matrix theory tools Rayleigh quotient

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please someone can explain to me or give me some details about what I found on a university article about the rayleigh quotient



Rayleigh quotient



Let $Bsucceq 0$ and $Csucc 0$.
Raleigh quotient is defined, for all $xin mathbbR^d | 0$ by



$fracx^T Bxx^T Cx$



Two quantities are interesting:



$bigg {$ sup of Raleigh quotient = $lambda_max (C^-1/2 BC^-1/2)$



$bigg {$ inf of Raleigh quotient = $lambda_min (C^-1/2 BC^-1/2)$



Special case: C=Id, the sup is $lambda_max(B)$ and the inf is $lambda_min(B)$



Construction of k



Let $x in X^in$



$x^T A^kTQA^kx leq lambda_max(Q) |A^kx |_2^2 $ From Rayleigh quotient



$leq lambda_max(Q) |A^kx |_2^2 |x |_2^2 $ From norm operator def.



$leq lambda_max(Q) ^k |x |_2^2 $ From matrix norm def.



Now let define, for $Bsucceq 0$, $mu(B)= sup x^TBx $



We impose for $K$ that $x^TA^kTQ A^kx leq undersetx in X^insup x^TQx=mu(Q)$ for all $kgeq K$.



We have to exhibit a lower bound on integers $k$:



$^k leq mu(Q) lambda_max(Q)^-1 mu(Id)^-1$



Using ln, if $|A |_2^2 < 1$ we get:



$ kgeq fracln(mu(Q) lambda_max(Q)^-1 mu(Id)^-1) _2^2)$




matrices eigenvalues-eigenvectors




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  • What’s the question?
    – user7530
    Jul 20 at 15:26










  • I do not know if you can give me more details about what I wrote below, I can not understand this part of rayleigh quotient @user7530
    – RàMi
    Jul 20 at 19:47














up vote
-3
down vote

favorite












please someone can explain to me or give me some details about what I found on a university article about the rayleigh quotient



Rayleigh quotient



Let $Bsucceq 0$ and $Csucc 0$.
Raleigh quotient is defined, for all $xin mathbbR^d | 0$ by



$fracx^T Bxx^T Cx$



Two quantities are interesting:



$bigg {$ sup of Raleigh quotient = $lambda_max (C^-1/2 BC^-1/2)$



$bigg {$ inf of Raleigh quotient = $lambda_min (C^-1/2 BC^-1/2)$



Special case: C=Id, the sup is $lambda_max(B)$ and the inf is $lambda_min(B)$



Construction of k



Let $x in X^in$



$x^T A^kTQA^kx leq lambda_max(Q) |A^kx |_2^2 $ From Rayleigh quotient



$leq lambda_max(Q) |A^kx |_2^2 |x |_2^2 $ From norm operator def.



$leq lambda_max(Q) ^k |x |_2^2 $ From matrix norm def.



Now let define, for $Bsucceq 0$, $mu(B)= sup x^TBx $



We impose for $K$ that $x^TA^kTQ A^kx leq undersetx in X^insup x^TQx=mu(Q)$ for all $kgeq K$.



We have to exhibit a lower bound on integers $k$:



$^k leq mu(Q) lambda_max(Q)^-1 mu(Id)^-1$



Using ln, if $|A |_2^2 < 1$ we get:



$ kgeq fracln(mu(Q) lambda_max(Q)^-1 mu(Id)^-1) _2^2)$




matrices eigenvalues-eigenvectors




share|cite|improve this question





















  • What’s the question?
    – user7530
    Jul 20 at 15:26










  • I do not know if you can give me more details about what I wrote below, I can not understand this part of rayleigh quotient @user7530
    – RàMi
    Jul 20 at 19:47












up vote
-3
down vote

favorite









up vote
-3
down vote

favorite











please someone can explain to me or give me some details about what I found on a university article about the rayleigh quotient



Rayleigh quotient



Let $Bsucceq 0$ and $Csucc 0$.
Raleigh quotient is defined, for all $xin mathbbR^d | 0$ by



$fracx^T Bxx^T Cx$



Two quantities are interesting:



$bigg {$ sup of Raleigh quotient = $lambda_max (C^-1/2 BC^-1/2)$



$bigg {$ inf of Raleigh quotient = $lambda_min (C^-1/2 BC^-1/2)$



Special case: C=Id, the sup is $lambda_max(B)$ and the inf is $lambda_min(B)$



Construction of k



Let $x in X^in$



$x^T A^kTQA^kx leq lambda_max(Q) |A^kx |_2^2 $ From Rayleigh quotient



$leq lambda_max(Q) |A^kx |_2^2 |x |_2^2 $ From norm operator def.



$leq lambda_max(Q) ^k |x |_2^2 $ From matrix norm def.



Now let define, for $Bsucceq 0$, $mu(B)= sup x^TBx $



We impose for $K$ that $x^TA^kTQ A^kx leq undersetx in X^insup x^TQx=mu(Q)$ for all $kgeq K$.



We have to exhibit a lower bound on integers $k$:



$^k leq mu(Q) lambda_max(Q)^-1 mu(Id)^-1$



Using ln, if $|A |_2^2 < 1$ we get:



$ kgeq fracln(mu(Q) lambda_max(Q)^-1 mu(Id)^-1) _2^2)$




matrices eigenvalues-eigenvectors




share|cite|improve this question













please someone can explain to me or give me some details about what I found on a university article about the rayleigh quotient



Rayleigh quotient



Let $Bsucceq 0$ and $Csucc 0$.
Raleigh quotient is defined, for all $xin mathbbR^d | 0$ by



$fracx^T Bxx^T Cx$



Two quantities are interesting:



$bigg {$ sup of Raleigh quotient = $lambda_max (C^-1/2 BC^-1/2)$



$bigg {$ inf of Raleigh quotient = $lambda_min (C^-1/2 BC^-1/2)$



Special case: C=Id, the sup is $lambda_max(B)$ and the inf is $lambda_min(B)$



Construction of k



Let $x in X^in$



$x^T A^kTQA^kx leq lambda_max(Q) |A^kx |_2^2 $ From Rayleigh quotient



$leq lambda_max(Q) |A^kx |_2^2 |x |_2^2 $ From norm operator def.



$leq lambda_max(Q) ^k |x |_2^2 $ From matrix norm def.



Now let define, for $Bsucceq 0$, $mu(B)= sup x^TBx $



We impose for $K$ that $x^TA^kTQ A^kx leq undersetx in X^insup x^TQx=mu(Q)$ for all $kgeq K$.



We have to exhibit a lower bound on integers $k$:



$^k leq mu(Q) lambda_max(Q)^-1 mu(Id)^-1$



Using ln, if $|A |_2^2 < 1$ we get:



$ kgeq fracln(mu(Q) lambda_max(Q)^-1 mu(Id)^-1) _2^2)$





matrices eigenvalues-eigenvectors





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edited Jul 20 at 14:41
























asked Jul 20 at 13:47









RàMi

11




11











  • What’s the question?
    – user7530
    Jul 20 at 15:26










  • I do not know if you can give me more details about what I wrote below, I can not understand this part of rayleigh quotient @user7530
    – RàMi
    Jul 20 at 19:47
















  • What’s the question?
    – user7530
    Jul 20 at 15:26










  • I do not know if you can give me more details about what I wrote below, I can not understand this part of rayleigh quotient @user7530
    – RàMi
    Jul 20 at 19:47















What’s the question?
– user7530
Jul 20 at 15:26




What’s the question?
– user7530
Jul 20 at 15:26












I do not know if you can give me more details about what I wrote below, I can not understand this part of rayleigh quotient @user7530
– RàMi
Jul 20 at 19:47




I do not know if you can give me more details about what I wrote below, I can not understand this part of rayleigh quotient @user7530
– RàMi
Jul 20 at 19:47















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