determinant_and_inverse.jl

### A Pluto.jl notebook ###
# v0.19.19

using Markdown
using InteractiveUtils

# ╔═╡ a050a2c0-7872-11ec-3a8e-43fb9497f3ee
# ╠═╡ show_logs = false
using Symbolics, LinearAlgebra, PlutoUI, ForwardDiff, Zygote

# ╔═╡ ee300cb9-dcfe-4d69-92d3-fd91af0b12a9
TableOfContents(title="Differentiating the Determinant and the Inverse", indent=true, depth=4, aside=true)

# ╔═╡ 410e74bd-8974-4b4a-b643-82f6e87d34e8
md"""
# Math: The gradient of the determinant
"""

# ╔═╡ 55d7b972-1aa4-4306-9a28-dcaa759514ee
md"""
Theorem:
``\nabla(\det A) = \text{cofactor}(A) =  (\det A)A^{-T} = \text{adj}(A^T)``  $(br)
``d(\det A) =  \text{tr}( \det(A)A^{-1}dA) = \text{tr(adj}(A) dA) = \text{tr(cofactor}(A)^T dA)`` $(br)
Note:the adjugate is defined as ``(\det A)A^{-1}``; it is the transpose of the cofactor.
"""

# ╔═╡ 6b91801a-bccb-49dc-a415-e041b2b89dac
adj(A) = det(A) * inv(A) # adjugate function

# ╔═╡ 2452e847-51c0-4682-a3b6-38ea6bc6ff9a
cofactor(A) = adj(A)'

# ╔═╡ cde90af2-53d7-4768-9435-60033a80bc72
md"""
**Formulas at a glance:** $(br)
``A^{-1}`` = adj(``A``)/det(``A``) = cofactor(``A``)``^T``/det(``A``) $(br)
adj(``A``) = det(``A``)``A^{-1}`` = cofactor(``A``)``^T`` $(br)
cofactor(``A``) = ``A^{-T}``/det(``A``) = adj(``A``)``^{T}``
"""

# ╔═╡ e00afc65-a062-4816-8ffe-35cc958eda86
md"""
The adjugate is the transpose of the cofactor matrix.
You may remember that the cofactor matrix has as (i,j) entry ``(-1)^{i+j}`` times the determinant obtained by deleting row i and column j.
"""

# ╔═╡ 6c70f6a2-eb0a-4554-9b25-f2ca9784afe7
@variables a b c d

# ╔═╡ cc69feac-6c63-4ae5-b58a-07761e2d1c2f
M = [ a c;b d]

# ╔═╡ 6a281990-c9ad-45f9-9072-2730921ec983
simplify.(cofactor(M))

# ╔═╡ 046dbf67-01b8-4f60-b529-6514452a1a66
simplify.(adj( M))

# ╔═╡ 0e0c347d-0107-44d0-9639-eb307660d193
simplify.(inv(M))

# ╔═╡ 2c25764e-91cc-4d18-9d5c-e684ef398d31
md"""
## Numerical Demonstration
"""

# ╔═╡ cd0836e7-ad52-433e-81f4-b2cdeb3a0da6
begin
  n  = 9
  A = rand(n,n); dA = rand(n,n) * .00001
end

# ╔═╡ 2cd45d14-9158-422a-9c77-a171c9a1ac5e
det(A+dA) - det(A),  tr(  adj(A) * dA )

# ╔═╡ bb9912e8-df10-4880-a289-9b8dd9d00706
md"""
## Forward Diff Autodiff
"""

# ╔═╡ c30eec99-ea46-4e48-bd51-fb27b1cb516e
ForwardDiff.gradient(A->det(A), A)

# ╔═╡ 01398311-2790-475b-adc0-572d27f96e29
cofactor(A)

# ╔═╡ 7bf51e17-106e-412a-b909-81161ed0a2a8
md"""
## Reverse Mode Autodiff
"""

# ╔═╡ 7fc6d585-177d-46f7-9fcd-ec9e74d88ef7
md"""
Zygote does reverse ad
"""

# ╔═╡ 2235f273-b10f-4bd2-a1f6-24a6d56a7b14
Zygote.gradient(A->det(A), A )

# ╔═╡ a49b7161-7015-416f-a8df-a393341ecd9e
md"""
## Symbolic Demonstration
"""

# ╔═╡ dafd2a0f-c416-48de-b176-eb54c8395d1a
reshape(Symbolics.gradient(det(M),vec(M)), 2, 2)

# ╔═╡ 04d3499f-748c-444d-b9c3-48722f094a42
simplify.(adj(M'))

# ╔═╡ 42a67677-6d6e-4219-920f-b59b4997c7e7
md"""
# Direct Proof
A direct proof where you just differentiate the scalar with respect to every input
can be obtained as a simple consequence from the cofactor expansion a.k.a. the [Laplace expansion](https://en.wikipedia.org/wiki/Laplace_expansion) of the determinant based on the ``i``th row.

``\det(A) = A_{i1} C_{i1} + A_{i2}C_{i2} + \ldots + A_{in}C_{in}``

Recall that the determinant is a linear (affine) function of any element with slope equal to the cofactor.

We then readily obtain
``\frac{\partial \det(A)}{\partial A_{ij}}=C_{ij}`` from which we conclude
``\nabla (\det A ) = C``.



$(br) **Example:**
"""

# ╔═╡ c38aed11-89d6-44fd-8489-850e6a37777a
begin
i,j = 2,1
Z= [Num(4) 1 5; 3 4 5 ; 6 7 8]
Z[i,j] = a
Z
end

# ╔═╡ 58313b6e-f163-41f3-b017-20da3b2a181f
det(Z)

# ╔═╡ b5d78163-b83e-4c6a-b603-5e126f314c2c
simplify( cofactor(Z)[i,j] )

# ╔═╡ eae089f9-9046-473c-983c-025f7eb1476a
md"""
# Fancy Proof
Figure out linearization near the identity I $(br)

``\det(I+dA)- 1 = \text{tr}(dA)``  (think of the ``n!`` terms in the determinant and drop higher terms) $(br)
"""

# ╔═╡ 8be99015-e89a-4f6f-8472-7f304e681318
md"""
``\det(A+ A(A^{-1}dA)) - \det(A) = \det(A)\det(I+A^{-1}dA) -\det(A) = ``
``\det(A) \text{tr}( A^{-1}dA) =
\text{tr}(\det(A) A^{-1}dA)``
"""

# ╔═╡ a54041d7-0107-4710-bf0d-d138abbf74c3
md"""
## Application to derivative of the characteristic polynomial evaluated at ``x``

``p(x) = \det(xI-A)``,  a scalar function of ``x``

Recall that ``p(x)`` can be written in terms of the eigenvalues ``\lambda_i``:


"""

# ╔═╡ 2022577f-a85b-4840-86a2-b2fc6dc0e5b1
md"""
### Direct derivative (freshman calculus):

``\frac{d}{dx} \prod_i (x-\lambda_i) \ = \ \sum_i \prod_{j\ne i} (x-\lambda _j) =
 \prod_i (x-\lambda_i)   \left\{ \sum_i (x-\lambda_i)^{-1} \right\}``
(as long as ``x \ne \lambda_i``)

$(br)
Perfectly good simple proof, but if you want to show off...
"""

# ╔═╡ 00c81732-90ae-45d2-8e0b-a5dba732d066
md"""
###  With our new technology:

``d(\det (xI-A))= \det(xI-A) \text{tr}( (xI-A)^{-1} d(xI-A)  )  ``$(br)
``  \hspace{1.2in}=  \det(xI-A) \text{tr}(xI-A)^{-1} dx``

Note: ``d(xI-A)=dxI`` when ``A`` is constant and  $(br) ``\text{tr}(Adx) = \text{tr}(A)dx`` since ``dx`` is a scalar.
"""

# ╔═╡ 5e1905cd-2687-401e-b913-cff5626707cc
md"""
### Check:
"""

# ╔═╡ 74c7703a-a5c8-4863-872c-7b3f9a0891ab
begin
	x = rand()
	ForwardDiff.derivative( x-> det(x*I-A), x),  # Automatic Differentiation
	det(x*I-A)*tr(inv(x*I-A))                    # Analytic Answer
end

# ╔═╡ b2777d02-9918-4721-84bb-33015a2c21a9
md"""
## Application d(log(det(A)))

``d(\log(\det(A))) = \frac{d(\det(A))}{\det A} =  \det(A^{-1}) d(\det(A)) =
\text{tr}( A^{-1} dA) ``  $(br)

The logarithmic derivative shows up a lot in applied mathematics. For example the  key term in Newton's method: ``f(x)/f'(x)`` may be written ``1/ \left\{ \log f(x) \right\}'``
"""

# ╔═╡ d610e914-6940-4a3f-83c2-fb426346dc23
md"""
# Math: The Jacobian of the Inverse
``A^{-1}A= I \rightarrow d(A^{-1}A)=0=d(A^{-1})A + A^{-1}dA`` from the product rule $(br)
``\rightarrow d(A^{-1}) = -A^{-1} (dA) A^{-1} = -(A^{-T} \otimes A^{-1}) dA``
"""

# ╔═╡ b56a6bf8-fa04-487d-8a62-d8bb28f1ec04
begin
	C = rand(4,4)
	ForwardDiff.jacobian(A->inv(A),C)
end

# ╔═╡ 1ba06d45-df23-4776-a154-65f8206a9a71
-kron(inv(C'),inv(C))

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"""

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