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Title: GitHub - mitmath/matrixcalc: MIT IAP short course: Matrix Calculus for Machine Learning and Beyond

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READMEhttps://patch-diff.githubusercontent.com/mitmath/matrixcalc
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#matrix-calculus-for-machine-learning-and-beyond
IAPhttps://elo.mit.edu/iap/
Alan Edelmanhttps://math.mit.edu/~edelman/
Steven G. Johnsonhttps://math.mit.edu/~stevenj/
Matrix Calculus in IAP 2023 (OCW)https://ocw.mit.edu/courses/18-s096-matrix-calculus-for-machine-learning-and-beyond-january-iap-2023/
on githubhttps://github.com/mitmath/matrixcalc/tree/iap2023
on YouTubehttps://www.youtube.com/playlist?list=PLUl4u3cNGP62EaLLH92E_VCN4izBKK6OE
Matrix Calculus in IAP 2022 (OCW)https://ocw.mit.edu/courses/18-s096-matrix-calculus-for-machine-learning-and-beyond-january-iap-2022/pages/lecture-notes-and-readings/
on githubhttps://github.com/mitmath/matrixcalc/tree/iap2022
Matrix Calculus 2024 (github)https://github.com/mitmath/matrixcalc/tree/iap2024
Matrix Calculus 2025 (github)https://github.com/mitmath/matrixcalc/tree/iap2025
via Canvashttps://canvas.mit.edu/courses/35760
18.063 COURSE NOTEShttps://www.dropbox.com/scl/fi/iq4plt8oqja845cuuosa4/Matrix-Calculus-latest.pdf?rlkey=nsnytdu28jje41nhh1bl2dbba&st=i6lfha0r&dl=0
Piazzahttps://piazza.com/class/mkab8649oo96qm/
18.06https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
18.02https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/
Julia languagehttps://github.com/mitmath/julia-mit
these instructionshttps://github.com/mitmath/julia-mit#installing-julia-and-ijulia-on-your-own-computer
https://mybinder.org/v2/gh/mitmath/binder-env/main
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-1-jan-12
slideshttps://docs.google.com/presentation/d/16uwYARbg4unaGU4Enp6uQvlBb6N21j1UINQW99om6R4/edit?usp=sharing
linear operatorhttps://en.wikipedia.org/wiki/Linear_map
Jacobian matrixhttps://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
Kronecker productshttps://en.wikipedia.org/wiki/Kronecker_product
matrixcalculus.orghttp://www.matrixcalculus.org/
Matrix Cookbookhttps://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf
notes on vector and matrix differentiationhttps://cdn-uploads.piazza.com/paste/j779e63owl53k6/04b2cb8c2f300212d723bea822a6b856085b28e28ca9debc75a05761a436499c/6.S087_Lecture_2.pdf
Fréchet derivativehttps://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative
asymptotic notationhttps://en.wikipedia.org/wiki/Big_O_notation
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-2-jan-14
slideshttps://docs.google.com/presentation/d/16uwYARbg4unaGU4Enp6uQvlBb6N21j1UINQW99om6R4/edit?usp=sharing
vectorizationhttps://en.wikipedia.org/wiki/Vectorization_(mathematics)
Kronecker productshttps://en.wikipedia.org/wiki/Kronecker_product
2×2 Matrix Jacobians (html)https://rawcdn.githack.com/mitmath/matrixcalc/3f6758996e40c5c1070279f89f7f65e76e08003d/notes/2x2Jacobians.jl.html
(pluto notebook source code)https://github.com/mitmath/matrixcalc/blob/main/notes/2x2Jacobians.jl
(jupyter notebook)https://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/notes/2x2Jacobians.ipynb
linear formhttps://en.wikipedia.org/wiki/Linear_form
Riesz representation theoremhttps://en.wikipedia.org/wiki/Riesz_representation_theorem
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-3-jan-16
slideshttps://docs.google.com/presentation/d/1U1lB5bhscjbxEuH5FcFwMl5xbHl0qIEkMf5rm0MO8uE/edit?usp=sharing
"Frobenius" inner producthttps://en.wikipedia.org/wiki/Frobenius_inner_product
pset 1https://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/psets/pset1.pdf
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-4-jan-21
handwritten noteshttps://www.dropbox.com/scl/fi/byg5mpcnnk4xh9tqjbjmk/Inner-Products-and-Norms.pdf?rlkey=egsdhyee9go9v17iuxxqx1edj&dl=0
gradient of the determinanthttps://rawcdn.githack.com/mitmath/matrixcalc/b08435612045b17745707f03900e4e4187a6f489/notes/determinant_and_inverse.html
Hilbert spacehttps://en.wikipedia.org/wiki/Hilbert_space
linear formhttps://en.wikipedia.org/wiki/Linear_form
must be a dot producthttps://en.wikipedia.org/wiki/Riesz_representation_theorem
steepest-ascenthttps://math.stackexchange.com/questions/223252/why-is-gradient-the-direction-of-steepest-ascent
requirements for an inner producthttps://en.wikipedia.org/wiki/Inner_product_space
Cauchy–Schwarz inequalityhttps://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
Frobenius inner producthttps://en.wikipedia.org/wiki/Frobenius_inner_product
adjugatehttps://en.wikipedia.org/wiki/Adjugate_matrix
normhttps://en.wikipedia.org/wiki/Norm_(mathematics)
in finite dimensions all norms are equivalent up to constant factorshttps://math.mit.edu/~stevenj/18.335/norm-equivalence.pdf
derivative of a determinanthttps://en.wikipedia.org/wiki/Jacobi%27s_formula
"adjugate" matrixhttps://en.wikipedia.org/wiki/Adjugate_matrix
Laplace expansionhttps://en.wikipedia.org/wiki/Laplace_expansion
derivative of log(det A)https://statisticaloddsandends.wordpress.com/2018/05/24/derivative-of-log-det-x/
Matrix Cookbookhttps://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf
nonlinear eigenvalueshttps://en.wikipedia.org/wiki/Nonlinear_eigenproblem
apply Newton's methodhttps://www.maths.manchester.ac.uk/~ftisseur/talks/FT_talk2.pdf
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-5-jan-23
handwritten backpropagation noteshttps://www.dropbox.com/scl/fi/bke4pbr342e1jhv9qytg1/NN-Backpropagation.pdf?rlkey=b7krtzdt4hgsj63zyq9ok2gqv&dl=0
dual numbershttps://en.wikipedia.org/wiki/Dual_number
Julia notebookhttps://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/notes/AutoDiff.ipynb
pset 1 solutionshttps://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/psets/pset1sol.pdf
pset 2https://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/psets/pset2.pdf
Strang (2019)https://math.mit.edu/~gs/learningfromdata/
18.065 OCW lecture 27https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-27-backpropagation-find-partial-derivatives/
backpropagationhttps://en.wikipedia.org/wiki/Backpropagation
backpropagation/adjoint methods for recurrence relations (course notes)https://math.mit.edu/~stevenj/18.336/recurrence2.pdf
computational graphs (blog post)https://colah.github.io/posts/2015-08-Backprop/
ForwardDiff.jlhttps://github.com/JuliaDiff/ForwardDiff.jl
this paperhttps://arxiv.org/abs/1607.07892
dual numberhttps://en.wikipedia.org/wiki/Dual_number
AMS review articlehttp://www.ams.org/publicoutreach/feature-column/fc-2017-12
lecture notes on automatic differentiation (Julia-based)https://adrhill.github.io/julia-ml-course/L6_Automatic_Differentiation/
TaylorDiff.jlhttps://github.com/JuliaDiff/TaylorDiff.jl
Zoom (MIT only)https://mit.zoom.us/j/98915152715?pwd=4GftZplphHYx7QIlDUL4vgiwzD7Rxc.1
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-6-jan-26-via-zoom-mit-only
slideshttps://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/notes/gilbert_autodiff_2023.pdf
"Backpropagation through back substitution with a backslash (2003)https://arxiv.org/abs/2303.15449
Julia notebookhttps://simeonschaub.github.io/ReverseModePluto/notebook.html
Prof. Edelman's posterhttps://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/notes/backprop_poster.pdf
calculus on computational graphshttps://colah.github.io/posts/2015-08-Backprop/
course noteshttp://www.cs.columbia.edu/~mcollins/ff2.pdf
tradeoffs in ADhttps://www.stochasticlifestyle.com/engineering-trade-offs-in-automatic-differentiation-from-tensorflow-and-pytorch-to-jax-and-julia/
AD limitationshttps://discourse.julialang.org/t/open-discussion-on-the-state-of-differentiable-physics-in-julia/72900/2
"calculus of variations"https://en.wikipedia.org/wiki/Calculus_of_variations
Euler–Lagrange equationhttps://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation
Lagrangian mechanicshttps://en.wikipedia.org/wiki/Lagrangian_mechanics
Brachistochrone problemhttps://en.wikipedia.org/wiki/Brachistochrone_curve
optimal controlhttps://en.wikipedia.org/wiki/Optimal_control
Calculus of Variations by Gelfand and Fominhttps://store.doverpublications.com/0486414485.html
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-7-jan-28
handwritten noteshttps://www.dropbox.com/scl/fi/cn2cmzf1q2anpeeg9s5a4/CR-Calculus.pdf?rlkey=b0qd9b3r8ldc9tu0s80nxpvir&dl=0
(html)https://rawcdn.githack.com/mitmath/matrixcalc/d11b747d70a5d9e1a3da8cdb68a7f8a220d3afae/notes/symeig.jl.html
(julia source)https://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/notes/symeig.jl
The Complex Gradient Operator and the CR-Calculushttps://arxiv.org/abs/0906.4835
Wirtinger derivativeshttps://en.wikipedia.org/wiki/Wirtinger_derivatives
complex functions in ChainRules.jlhttps://juliadiff.org/ChainRulesCore.jl/dev/maths/complex.html
complex functions in JAXhttps://jax.readthedocs.io/en/latest/notebooks/autodiff_cookbook.html#complex-numbers-and-differentiation
holomorphichttps://en.wikipedia.org/wiki/Holomorphic_function
analytichttps://en.wikipedia.org/wiki/Analytic_function
complex analysishttps://en.wikipedia.org/wiki/Complex_analysis
18.04https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/
tangent spaceshttps://en.wikipedia.org/wiki/Tangent_space
Lagrange multipliershttps://en.wikipedia.org/wiki/Lagrange_multiplier
Convex Optimizationhttps://web.stanford.edu/~boyd/cvxbook/
"time-independent" perturbation theoryhttps://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#Time-independent_perturbation_theory
Hellmann–Feynmann theoremhttps://en.wikipedia.org/wiki/Hellmann%E2%80%93Feynman_theorem
18.335 notes on adjoint methodshttps://github.com/mitmath/18335/blob/spring21/notes/adjoint/adjoint.pdf
implicit functionhttps://en.wikipedia.org/wiki/Implicit_function
Cox (1995)https://doi.org/10.1006/jfan.1995.1117
Seyranian et al. (1994)https://doi.org/10.1007/BF01742705
Stechlinski (2022)https://doi.org/10.1016/j.laa.2022.04.019
degenerate perturbation theoryhttps://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/a0889c5ca8a479c3e56c544d646fb770_MIT8_06S18ch1.pdf
Nesterov (2005)https://doi.org/10.1007/s10107-005-0633-0
Barton et al (2017)https://doi.org/10.1080/10556788.2017.1374385
SDPhttps://en.wikipedia.org/wiki/Semidefinite_programming
(Men et al., 2014)http://doi.org/10.1364/OE.22.022632
defective matrixhttps://en.wikipedia.org/wiki/Defective_matrix
https://patch-diff.githubusercontent.com/mitmath/matrixcalc#lecture-9-jan-30
delta functions and distributional derivativeshttps://math.mit.edu/~stevenj/18.303/delta-notes.pdf
pset 2 solutionshttps://patch-diff.githubusercontent.com/mitmath/matrixcalc/blob/main/psets/pset2sol.pdf
Bilinear formshttps://en.wikipedia.org/wiki/Bilinear_form
symmetric bilinear formshttps://en.wikipedia.org/wiki/Symmetric_bilinear_form
quadratic formhttps://en.wikipedia.org/wiki/Quadratic_form
Hessian matrixhttps://en.wikipedia.org/wiki/Hessian_matrix
introduction to quadratic approximationhttps://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/quadratic-approximations/a/quadratic-approximation
Positive-definitehttps://en.wikipedia.org/wiki/Definite_matrix
definite quadratic formshttps://en.wikipedia.org/wiki/Definite_quadratic_form
more formal treatmentshttp://www.columbia.edu/~md3405/Unconstrained_Optimization.pdf
simple 2-variable versionhttps://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/optimizing-multivariable-functions/a/second-partial-derivative-test
stackexchange discussionhttps://math.stackexchange.com/questions/2285282/relating-condition-number-of-hessian-to-the-rate-of-convergence
ill-conditionedhttps://nhigham.com/2020/03/19/what-is-a-condition-number/
course notes on the topichttps://www.cs.toronto.edu/~rgrosse/courses/csc421_2019/slides/lec07.pdf
sequential quadratic optimizationhttps://web.stanford.edu/class/ee364b/lectures/seq_notes.pdf
notes on BFGS quasi-Newton methodshttps://github.com/mitmath/18335/blob/spring21/notes/BFGS.pdf
videohttps://mit.zoom.us/rec/share/naqcRgSkZ0VNeDp0ht8QmB566mPowuHJ8k0LcaAmZ7XxaCT1ch4j_O4Khzi-taXm.CXI8xFthag4RvvoC?startTime=1620241284000
strong dualityhttps://en.wikipedia.org/wiki/Strong_duality
Convex Optimization bookhttps://web.stanford.edu/~boyd/cvxbook/
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