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Specify how to obtain all inverses #1835

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Oct 7, 2024
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Specify how to obtain all inverses #1835

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7298a3d
Clarify how to get all left/right inverses
albinahlback Oct 7, 2024
2c1675e
Update src/Matrix.jl
fingolfin Oct 7, 2024
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14 changes: 9 additions & 5 deletions src/Matrix.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -3553,11 +3553,15 @@ end
@doc raw"""
is_invertible_with_inverse(A::MatrixElem{T}; side::Symbol = :left) where {T <: RingElement}

Given an $n\times m$ matrix $A$ over a ring, return a tuple `(flag, B)`.
If `side` is `:right` and `flag` is true, $B$ is the right inverse of $A$
i.e. $AB$ is the $n\times n$ unit matrix. If `side` is `:left` and `flag` is
true, $B$ is the left inverse of $A$ i.e. $BA$ is the $m\times m$ unit matrix.
If `flag` is false, no right or left inverse exists.
Given an $n \times m$ matrix $A$ over a ring, return a tuple `(flag, B)`. If
`side` is `:right` and `flag` is true, $B$ is a right inverse of $A$ i.e. $A B$
is the $n \times n$ unit matrix. If `side` is `:left` and `flag` is true, $B$
is a left inverse of $A$ i.e. $B A$ is the $m \times m$ unit matrix. If `flag`
is false, no right or left inverse exists.
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To get the space of all inverses, note that if $B$ and $C$ are both right
inverses, then $A (B - C) = 0$, and similar for left inverses. Hence from one
inverse one can find all by making suitable use of [`kernel`](@ref).
"""
function is_invertible_with_inverse(A::MatrixElem{T}; side::Symbol = :left) where {T <: RingElement}
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Not an issue of this PR, but for the record: I am surprised that this function can do strictly more (namely handle non-square matrices) than its is_invertible sibling.

if (side == :left && nrows(A) < ncols(A)) || (side == :right && ncols(A) < nrows(A))
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