Two by Two Matrix Jacobians

md"""

# Two by Two Matrix Jacobians

"""

168 μs

This notebook emphasizes the multiple views of Jacobians with examples of 2x2 matrix functions.

In particular we will see the

Symbolic "vec" format producing 4x4 matrices (generally n² by n² or mn by mn)
Numerical formats
The important Linear Transformation view
Kronecker notation
An example using ForwardDiff automatic differentiation

We also emphasize that matrix factorizations are also matrix functions, just as much as the square and the cube.

md"""

This notebook emphasizes the multiple views of Jacobians with examples of 2x2 matrix functions.

In particular we will see the

* Symbolic "vec" format producing 4x4 matrices (generally n² by n² or mn by mn)

* Numerical formats

* The important Linear Transformation view

* Kronecker notation

* An example using ForwardDiff automatic differentiation

We also emphasize that matrix factorizations are also matrix functions, just as much as the square and the cube.

"""

2.4 ms

using Symbolics

, LinearAlgebra

, PlutoUI

3.6 s

TableOfContents(title="Two by Two Matrix Jacobians", indent=true,aside=true)

4.3 ms

Symbolic Matrices

md"""

# Symbolic Matrices

"""

134 μs

Symbolics.Num

$p$

$q$

$r$

$s$

$θ$

@variables p,q,r,s,θ

101 ms

$[\begin{array}{cc} r \cos (θ) & r \sin (θ) \\ r \sin (θ) & - r \cos (θ) \end{array}]$

simplify.( Q*[r 0 ; 0 -r]*Q' )

10.8 s

$[\begin{array}{cc} p & r \\ q & s \end{array}]$

X = [p r;q s]

18.1 ms

vec

The vec command in Julia and in standard mathematics flattens a matrix column by column.

md"""

## vec

The `vec` command in Julia and in standard mathematics flattens a matrix column by column.

"""

186 μs

Symbolics.Num

$p$

$q$

$r$

$s$

vec(X)

8.7 μs

1) The matrix square function

md"""

# 1) The matrix square function

"""

135 μs

$[\begin{array}{cc} p^{2} + q r & p r + r s \\ p q + q s & q r + s^{2} \end{array}]$

X^2

674 ms

Symbolics.Num

$p^{2} + q r$

$p q + q s$

$p r + r s$

$q r + s^{2}$

vec(X^2)

2.5 ms

Symbolic Jacobian

The Jacobian of the (flattened) matrix function X² symbolically

md"""

## Symbolic Jacobian

The Jacobian of the (flattened) matrix function X² symbolically

"""

166 μs

jac (generic function with 1 method)

jac(Y,X) = Symbolics.jacobian(vec(Y),vec(X))

292 μs

$[\begin{array}{cccc} 2 p & r & q & 0 \\ q & p + s & 0 & q \\ r & 0 & p + s & r \\ 0 & r & q & 2 s \end{array}]$

J = jac(X^2, X)

1.2 s

Numerical Jacobian

md"""

## Numerical Jacobian

"""

125 μs

$[\begin{array}{cccc} 2 & 2 & 3 & 0 \\ 3 & 5 & 0 & 3 \\ 2 & 0 & 5 & 2 \\ 0 & 2 & 3 & 8 \end{array}]$

begin

M = [1 2;3 4]

E = [.0001 .0002;.0003 .0004]

substitute(J,Dict(p=>1,q=>3,r=>2,s=>4))

end

985 ms

Symbolics.Num

$0.0014$

$0.003$

$0.002$

$0.0044$

substitute(J,Dict(p=>1,q=>3,r=>2,s=>4)) * vec(E)

92.4 ms

2×2 Matrix{Float64}:
 0.00140007  0.0020001
 0.00300015  0.00440022

(M+E)^2 - M^2

67.0 ms

Linear Transformation Jacobian

Notice: there is no flattening; this is just matrix to matrix.

md"""

## Linear Transformation Jacobian

Notice: there is no flattening; this is just matrix to matrix.

"""

160 μs