Matrix

Application

Matrix as a way to solve multi-equations

National income model in two endogenous variables Y and C.

(1)
\begin{cases} & Y=C+I+G \\ & C=a+bY \end{cases}
(2)
\begin{cases} & Y=C+I+G \\ & C=a+bY \end{cases}
(3)
\begin{cases} & Y-C=I+G \\ & -bY+C=a \end{cases}

Define

(4)
\begin{align} A=\begin{bmatrix} 1 & -1 \\ -b & 1 \end{bmatrix}\;\; x= \begin{bmatrix} Y \\ C \end{bmatrix}}&\;\; d= \begin{bmatrix} I+G \\ a \end{bmatrix} \end{align}

We have

(5)
\begin{equation} Ax=d \end{equation}
(6)
\begin{equation} x={A}^{-1}d \end{equation}

Leontief Input-Output model

Matrix as a linear function from one vector space to another

(7)
\begin{pmatrix} {x}_{11} & {x}_{12} \end{pmatrix} \cdot \begin{bmatrix} {a}_{11} & {a}_{12} \\ {a}_{21} & {a}_{22} \end{bmatrix} = \begin{pmatrix} {y}_{11} & {y}_{12} \end{pmatrix}
${x}_{11}$
corn
${x}_{12}$
PC
${a}_{11}$
corn to PC
${a}_{21}$
PC to corn
${y}_{11}$
corn
${y}_{12}$
PC

Markov Chains (or Markov transition matrix)

(8)
\begin{align} m= \begin{bmatrix} 0.7 & 0.3 \\ 0.6 & 0.4 \end{bmatrix} \end{align}
(9)
\begin{align} \lim_{n \to \infty} {m}^{n}=\text{steady state} \end{align}

Absorbing Markov chains

(10)
\begin{pmatrix} {A}_{0} & {B}_{0} \end{pmatrix} \cdot \lim_{n \to \infty} {\begin{bmatrix} {P}_{AA} & {P}_{AB} \\ 0 & 1 \end{bmatrix}}^{n} = \begin{pmatrix} 0 & {A}_{0} + {B}_{0} \end{pmatrix}

Mathematical Concepts of Matrix

see here

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