Symmetric matrices
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- Опубліковано 9 лют 2025
- Symmetric matrices have nice properties. We show that all the eigenvalues of a real symmetric matrix are real, and the eigenvectors corresponding to different eigenvalues are orthogonal. We further show the diagonalization and the spectral decomposition of a symmetric matrix.
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The eigen basis is dual to the standard basis -- conjugacy is dual, spectral decomposition.
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Real is dual to imaginary -- complex numbers are dual.
Antipodal points identify for the rotation group SO(3) -- stereographic projection.
You really love duality!
@@BruneiMathClub Yes, duality means that there is a 4th law of thermodynamics.
Anything which is dual to entropy is by definition the 4th law of thermodynamics:-
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Syntax is dual to semantics -- languages, communication.
If mathematics is a language then it is dual.
All observers make predictions to track targets, goals and objectives and this is a syntropic process -- teleological.
The Einstein reality criterion:-
"If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity)
the value of a physical quantity, then there exists an element of reality corresponding to that quantity."
(Einstein, Podolsky, Rosen 1935, p. 777)
Internet Encyclopedia of Philosophy:-
www.iep.utm.edu/epr/
According to Einstein reality is predicted into existence -- a syntropic process, teleological.
Your brain/mind creates models or predictions of reality hence your mind is syntropic (convergent).
Here is a video of some well known physicists talking about duality, watch at 1 hour 4 minutes:-
ua-cam.com/video/UjDxk9ZnYJQ/v-deo.html
Mathematics is full of dualities (see next comment).
Once you accept the 4th law this means that that there is a 5th law of thermodynamics:-
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is a symmetry and it is being conserved according to Noether's theorem.
Energy is duality, duality is energy -- the 5th law of thermodynamics!
Potential energy is dual to kinetic energy -- gravitational energy is dual.
@@BruneiMathClub Here are some examples of duality in mathematics:-
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Homology is dual to co homology -- the word co means mutual and implies duality.
Sine is dual to cosine or dual sine -- perpendicularity.
Sinh is dual to cosh -- hyperbolic functions.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy).
Convergence (syntropy) is dual to divergence (entropy).
Injective is dual to surjective synthesizes bijective or isomorphism.
The word isomorphism actually means duality.
Subgroups are dual to subfields -- the Galois correspondence.
Positive is dual to negative -- electric charge or numbers.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Professor Norman J. Wildberger, universal hyperbolic geometry:-
ua-cam.com/video/EvP8VtyhzXs/v-deo.html
All observers have a syntropic or hyperbolic perspective of reality.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The icosahedron is dual to the dodecahedron.
Waves are dual to particles -- quantum duality or pure energy is dual.
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem.
Bosons are dual to Fermions -- atomic duality.
Pure energy is dual and it is being conserved -- the 5th law of thermodynamics!
@@BruneiMathClub Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new ideas or concepts all the time from their perceptions, measurements, observations or intuitions -- they are using duality!
The bad news is that Immanuel Kant has been completely ignored for over 200 years and this is why you need new laws of physics!
Antinomy (duality) is two truths that contradict each other -- Immanuel Kant.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.