Re: det(A+ tB)


"Virgil" <virgil@xxxxxxxxxxx> wrote in message
news:virgil-8366EB.13245828012007@xxxxxxxxxxxxxxxxxxxxxxxxxxx
In article <hPYuh.2501$MN.643@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx> wrote:

"Virgil" <virgil@xxxxxxxxxxx> wrote in message
news:virgil-210E00.00511828012007@xxxxxxxxxxxxxxxxxxxxxxxxxxx
In article <YSXuh.4980$O02.3766@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx> wrote:

"Virgil" <virgil@xxxxxxxxxxx> wrote in message
news:virgil-F026CE.23015527012007@xxxxxxxxxxxxxxxxxxxxxxxxxxx
In article <DnWuh.56425$wc5.39323@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx> wrote:

"Virgil" <virgil@xxxxxxxxxxx> wrote in message
news:virgil-FFAAEE.14342627012007@xxxxxxxxxxxxxxxxxxxxxxxxxxx
In article <CRIuh.23117$yC5.3776@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Jon Slaughter" <Jon_Slaughter@xxxxxxxxxxx> wrote:

<caos.snow@xxxxxxxxx> wrote in message
news:1169903338.954668.85220@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Sorry, the expansions have a negative sign in from I think.
And I
would say that n=4 the thing should be like

det(A+tB) = - ( w0 + w1*t + w2*t^2 + w3*t^3 + w4*t^4 )

w0 = det(A)
w1 = det(baaa) + det(abaa) + det(aaba) + det(aaab)
w2 = det(bbaa) + det(baba) + det(baab) + det(abba) + det(abab)
+
det(aabb)
w3 = det(abbb) + det(babb) + det(bbab) + det(bbba)
w4 = det(B)

I can see already the familiar {1,1} {1,2,1}, {1,3,3,1},
{1,4,6,4,1}
pattern here.


Note that what your essentially trying to find is the char.
poly.
of -A/B
since

det(A + t*B) = det(B)*det(t*I - A/B) = det(B)*p(t)

where p(t) is the char. poly. of -A/B.


The char. poly. of -A/B is

p(t) = -det(A/B) - c2*t - tr(A/B)*t^2 - t^3

where c2 = sum(principal minors)

These methods are very useful if A and B are fixed for your
problem
as
they
can be computed beforehand.

This analysis presumes that det(B) <> 0, which is not implicit in
the
original problem.


Not really. You can easily see this if you think about it a
little(and
it
doesn't require anything more than just basic algebra and a little
common
sense). Hint: It only requires that both A and B not be singular...
but
its
obvious what the answer is if they are.

Being non-singular and having a non-zero determinant are equivalent
for
square matrices.


and?

The analysis mentions such things as A/B and det(A/B), which
requires
that B have an inverse and a non-zero determinant.

Yes, because I used that notation. The method I used is entirely
general.

So you say that det(B) != 0?

I merely say that your use of A/B implies that B is invertible, which
means that it is you requiring det(B) != 0.

Do you realize that

det(A + tB) = 1/t^3*det(B + tA)?

Not if t = 0 is a solution.

So it has nothing to do with just B because we can use A if B is
singular.

The only requirement is that A and B both be non singular.

How is "non singular" different from "determinant not zero"?


No one said they were? You said only that B had to be non-singular.

I said that according to the analysis, which used det(A/B), one would
need det(B) <> 0.

Yes, but what you don't realize is that det(B)*det(A/B) = det(A). det(A/B)
may look like you need det(B) != 0 but you don't because its not det(A/B)
that your looking at but det(B)*det(A/B). Your not seeing the full picture.
its kinda like saying that x/x must have x != 0 but would you say that?


.

Relevant Pages

  • Re: det(A+ tB)
    ... Note that what your essentially trying to find is the char. ... It only requires that both A and B not be singular... ... that B have an inverse and a non-zero determinant. ... means that it is you requiring det!= 0. ...
    (sci.math)
  • Re: det(A+ tB)
    ... Note that what your essentially trying to find is the char. ... that B have an inverse and a non-zero determinant. ... means that it is you requiring det!= 0. ... So it has nothing to do with just B because we can use A if B is singular. ...
    (sci.math)
  • Re: det(A+ tB)
    ... Note that what your essentially trying to find is the char. ... means that it is you requiring det!= 0. ... The only requirement is that A and B both be non singular. ... I certainly would say that x/x must have x 0 in order to be sensible. ...
    (sci.math)
  • Re: det(A+ tB)
    ... Note that what your essentially trying to find is the char. ... This analysis presumes that det0, which is not implicit in the ... original problem. ... that B have an inverse and a non-zero determinant. ...
    (sci.math)