I'm new to julia and I have not been able to find an explanation of where that fully makes sense to me.
So I understand that for example
function f(x::T, y::T) where {T<:Number} = true is requiring that both the x and y parameters be numeric types of some sort.
But then there are other functions that will look like function f(X::AbstractMatrix{T}) where T without any specification of what T is supposed to be. Can someone explain what that is doing and when I would want to use this syntax?
where T without anything else is just where {T<:Any} in other words, it's true for all T, but Julia needs you to write something so that `T is defined.
Oscar's answer is correct, but to answer your second question,
When I would want to use this syntax?
like mbauman suggested in the comments, a typical use for this syntax is when you want to use the type in the body of the function. Maybe the example in the docs helps:
julia> mytypeof(x::T) where {T} = T
mytypeof (generic function with 1 method)
julia> mytypeof(1)
Int64
julia> mytypeof(1.0)
Float64