Nathan Benedetto Proença

Function matrices

Outline

  • What am I trying to do
  • Why do I care
  • How did I do it
  • Consequences

Properly define matrice operations I will use in my work.

Avoid some approaches

Kirchhoff's Matrix Tree Theorem

Kirchhoff's Matrix Tree Theorem

Let \(G = (V,E, \psi,w)\) be a

weighted graph. Let \(r \in V\). Then

\det(L_G[\{r\}^c,\{r\}^c]) = \sum_{S \in \mathcal{T}_G} \prod w(S).
det⁡(LG[{r}c,{r}c])=∑S∈TG∏w(S).\det(L_G[\{r\}^c,\{r\}^c]) = \sum_{S \in \mathcal{T}_G} \prod w(S).

Tutte's Matrix Tree Theorem

Let \(D = (V,A, \psi,w)\) be a

weighted digraph. Let \(r \in V\). Then

\det(L_D[\{r\}^c,\{r\}^c]) = \sum_{S \in \mathcal{T}_D(r)} \prod w(S).
det⁡(LD[{r}c,{r}c])=∑S∈TD(r)∏w(S).\det(L_D[\{r\}^c,\{r\}^c]) = \sum_{S \in \mathcal{T}_D(r)} \prod w(S).

Cauchy Binet Formula

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Cauchy Binet Formula

Deus Ex Machina

How do I even state it?

Let \(U\) and \(V\) be finite sets. A matrix is a function \(A \colon V\times U \to \mathbb{R}\).

The elements of \(V\) are the row indices
of \(A\), and the elements of \(U\) are the column indices of \(A\).

Let \(U\) be a finite set. The determinant of a matrix \(A \in \mathbb{R}^{U\times U}\) is

\det(A) \coloneqq \sum_{\sigma \in \text{sym}{U}} \text{sgn}(\sigma)\prod_{i \in U} A_{i, \sigma(i)}.
det⁡(A):=∑σ∈symUsgn(σ)∏i∈UAi,σ(i).\det(A) \coloneqq \sum_{\sigma \in \text{sym}{U}} \text{sgn}(\sigma)\prod_{i \in U} A_{i, \sigma(i)}.

Let \(U\) be a finite set. The determinant of a matrix \(A \in \mathbb{R}^{U\times U}\) is

\sum_{\sigma \in \text{sym}{U}} \text{sgn}(\sigma)\prod_{i \in U} A_{i, \sigma(i)} = \det(A).
∑σ∈symUsgn(σ)∏i∈UAi,σ(i)=det⁡(A).\sum_{\sigma \in \text{sym}{U}} \text{sgn}(\sigma)\prod_{i \in U} A_{i, \sigma(i)} = \det(A).

Function Matrix

Function Matrix

Let \(U\) and \(V\) be finite sets. Let \(f\colon U \to V\) be a function. The function matrix \(P_f \in \mathbb{R}^{V\times U}\) is defined as

P_f \coloneqq \sum_{i \in U} e_{f(i)}e_i^\mathsf{T}.
Pf:=∑i∈Uef(i)eiT.P_f \coloneqq \sum_{i \in U} e_{f(i)}e_i^\mathsf{T}.

Function Matrix

P_f e_i = e_{f(i)}.
Pfei=ef(i).P_f e_i = e_{f(i)}.

Composition

Let \(T\), \(U\) and \(V\) be finite sets. Let \( f \colon U \to V\) and \(g \colon T \to U\). Then

P_{f\circ g} = P_fP_g.
Pf∘g=PfPg.P_{f\circ g} = P_fP_g.

Composition

Let \(T\), \(U\) and \(V\) be finite sets. Let \( f \colon U \to V\) and \(g \colon T \to U\). Then

P_{f g} = P_fP_g.
Pfg=PfPg.P_{f g} = P_fP_g.

Inversion

Let \(U\) and \(V\) be finite sets. Let \( f \colon U \to V\) be a bijective function. Then

P_{f^{-1}} = P_f^\mathsf{T}.
Pf−1=PfT.P_{f^{-1}} = P_f^\mathsf{T}.

Nonsquare determinants

Let \(U\) be a finite set. The determinant of a matrix \(A \in \mathbb{R}^{U\times U}\) is

\det(A) \coloneqq \sum_{\sigma \in \text{sym}{U}} \text{sgn}(\sigma)\prod_{i \in U} A_{i, \sigma(i)}.
det⁡(A):=∑σ∈symUsgn(σ)∏i∈UAi,σ(i).\det(A) \coloneqq \sum_{\sigma \in \text{sym}{U}} \text{sgn}(\sigma)\prod_{i \in U} A_{i, \sigma(i)}.

Let \(V\) and \(U\) be finite sets. Let \(A \in \mathbb{R}^{V\times U}\) and \(\phi \colon V \to U\) be bijective. Note that

P_\phi \in \mathbb{R}^{U\times V}\\ AP_\phi \in \mathbb{R}^{V\times V}\\ P_\phi A \in \Reals^{U\times U}
Pϕ∈RU×VAPϕ∈RV×VPϕA∈RU×UP_\phi \in \mathbb{R}^{U\times V}\\ AP_\phi \in \mathbb{R}^{V\times V}\\ P_\phi A \in \Reals^{U\times U}

Let \(U\) and \(V\) be finite sets. Let \(\phi\colon V \to U\) be a bijective function.

If \(A \in \mathbb{R}^{V\times U}\), then

\det(AP_\phi) = \det(P_\phi A).
det⁡(APϕ)=det⁡(PϕA).\det(AP_\phi) = \det(P_\phi A).

Let \(U\) and \(V\) be finite sets. Let \(f,g \colon U \to V\) be bijective functions. Then

\text{sgn}(gf^{-1}) = \text{sgn}(f^{-1}g).
sgn(gf−1)=sgn(f−1g).\text{sgn}(gf^{-1}) = \text{sgn}(f^{-1}g).

Let \(U\) and \(V\) be finite sets. Let \(\phi\colon V \to U\) be a bijective function. Let \(A \in \mathbb{R}^{V\times U}\).

The determinant (with respect to \(\phi\)) is defined as

\det_\phi(A) \coloneqq \det(AP_\phi).
det⁡ϕ(A):=det⁡(APϕ).\det_\phi(A) \coloneqq \det(AP_\phi).

Let \(U\) and \(V\) be finite sets. Let \(\phi\colon V \to U\) be a bijective function. Let \(A \in \mathbb{R}^{V\times U}\). Then

 

\det_\phi(A) = \det_{\phi^{-1}}(A^\mathsf{T}).
det⁡ϕ(A)=det⁡ϕ−1(AT).\det_\phi(A) = \det_{\phi^{-1}}(A^\mathsf{T}).

Algebraic Graph Theory

Graph

A graph \(G\) is an ordered triple \((V,E,\psi)\), where \(V\) and \(E\) are finite sets and

\(\psi \colon E \to \binom{V}{2} \cup \binom{V}{1}\).

Digraph

A digraph \(D\) is an ordered triple \((V,A,\psi)\), where \(V\) and \(A\) are finite sets and

\(\psi \colon A \to V\times V\).

Let \(D = (V,A,\psi)\) be a digraph.

What is \(P_\psi\)?

Let \(D = (V,A,\psi)\) be a digraph.

What is \(P_\psi\)?

P_\psi \in \mathbb{R}^{(V\times V)\times A}.
Pψ∈R(V×V)×A.P_\psi \in \mathbb{R}^{(V\times V)\times A}.

Let \(V\) be a finite set. Denote by \(\lambda, \rho \colon V\times V \to V\) functions such that for every \((i, j) \in V\times V\)

\lambda(i,j) \coloneqq i,\\ \rho(i,j) \coloneqq j.
λ(i,j):=i,ρ(i,j):=j.\lambda(i,j) \coloneqq i,\\ \rho(i,j) \coloneqq j.

Let \(D = (V,A,\psi)\) be a digraph.

Note that

\(\lambda\psi \colon A \to V\) maps arcs to their tails,

\(\rho\psi \colon A \to V\) maps arcs to their heads.

The tail matrix of \(D\) is defined as

 

\(T_D \coloneqq P_{\lambda \psi}\).

Let \(D = (V,A,\psi)\) be a digraph.

The head matrix of \(D\) is defined as

 

\(H_D \coloneqq P_\rho\psi\).

The tail matrix of \(D\) is defined as

\(T_D \coloneqq P_{\lambda \psi}\).

Let \(D = (V,A,\psi)\) be a digraph.

The head matrix of \(D\) is defined as

\(H_D \coloneqq P_\rho\psi\).

\text{if } \psi(a) = ij,\\ H_D e_a = e_j,\\ T_D e_a = e_i.
if ψ(a)=ij,HDea=ej,TDea=ei.\text{if } \psi(a) = ij,\\ H_D e_a = e_j,\\ T_D e_a = e_i.

All the matrices can be defined from \(H_D\) and \(T_D\)

  • Incidence matrix:
    \(B_D \coloneqq H_D - T_D\).
  • Adjecency matrix:
    \(A_D \coloneqq H_D W T_D^\mathsf{T}\).
  • Degree matrix:
    \(D_D \coloneqq H_D W H_D^\mathsf{T}\).
  • Laplacian
    \(L_D \coloneqq H_D W B_D^\mathsf{T}\).

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