I discovered this with matrices that were not random, but this simple test
illustrates the problem:   julia aborts when it tries to diagonalize a 
705x705 matrix

mat = rand(704,704)
mat = mat' + mat
e = eigfact(mat)
@show "done 704"

mat = rand(705,705)
mat = mat' + mat
e = eigfact(mat)
@show "done 705"

Results:
"done 704" = "done 704"
Abort

I also get this when I try an svd:
mat = rand(705,705)
svd(mat)

This gives 
Abort

So, to find what the key problem sizes are:
for i=1:705
       mat = rand(i,i)
       svd(mat)
       @show i
 end

This dies on i=129.   Note that 129 is odd.
But replacing the for loop to count by twos is fine:
for i=2:2:1705
       mat = rand(i,i)
       svd(mat)
       @show i
 end
No problem here. 

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