Applications of eigenvalues and eigenvectors by 
Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01- math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors/3503875#3503875
- math.stackexchange.com/questions/1520832/real-life-examples-for-eigenvalues-eigenvectors
- matheducators.stackexchange.com/questions/520/what-is-a-good-motivation-showcase-for-a-student-for-the-study-of-eigenvalues
Set of eigenvalues of a linear operator.
Continuous spectrum (functional analysis) by Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01Unlike the simple case of a matrix, in infinite dimensional vector spaces, the spectrum may be continuous.
The quintessential example of that is the spectrum of the position operator in quantum mechanics, in which any real number is a possible eigenvalue, since the particle may be found in any position. The associated eigenvectors are the corresponding Dirac delta functions.
Every invertible matrix can be written as:where:Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.
- is a diagonal matrix containing the eigenvalues of
- columns of are eigenvectors of
Intuitively, Note that this is just the change of basis formula, and so:
- changes basis to align to the eigenvectors
- multiplies eigenvectors simply by eigenvalues
- changes back to the original basis
Eigendecomposition of a real symmetric matrix by Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01The general result from eigendecomposition of a matrix:becomes:where is an orthogonal matrix, and therefore has .
Matrix congruence can be seen as the change of basis of a bilinear form by Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01From effect of a change of basis on the matrix of a bilinear form, remember that a change of basis modifies the matrix representation of a bilinear form as:
So, by taking , we understand that two matrices being congruent means that they can both correspond to the same bilinear form in different bases.
Eigenvectors and eigenvalues of the identity matrix by Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-20 +Created 1970-01-01This is the possibly infinite dimensional version of a Hermitian matrix, since linear operators are the possibly infinite dimensional version of matrices.
There's a catch though: now we don't have explicit matrix indices here however in general, the generalized definition is shown at: en.wikipedia.org/w/index.php?title=Hermitian_adjoint&oldid=1032475701#Definition_for_bounded_operators_between_Hilbert_spaces
A good definition is that the sparse matrix has non-zero entries proportional the number of rows. Therefore this is Big O notation less than something that has non zero entries. Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.
Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.
Co-founder of PowerDNS, an open source dNS implementation.
Homepage: berthub.eu/ says:.
All stuff Ciro cares about too! Cool dude! In particular Ciro loved his quote of I should have loved biology.
All stuff Ciro cares about too! Cool dude! In particular Ciro loved his quote of I should have loved biology.
He's writing a fun-sounding book about molecular biology as of 2022: berthub.eu/dna-book. Appears to be closed source though. Ciro wonders if he really needs to sell the book for money after all those years though, rather than just publishing it online for free.
Looking at Bert just brings the Dutch Golden Age, and more in particular Antonie van Leeuwenhoek to mind. Epic.
How life is Digital by Bert Hubert (2021)
Source. Just a "boring" overview of the central dogma of molecular biology ;-)A monopolistic operating system that only exists in the 2010's because of the IBM-linked historical lock-in and constant useless changes of the Microsoft Word document format to prevent cross operability.
It offers no technical advantages over free Linux distros in the late 2010's, and it is barely impossible to buy a non-Mac computer without paying for it, which should be illegal. European Union, time to use your regulatory powers.
The following anecdote illustrates Windows' pervasiveness. Ciro Santilli was once tutoring a high school student in Brazil, and decided to try and get her into programming. When the "Windows is not free" subject came up, the high school student was shocked: "I paid 100 dollars for this?". She never even knew it was there. To her, it was "just a computer".
Laws should really be passed forcing OEMs to allow you to not buy Microsoft Windows when buying a computer, European Union, why have you failed me in 2016??? en.wikipedia.org/wiki/Bundling_of_Microsoft_Windows
blog.zorinaq.com/i-contribute-to-the-windows-kernel-we-are-slower-than-other-oper/ I Contribute to the Windows Kernel. We Are Slower Than Other Operating Systems. Here Is Why. by Marc Bevand (2013) has some interesting remarks:
There's also little incentive to create changes in the first place. On linux-kernel, if you improve the performance of directory traversal by a consistent 5%, you're praised and thanked. Here, if you do that and you're not on the object manager team, then even if you do get your code past the Ob owners and into the tree, your own management doesn't care. Yes, making a massive improvement will get you noticed by senior people and could be a boon for your career, but the improvement has to be very large to attract that kind of attention.
Bomb disposal robot by The IT Crowd
. Source. Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
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Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Internal cross file references done right:
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