Hermitian Matrix Puns

Hello,

I've come upon this question and I'm not sure how to approach it. I know that the algebraic multiplicity is how many times an eigenvalue appears as an answer to the characteristic equation, and geometric multiplicity is how many linearly independent eigenvectors correspond to each eigenvalue. Hermitian matrices are symmetric and equal to their conjugate transpose. Where do I go from here in order to prove this?

Is there a LAPACK subroutine where it checks the matrix if it's hermitian or not?

This is really important to me because my Hamiltonian matrix yielded complex eigenvalues and I assumed that the matrix was hermitian (96x96 matrix). I wanted to make sure that my matrix is hermitian. Thank you!

Call a and b two real vectors, and A is a hermitian matrix.

Is a^T A b = b^T A a ?

Thank you!

If there's an observable (a hermitian matrix M) that has multiple eigenvectors |λ_i> that each share the same eigenvalue λ, and you measure the observable and discover λ, the system must change to one of the eigenvectors |λ_i>. How do you know which one?

Hi there

I am used to MATLAB, but for my current purpose it is too slow, and I am therefore trying to use FORTRAN 95, which is giving me some headaches. I need to calculate the eigenvalues of a matrix, and for that purpose I installed lapack using apt-get install liblapack-dev

I can compile and run the programs on this web site: http://people.sc.fsu.edu/~jburkardt%20/f_src/lapack_examples/lapack_examples.html (the one named apack_prb.f90, near the bottom), so I would think that everything is installed correctly.

To find the eigenvalues and eigenvectors, I followed the recommendations here: http://software.intel.com/sites/products/documentation/hpc/mkl/mklman/index.htm#GUID-A37AAAD0-DBF8-48BD-91AB-446CCAB4537F.htm and am trying to use the heevd routine (http://software.intel.com/sites/products/documentation/hpc/mkl/mklman/index.htm#GUID-A37AAAD0-DBF8-48BD-91AB-446CCAB4537F.htm)

In my code, I implement this simply as call heevd(Hup, Evecup, 'V'), where Hup is a 16x16 matrix and Evecup is declared as an array with 16 entries.

I try to compile the code with gfortran -o test test.f90 -L$HOME/libf77/$ARCH -lblas -llapack, and get the following error: /tmp/ccBaf67l.o: In function MAIN__': test.f90:(.text+0x33ff): undefined reference toheevd_' collect2: error: ld returned 1 exit status

I have no idea what to do - can anyone enlighten me?

I use Ubuntu 12.10.

Edit: Rentheil's solution seems to work! Thank you very much.

Edit2: I get errors when trying to run the code. Some example code is below in the comments

Hi,

I was wondering if I have two matrices of complex numbers A, B and I wish to multiply them, do I just follow a standard matrix multiplication procedure? I have seen some posts which treat this as a standard multiplication, however, others specify that the Hermitian inner product needs to be computed i.e. AxB^H where B^H is the conjugate transpose (Hermitian transpose) of B.

Under what circumstances would the Hermitian transpose need to be taken before multiplying two complex matrices?

I have an operator Q = e^iA where A is a matrix. I want to compute Q^†. What is the proper way to think about this operation? Do i take the complex conjugate of the exponential with A^† as the argument, i.e. e^-iA^† ? Do I need to take the taylor expansion into account? Q^† = 1 - iA^† - A^(2†)/2! + ... ?

Show that the eigenvalues of a skew-Hermitian matrix are either zero or purely imaginary.

I am currently taking the second level Linear Algebra course which has this description:

>Vector spaces. Inner product spaces. Examples of n-space and the space of continuous functions. Gram-Schmidt process, QR factorization of a matrix and least squares. Linear transformations, change of basis, similarity and diagonalization. Orthogonal diagonalization, quadratic forms. Applications in a variety of fields.

And I'm wondering if it would also be worth taking this third course, Linear Algebra III, which has the description:

>Hermitian and unitary matrices, spectral theorem. Jordan canonical form. Cayley-Hamilton Theorem. Bilinear forms, positive-definiteness, Sylvester’s Law of inertia, geometric lattices. Numerical methods. Application to discrete system evolution, matrix exponentials and differential equations.

It's either this or Graph Theory, which would you recommend? Thanks in advance.

This review consists of two parts: an overview with marked spoilers and unmarked mild thematic spoilers to help potential readers decide if this work is right for them, and a more in-depth analysis, which contains unmarked moderate spoilers for Orthogonal.

#Overview

Tenet: Don't try to understand it, feel it.

Me: Oh come on, Nolan. That's just lazy, now. You call this sci-fi? You know you can't just slap some vague technobabble on it and call it a day, right? I hate how so much science fiction feels the need to dumb it down for the masses. What's the point of hiring renowned physicists to consult on your movies if you end up with scenes where your pilot has to have wormholes explained to him when he's literally in sight of the one he's about to fly through? I wish there was something out there that puts in some actual effort in the worldbuilding, something smart. Something that doesn't treat its audience like idiots. Such a shame, too. Inverted matter looks really cool.

Monkey's paw: *curls*

...

Well, the good news is, I found a story like that. The bad news is, having read it, I now feel like an idiot.

Welcome to the world(s) of Greg Egan, who puts the 'hard' in 'hard science fiction' in more ways than one.^1 He's the author of works such as Permutation City and Sqchild's Ladder, which are considered some of the hardest science fiction novels ever. But while those are set in more-or-less our world, Egan's most valuable contribution to the genre (at least, in my opinion), is his alternate universes, which run on different laws of physics from ours.

Orthogonal is a trilogy of novels (The Clockwork Rocket, The Eternal Flame, & The Arrows of Time; roughly 1100 pages total) set in one such universe. I chose it as my first Egan work to read largely because of its concept, which was just too intriguing for me to resist. From the blurb on the website:^2

>In Yalda’s universe, light has no universal speed and its creation generates energy.

>On Yalda’s world, plants make food by emitting their own light into the dark night sky.

>As a child Yalda witnesses one of a series of strange meteors, the Hurtlers, that are entering the planetary sys

Have you heard of the Neumann series? 🤔

"Neumann Series for Matrix Inversion and Its Approximation"
(6 min 45 sec)
https://youtu.be/DM6brJrJYbM

A short summary:

• Neumann series is the matrix generalization of geometric series, which can be used for the approximation of matrix inversion.
• But, the caveat is that the contraction condition and the time complexity are important for successful applications.
• As a real-world example, we briefly show that the Neumann series can be used to optimize the millions of hyperparameters in deep neural networks through the work from Lorraine et al. (AISTAT 2020).

Cheers!

Do your worst!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

I'm surprised it hasn't decade.

They’re on standbi

Buenosdillas

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

I won't be doing that today!

So SU(2) has three generators (associated in some way with the weak interaction bosons). These generators are the said to be the Pauli matrices as they have determinant =1 and they form a basis for the 2x2 unitary matrices. Please correct me if I'm wrong.

However when I try to form a basis for U(2), I end up with 4 matrices, the identity and the 3 pauli matrices. Why is the identity not one of the generators of SU(2)?

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "