C`

Calculates the matrix representation of `A*B`

and stores the result in `C`

, overwriting the existing value of `C`

. Note that `C`

must not be aliased with either `A`

or `B`

. The computation `C = A*B`

is performed via `C' = B'A'`

.

**Examples**

```
julia> A=[1.0 1.0; 1.0 1.0]; B=LinearMap([1.0 2.0; 3.0 4.0]); C = similar(A); mul!(C, A, B);
julia> C
2×2 Matrix{Float64}:
4.0 6.0
4.0 6.0
```

`LinearAlgebra.mul!`

— Method`mul!(C::AbstractMatrix, A::AbstractMatrix, B::LinearMap, α, β) -> C`

Combined inplace multiply-add $A B α + C β$. The result is stored in `C`

by overwriting it. Note that `C`

must not be aliased with either `A`

or `B`

.

**Examples**

```
julia> A=[1.0 1.0; 1.0 1.0]; B=LinearMap([1.0 2.0; 3.0 4.0]); C = copy(A);
julia> mul!(C, A, B, 1, 1)
2×2 Matrix{Float64}:
5.0 7.0
5.0 7.0
```

`LinearAlgebra.mul!`

— Method`mul!(Y::AbstractMatrix, A::LinearMap, b::Number) -> Y`

Scales the matrix representation of the linear map `A`

by `b`

and stores the result in `Y`

, overwriting the existing value of `Y`

.

**Examples**

```
julia> A = LinearMap{Int}(cumsum, 3); b = 2; Y = Matrix{Int}(undef, (3,3));
julia> mul!(Y, A, b)
3×3 Matrix{Int64}:
2 0 0
2 2 0
2 2 2
```

`LinearAlgebra.mul!`

— Method`mul!(Y::AbstractMatrix, A::LinearMap, b::Number, α::Number, β::Number) -> Y`

Scales the matrix representation of the linear map `A`

by `b*α`

, adds the result to `Y*β`

and stores the final result in `Y`

, overwriting the existing value of `Y`

.

**Examples**

```
julia> A = LinearMap{Int}(cumsum, 3); b = 2; Y = ones(Int, (3,3));
julia> mul!(Y, A, b, 2, 1)
3×3 Matrix{Int64}:
5 1 1
5 5 1
5 5 5
```

`Base.:*`

— Method`*(x::LinearAlgebra.AdjointAbsVec, A::LinearMap)::AdjointAbsVec`

Compute the right-action of the linear map `A`

on the adjoint vector `x`

and return an adjoint vector.

**Examples**

```
julia> A=LinearMap([1.0 2.0; 3.0 4.0]); x=[1.0, 1.0]; x'A
1×2 adjoint(::Vector{Float64}) with eltype Float64:
4.0 6.0
```

`Base.:*`

— Method`*(x::LinearAlgebra.TransposeAbsVec, A::LinearMap)::TransposeAbsVec`

Compute the right-action of the linear map `A`

on the transpose vector `x`

and return a transpose vector.

**Examples**

```
julia> A=LinearMap([1.0 2.0; 3.0 4.0]); x=[1.0, 1.0]; transpose(x)*A
1×2 transpose(::Vector{Float64}) with eltype Float64:
4.0 6.0
```

Applying the adjoint or transpose of `A`

(if defined) to `x`

works exactly as in the usual matrix case: `transpose(A) * x`

and `mul!(y, A', x)`

, for instance.

### Conversion methods

`Array`

,`Matrix`

and associated`convert`

methodsCreate a dense matrix representation of the

`LinearMap`

object, by multiplying it with the successive basis vectors. This is mostly for testing purposes or if you want to have the explicit matrix representation of a linear map for which you only have a function definition (e.g. to be able to use its`transpose`

or`adjoint`

). This way, one may conveniently make`A`

act on the columns of a matrix`X`

, instead of interpreting`A * X`

as a composed linear map:`Matrix(A * X)`

. For generic code, that is supposed to handle both`A::AbstractMatrix`

and`A::LinearMap`

, it is recommended to use`convert(Matrix, A*X)`

.`convert(Abstract[Matrix/Array], A::LinearMap)`

Create an

`AbstractMatrix`

representation of the`LinearMap`

. This falls back to`Matrix(A)`

, but avoids explicit construction in case the`LinearMap`

object is matrix-based.`SparseArrays.sparse(A::LinearMap)`

and`convert(SparseMatrixCSC, A::LinearMap)`

Create a sparse matrix representation of the

`LinearMap`

object, by multiplying it with the successive basis vectors. This is mostly for testing purposes or if you want to have the explicit sparse matrix representation of a linear map for which you only have a function definition (e.g. to be able to use its`transpose`

or`adjoint`

).

In Julia versions v1.9 and higher, conversion to sparse matrices requires loading `SparseArrays.jl`

by the user in advance.

### Slicing methods

Complete slicing, i.e., `A[:,j]`

, `A[:,J]`

, `A[i,:]`

, `A[I,:]`

and `A[:,:]`

for `i`

, `j`

`Integer`

and `I`

, `J`

`AbstractVector{<:Integer}`

is generically available for any `A::LinearMap`

subtype via application of `A`

(or `A'`

for (predominantly) horizontal slicing) to standard unit vectors of appropriate length. By complete slicing we refer two-dimensional Cartesian indexing where at least one of the "indices" is a colon. This is facilitated by overloads of `Base.getindex`

. Partial slicing à la `A[I,J]`

and scalar or linear indexing are *not* supported.

### Sum, product, mean and trace

Natural function overloads for `Base.sum`

, `Base.prod`

, `Statistics.mean`

and `LinearAlgebra.tr`

exist.

In Julia versions v1.9 and higher, creating the mean linear operator requires loading `Statistics.jl`

by the user in advance.

- 1Fernandes, P. and Plateau, B. and Stewart, W. J. "Efficient Descriptor-Vector Multiplications in Stochastic Automata Networks",
*Journal of the ACM*, 45(3), 381–414, 1998. - 1Fernandes, P. and Plateau, B. and Stewart, W. J. "Efficient Descriptor-Vector Multiplications in Stochastic Automata Networks",
*Journal of the ACM*, 45(3), 381–414, 1998.