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Location of North and South Celestial Poles

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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Classical Physics

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Archimedes Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Refractive Index
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Rotational Dynamics
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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Wien's Displacement Law
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Climate Change

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Cosmology

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Penrose Diagrams
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Geometries of the Universe
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Inflation Theory
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Introduction to Black Holes
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Finance and Accounting

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Periodically and Continuously Compounded Interest
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General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Tensors
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The Area Metric
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The Metric Tensor
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Macroeconomics

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Mathematics

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Amplitude, Period and Phase
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Building Groups From Other Groups
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Complex Numbers
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Vector Calculus
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Volume Integrals

Microeconomics

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Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Spin 1 Eigenvectors
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Probability and Statistics

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Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Qubit
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Newton versus Einstein
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The Lorentz Group
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Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

test

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test

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: December 30, 2019

Similar Matrices and Diagonalization ------------------------------------ Change of Basis --------------- Consider ℝ2 as a subspace of ℝ2 with the following basis, B: - - - - B = {| 2 |, | 1 |} | 1 | | 2 | - - - - Consider an arbitrary vector in this basis, v: - - - - - - [v]B = 3| 2 | + 2| 1 | = | 8 |   | 1 | | 2 | | 7 | - - - - - - This can be written as: - - - - [v]B = | 3 || [v1]B |   | 2 || [v2]B | - - - - - - Where | 3 | is called the CHANGE OF BASIS matrix. | 2 | - - - - Now, | 8 | is just a vector in the STANDARD BASIS. | 7 | - - Therefore, - - - - [v]S = 8| 1 | + 7| 0 |   | 0 | | 1 | - - - - - - - - = | 8 || [v1]S | | 7 || [v2]S | - - - - The following diagram shows this: In general we can write: C[v]B = [v]S Where, C = Change of basis matrix. [v]B = The vector in the B basis [v]B = The vector in the Standard Basis. Here is another example. Consider the 3-dimensional subspace of ℝ4 (which is not the same as ℝ3 because the dimension of the vectors is 4). - - - - - - | -1 | | -1 | | -1 | B = {| 1 |,| 0 |,| 0 |} | 0 | | 1 | | 0 | | 0 | | 0 | | 1 | - - - - - - - - - - - -   | -1 | | -1 | | -1 | [v]B = 3| 1 | + 2| 0 | + 4| 0 |   | 0 | | 1 | | 0 |   | 0 | | 0 | | 1 | - - - - - - - - - - | -1 -1 -1 || 3 | = | 1 0 0 || 2 | | 0 1 0 || 4 | | 0 0 1 | - - - - - - | -9 | = | 3 | | 2 | | 4 | - - - - - - - - - - | 1 | | 0 | | 0 | | 0 | = -9| 0 | + 3| 1 | + 2| 0 | + 4| 0 | | 0 | | 0 | | 1 | | 0 | | 0 | | 0 | | 0 | | 1 | - - - - - - - - - - - - | -9 || [v1]S | = | 3 || [v2]S | | 2 || [v3]S | | 4 || [v4]S | - - - - Now suppose we are only given [v]B and [v]S and want to find C. - - - - - - | -1 -1 -1 || c1 | | -9 | | 1 0 0 || c2 | = | 3 | | 0 1 0 || c3 | | 2 | | 0 0 1 | - - | 4 | - - - - At first sight we might think that we could rearrange [v]BC = [v]S to get C = [v]B-1[v]S However, [v]B is not invertible. To overcome this we can write the equation in augmented matrix form and use Gauss-Jordan elimination to get the answer. - - | -1 -1 -1 | -9 | | 1 0 0 | 3 | => c1 = 3, c2 = 2 and c3 = 4 | 0 1 0 | 2 | | 0 0 1 | 4 | - - Alternatively we can compute the reduced row echelon form (rref): - - | 1 0 0 | 3 | | 0 1 0 | 2 | => c1 = 3, c2 = 2 and c3 = 4 | 0 0 1 | 4 | - - Transformations --------------- We have seen how vectors are affected by a change in basis. We now want to see what happens with transformations. Consider a transformation, T. This can be written in matrix form as: T(v) = Mv Where M is the transformation matrix and T: ℝn -> ℝn In the B basis we can write: [T(v)]B = [N]B[v]B Where [N]B is the transformation matrix in the B basis. C[v]B = [v]S ∴ [v]B = C-1[v]S [N]B[v]B = [T(v)]B = [Mv]B since [T(v)]B = [Mv]B = C-1[Mv]S = C-1[M]S[v]S = C-1[M]SC[v]B We conclude: [N]B = C-1[M]SC Where, [N]B is the transformation matrix in the B basis. [M]S is the transformation matrix in the S basis. C is the change of basis matrix for B. Schematically: [M]S Standard basis: v ---------> T(v) |    | | C-1 | C-1 | [N]B   | B basis: v ---------> v [v]B [T(v)]B Similar Matrices and Diagonalization ------------------------------------ Two matrices are similar if they satisfy the above relationship: P-1AP = Q or, AP = PQ Where P is any invertible n x n matrix. Proof: P-1AP = Q If we consider a basis consisting of eigenvectors of the space, we can apply the the eigenvalue equation Qv = λv. Therefore, after substitution: (P-1AP)v = λv (P-1AP)P-1v = λP-1v P-1Av = λP-1v Qv = λv Therefore, the eigenvalues of A and Q are the same. Also consider the characteristic polynomials of A and Q: Ax = λx Therefore, (A - λI)x = 0 This has a solution only if det(A - λI) = 0 We can also write a similar equation for B: det(Q - λI)x = 0 Since λ can take the value 0 we can write: det(A) = det(Q) Thus, having the same eigenvalues implies that the determinants are also equal. The important result of these proofs is that similar matrices have the same trace, eigenvalues, determinant and dimension. However, they can have different eigenvectors. Example 1: Matrices with unique eigenvectors. - - - - W = | -2 6 | Q = | -4 3 | | -2 5 | | -10 7 | - - - - W and Q are certainly similar. - - - - - - | -2 6 | has eigenvectors | 3 | and | 2 | | -2 5 | | 2 | | 1 | - - - - - - - - - - - - | -4 3 | has eigenvectors | 1 | and | 3 | | -10 7 | | 2 | | 5 | - - - - - - We can find P as follows: - - - - - - - - | -2 6 || a b | = | a b || -4 3 | | -2 5 || c d | | c d || 10 7 | - - - - - - - - - - - - | -2a + 6c -2b + 6d | = | -4a - 10b 3a + 7b | | -2a + 5c -2b + 5d | | -4c - 10d 3c + 7d | - - - - Therefore, a = -3c - 5b and d = -b - 3c/2 Choosing b = 1 and c = 0 gives: - - P = | -5 1 | | 0 -1 | - - Confirming: - - - - - - | -2 6 || -5 1 | = | 10 -8 | | -2 5 || 0 -1 | | 10 -7 | - - - - - - and, - - - - - - | -5 1 || -4 3 | = | 10 -8 | | 0 -1 || -10 7 | | 10 -7 | - - - - - - P-1WP gives: - - - - - - - - | -5 1 |-1| -2 6 || -5 1 | = | -4 3 | | 0 -1 |  | -2 5 || 0 -1 | | -10 7 | - - - - - - - - Trace = 3, det = 2, λ = 2 and 1 and, - - - - - - - - | -5 1 |-1| -4 3 || -5 1 | = | -14 24/5 | | 0 -1 |  | -10 7 || 0 -1 | | -50 17 | - - - - - - - - Trace = 3, det = 2, λ = 2 and 1 It is easily seen that there are many versions of P that satisfy WP = PB and hence A has a whole family of matrices that are similar to it! Example 2: Matrices with non-unique eigenvectors. - - - - W = | 2 1 | Q = | 0 2 | | 0 2 | | -2 4 | - - - - W and Q are certainly similar. - - - - | 2 1 | has 2 identical eigenvectors | 1 | | 0 2 | | 0 | - - - - - - - - | 0 2 | has 2 identical eigenvectors | 1 | | -2 4 | | 1 | - - - - We can find P as before: - - - - - - - - | 2 1 || a b | = | a b || 0 2 | | 0 2 || c d | | c d || -2 4 | - - - - - - - - This leads to the system of equations: 2a + c + 2b = 0 ∴ c = -2a - 2b -2b - 2a + d = 0 2c + 2d = 0 ∴ c = -d -2d - 2c = 0 Choosing a = 0 and b = 1 gives: - - P = | 0 1 | | -2 2 | - - P-1AP gives: - - - - - - - - | 0 1 |-1| 2 1 || 0 1 | = | 0 2 | | -2 2 |  | 0 2 || -2 2 | | -2 4 | - - - - - - - - Again, it is easily seen that there are many versions of P that satisfy WP = PB and hence A has a whole family of matrices that are similar to it also. Diagonalization --------------- In the case of a matrix that has a set of n linearly independent (unique) eigenvectors, the similarity transformation can also be used to create an another similar matrix that has a diagonal form. In this case the diagonal form is obtained by constructing S such that the columns are the eigenvectors of the matrix being transformed. Again, diagonalization is only possible if and only if the n x n matrix posesses n eigenvectors that are unique. Proof: We start with the eigenvalue equation and write: Av1 = λ1v1, Av2 = λ2v2 ... Avn = λnvn where the v's are column (eigen) vectors with n entries. Let us write P as: - - |   .   . .  | | v1 . v2 . --- . vn | |   .   . .  | |   .   . .  | - - Therefore, - - |   .   . .  | AP = A| v1 . v2 . --- . vn | |   .   . .  | |   .   . .  | - - - - |   .   . .  | = | Av1 . Av2 . --- . Avn | |   .   . .  | |   .   . .  | - - - - |    .    . .   | = | λ1v1 . λ2v2 . --- . λnvn | |    .    . .   | |    .    . .   | - - - - - - |   .   . .  || λ1 | = | v1 . v2 . --- . vn || λ2 | |   .   . .  || .  | |   .   . .  || λn | - - - - So we have: AP = PΛ or, P-1AP = P-1PΛ = Λ - the diagonal matrix of eigenvalues. Example 3: Consider the example from before that had unique eigenvectors: - - - - - - | -2 6 | with eigenvectors | 3 | and | 2 | | -2 5 | | 2 | | 1 | - - - - - - Thus, - - - - PW = | 3 2 | PW-1 = | -1 2 |   | 2 1 |    | 2 -3 | - - - - and, PW-1WPW gives: - - - - - - - - | -1 2 || -2 6 || 3 2 | = | 2 0 | | 2 -3 || -2 5 || 2 1 | | 0 1 | - - - - - - - - Likewise, - - - - - - | -4 3 | with eigenvectors | 1 | and | 3 | | -10 7 | | 2 | | 5 | - - - - - - Thus, - - - - PQ = | 1 3 | PQ-1 = | -5 3 |   | 2 5 |    | 2 -1 | - - - - and PQ-1QPQ gives: - - - - - - - - | -5 3 || -4 3 || 1 3 | = | 2 0 | | 2 -1 || -10 7 || 2 5 | | 0 1 | - - - - - - - - Note that: - - - - - - - - | -2 6 || 3 2 | - | 3 2 || 2 0 | = 0 | -2 5 || 2 1 | | 2 1 || 0 1 | - - - - - - - - and, - - - - - - - - | -4 3 || 1 3 | - | 1 3 || 2 0 | = 0 | -10 7 || 2 5 | | 2 5 || 0 1 | - - - - - - - - The fact that diagonalization results in identical matrices is another way of confirming the similarity of W and Q. If we are given the eigenvalues and eigenvectors we can construct the original matrix, W, using the transformation W = PΛP-1. Using the above example, - - - - - - - - W = | 3 2 || 2 0 || 3 2 |-1 = | -2 6 | | 2 1 || 0 1 || 2 1 |   | -2 5 | - - - - - - - - Simultaneous Diagonalization ---------------------------- If 2 or more matrices have the same eigenvectors with each eigenvector being unique, then the matrices can be simultaneous diagonalized using the same transformation matrix. For example, - - - - W = | 2 1 | and Q = | 3 4 | | 1 2 | | 4 3 | - - - - have the same eigenvectors. They are: - - - - | 1 | and | 1 | | 1 | | -1 | - - - - - - W can be diagonalized to | 3 0 | = W' | 0 1 | - - - - Q can be diagonalized to | 7 0 | = Q' | 0 -1 | - - The product of WQ and QW is: - - WQ = QW = | 10 11 | therefore [W,Q] = 0 | 11 10 | - - From this we can conclude that matrices that can be simultaneously diagonalized commute with each other.