#Sylvester's criterion for 2x2 matrices

33 messages · Page 1 of 1 (latest)

flat lagoon
#

I'm having a hard time with the proof for Sylvester's criterion, and am wondering if there's a more intuitive version for only 2x2 matrices. Why is it that for a 2x2 matrix M to be positive definite (i.e. all eigenvalues positive), a_11 > 0 and det(M) > 0?

onyx slateBOT
#
  1. Do not ping the Moderators, unless someone is breaking the rules.
  2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
  3. Do not ping other members randomly for help.
  4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
  5. Wait patiently for a helper to come along.
  6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

grave summit
#

for 2x2 this is very quick to verify by hand

#

remember that by definition A is positive definite if x'Ax > 0 for all nonzero x

#

as for "intuitive"

#

whenever you multiply a vector with a positive semi definite matrix, the angle between the original vector and the result does not exceed pi/2

#

so "inuitively", a positive semi definite transformation attempts to keep the vector within certain boundaries

#

@flat lagoon

#

tried to illustrate

#

the blue region is allowed for the vector Ax

#

but the red one is impossible

#

this is precisely the condition x'Ax > 0 (or nonnegative for positive semi definite case)

#

i think you can extrapolate then what positive definite means

flat lagoon
#

i'm assuming you mean show that this > 0 given ad - bc > 0 and a > 0

image.png
#

well assuming x1 and x2 aren't both 0

#

got up to this step of completing the square and now i'm lost

image.png
flat lagoon
flat lagoon
grave summit
#

you are forgetting something

#

Sylvester is guaranteed to work on Hermitian matrices

#

you have a,b,c,d all freely picked right now

#

@flat lagoon

flat lagoon
#

ooh

#

yeah that makes sense now

#

thanks a lot

grave summit
#

that's why it works in analysis optimization problems, we assume the function is sufficiently smooth hence the hessian is hermitian

flat lagoon
#

right, makes sense

#

+close

strong larkBOT
# flat lagoon +close
Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

strong larkBOT
# strong lark

aL Thank you for helping our users, however our Mathematics Team cares about the health of its members, so you can no longer receive helper_points since you surpassed your daily limit, but don't worry, it will be reset tomorrow.