As an undergrad in France, I learned the definition of charpoly as det (
M- xI ) and remember, that our professor mentionned the other convention
as exotic.
Since then, I've worked on compting the charpoly during my phd thesis
and always chose to use the definition det (xI-M) which I found much
more convenient in almost every aspects.
It seems to me that this painful convention was only used in France but
I may be wrong. For example in the reference Russian book "Theory of
Matrices" by Gantmacher: det (xI-M) is also used.
I don't know the reasons for this convention, except maybe that the
determinant there is exactly the constant coefficient.
So I see no inconsistency there.
Btw: in a chapter about linear algebra using sage dedicated to french
undergrads that I'm currently writing, I also chose the det(xI-M)
convention.
Clément
John Cremona a écrit :
On 15 June 2010 13:25, Jason Grout <[email protected]> wrote:
On 6/15/10 6:21 AM, Minh Nguyen wrote:
As you can see, these two characteristic polynomials differ in only
their signs. One can be obtained from the other by multiplying through
by -1. What I would like to know is: Is there some reason for this
inconsistency? Or are the two characteristic polynomials above
"essentially" the same?
One is computed using x*Id-M, the other by M-x*Id. This will lead to a sign
difference for odd-sized matrices. In my graduate abstract algebra course,
we defined the characteristic polynomial using x*Id-M specifically so that
we'd always have a monic polynomial as the output.
Sure, though in undergraduate teaching one advantage is using M-x*Id
is that they are less likely to make sign errors if they subtract x
from the diagonal entries than if they have to negate all entries and
add x to the diagonal.... Having said that, I would definitely agree
that the *definition" of the char poly should be something monic, even
if in practice it may be convenient to work with its negative.
John
Jason
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