0
$\begingroup$
  1. I have a 3 x 3 matrix [a+8,1,0;1,a+2,6;0,6,1] with two unknown variables. I have to use Sylvester's criterion to find set values of parameter a where the matrix is positive definite.

What I thought to do is to find the determinant of the matrix and a would be the boundaries between the two values?

  1. Letting a = -2, the matrix is [6,1,0;1,0,6;0,6,1] which is indefinite. Without finding the eigenvectors, I have to find vector x such that x^t Ax > 0 and vector y y^t Ay<0

What I thought is that you make a 1 x 3 matrix for both x and y then mutliply with A. From that you calculate for det < 0 and det >0

Would this be the correct way of thinking for this question

https://i.stack.imgur.com/MFKtN.png

$\endgroup$
  • 2
    $\begingroup$ "2 unknown variables" ? Where is the second one ? $\endgroup$ – Jean Marie May 30 at 0:48
  • $\begingroup$ How do you propose to compute the determinant of a non-square matrix in part 2? $\endgroup$ – amd May 30 at 1:20
  • $\begingroup$ Not sure that was what I thought you would do. Mind sharing how you would do it? $\endgroup$ – Mr A May 30 at 1:39
3
$\begingroup$

If your matrix is $$\begin{pmatrix}a+8 & 1 & 0 \\ 1 & a+2 & 6 \\ 0 & 6 &1 \end{pmatrix},$$Sylvester's criterion says that the above will be positive-definite if and only if $$\det(a+8) > 0, \qquad \det\begin{pmatrix} a+8 & 1 \\ 1 & a+2\end{pmatrix}>0 \qquad\mbox{and}\qquad \det\begin{pmatrix}a+8 & 1 & 0 \\ 1 & a+2 & 6 \\ 0 & 6 &1 \end{pmatrix}>0.$$Now it boils down to an algebra/pre-calc problem: find the values of $a$ satisfying all three inequalities at the same time.

$\endgroup$
1
$\begingroup$

For the first problem, we note that Sylvester's Criterion states that a matrix is positive definite if and only if each of its principal minors are positive. So it suffices to compute formulas for each of the principal minors and restrict $a$ to take values only within the range that would yield the desired positive determinants.

$\endgroup$
  • $\begingroup$ " So it suffices to compute formulas for each of the principal minors" $\endgroup$ – Mr A May 30 at 0:34
  • $\begingroup$ Would you mean that I calculate each determinant value for a total of 3 of them and take positive values? Sorry im not too sure what you meant $\endgroup$ – Mr A May 30 at 0:35
  • $\begingroup$ “The principal minors” refers to the determinants of the following two matrices: the 1x1 matrix in the upper left hand corner and the 2x2 matrix in the upper left hand corner. $\endgroup$ – lagicol May 30 at 14:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.