Timeline for Bounding the norm of a contraction matrix
Current License: CC BY-SA 3.0
14 events
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| May 16, 2017 at 0:47 | vote | accept | user3007505 | ||
| May 15, 2017 at 14:52 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
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| May 15, 2017 at 14:44 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Minor improvements
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| May 15, 2017 at 14:03 | comment | added | Rodrigo de Azevedo | I edited the answer. | |
| May 15, 2017 at 14:03 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Improved the answer
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| May 15, 2017 at 12:37 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
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| May 15, 2017 at 12:26 | comment | added | user3007505 | The only hard constraint here is that the Joint covariance matrix should be positive semi-definite. Now C could be any matrix resulting in X which can ensure the positive semi-definiteness of joint matrix. For instance, X = $rA^(1/2)B^(1/2)$ (for r = [-1,1]) is a valid representation of X but there are some choices of X for some valid C that is not bounded by the said definition. I want to characterize the matrix C so that I can get maximum bound of X. | |
| May 15, 2017 at 12:11 | comment | added | Rodrigo de Azevedo | I still do not understand your question. Can a symmetric $\rm C$ have negative eigenvalues? Sure, provided that they are in $(-1,0)$, which guarantees that $\rm I - \Lambda^2 \succeq O$. Bounds on $\rm C$? I don't know what that means. Do you want to bound $\| \rm C \|_2$? | |
| May 15, 2017 at 11:55 | comment | added | user3007505 | I am sorry, the (1,2) block is X and (2,1) block is X^T. The joint matrix is a co variance matrix. | |
| May 15, 2017 at 10:03 | comment | added | Rodrigo de Azevedo | If the $(2,1)$ block is $\rm X$ then the block matrix is not symmetric. Only the symmetric part contributes to a quadratic form. | |
| May 15, 2017 at 10:01 | comment | added | user3007505 | The (2,1) block is X. Since, I assume that matrix X is positive semi definite, I can make the symmetric assumption on contraction matrix C. | |
| May 15, 2017 at 9:18 | comment | added | Rodrigo de Azevedo | What is the definition of "contraction matrix"? Is it symmetric? Are the eigenvalues real? Is the $(2,1)$-th block $\rm X$ or $\rm X^{\top}$, after all? | |
| May 15, 2017 at 2:02 | comment | added | user3007505 | Thank you very much for explaining the detail. Actually in my case, I have two known positive semi-definite matrices A and B and the matrix X is unknown. I also have the representation of X in terms of contraction matrix C. What I need is some kind of parametric form of matrix C which can give me the bounds of matrix X? | |
| May 14, 2017 at 14:33 | history | answered | Rodrigo de Azevedo | CC BY-SA 3.0 |