To install click the Add extension button. That's it.
The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.
How to transfigure the Wikipedia
Would you like Wikipedia to always look as professional and up-to-date? We have created a browser extension. It will enhance any encyclopedic page you visit with the magic of the WIKI 2 technology.
Try it — you can delete it anytime.
Install in 5 seconds
Yep, but later
4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
We remark that the converse of the theorem holds in the following sense. If is a symmetric matrix and the Hadamard product is positive definite for all positive definite matrices , then itself is positive definite.
YouTube Encyclopedic
1/5
Views:
14 701
6 462
8 923
505
112 836
Linear Algebra 18a: Introduction to the Eigenvalue Decomposition
RT4.2. Schur's Lemma (Expanded)
Polynomial Function - Complex Factorization Theorem
Complex Analysis: Lecture 34 part 2: some infinite product calculations
Solve a System of Linear Equations Using LU Decomposition
Transcription
Proof
Proof using the trace formula
For any matrices and , the Hadamard product considered as a bilinear form acts on vectors as
where is the matrix trace and is the diagonal matrix having as diagonal entries the elements of .
Suppose and are positive definite, and so Hermitian. We can consider their square-roots and , which are also Hermitian, and write
Then, for , this is written as for and thus is strictly positive for , which occurs if and only if . This shows that is a positive definite matrix.
Since a covariance matrix is positive definite, this proves that the matrix with elements is a positive definite matrix.
Proof using eigendecomposition
Proof of positive semidefiniteness
Let and . Then
Each is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, thus the sum is also positive semidefinite.
Proof of definiteness
To show that the result is positive definite requires even further proof. We shall show that for any vector , we have . Continuing as above, each , so it remains to show that there exist and for which corresponding term above is nonzero. For this we observe that
Since is positive definite, there is a for which (since otherwise for all ), and likewise since is positive definite there exists an for which However, this last sum is just . Thus its square is positive. This completes the proof.
References
^Schur, J. (1911). "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen". Journal für die reine und angewandte Mathematik. 1911 (140): 1–28. doi:10.1515/crll.1911.140.1. S2CID120411177.
^Zhang, Fuzhen, ed. (2005). The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. doi:10.1007/b105056. ISBN0-387-24271-6., page 9, Ch. 0.6 Publication under J. Schur
^Ledermann, W. (1983). "Issai Schur and His School in Berlin". Bulletin of the London Mathematical Society. 15 (2): 97–106. doi:10.1112/blms/15.2.97.