matrix

theme/javascript-source/convert.js

@@ -7,18 +7,17 @@ function code_render(obj) {
       code_render(obj[key]);
     }
     if (key === "html:code") {
-      if (obj['html:code'].length === 0) return;
-      const code = obj['html:code'][0]["#text"];
+      if (obj["html:code"].length === 0) return;
+      const code = obj["html:code"][0]["#text"];
       const lines = code.split("\n");
       let i = 0;
-      let res = ''
-      while (code[i] === ' ') {
-        i++;
-      }
+      let res = "";
+      while (code[i] === " ") i++;
       for (let j = 0; j < lines.length; j++) {
-        res += lines[j].replace(' '.repeat(i), '') + '\n'
+        res += lines[j].replace(" ".repeat(i), "") + "\n";
       }
-      obj['html:code'][0]['#text'] = res.trim()
+
+      obj["html:code"][0]['#text'] = res.trim()
     }
   }
 }
@@ -52,4 +51,4 @@ async function start() {
   }
 }
 
-start();
+start();

trees/def/def-0043.tree

@@ -0,0 +1,13 @@
+\title{Transpose}
+\taxon{Definition}
+\p{
+    The \strong{transpose} of a matrix #{A}, denoted by #{A^T} is
+    the matrix obtained by swapping the rows and columns of #{A}.
+    It satisfies the following properties:
+    \ul{
+        \li{#{(A^T)^T = A}}
+        \li{#{(A + B)^T = A^T + B^T}}
+        \li{#{(cA)^T = cA^T}}
+        \li{#{(AB)^T = B^TA^T}}
+    }
+}

trees/def/def-0044.tree

@@ -0,0 +1,11 @@
+\title{Determinant}
+\taxon{Definition}
+\p{
+    The determinant of a #{n\times n} square matrix #{A} is commonly denoted #{\det A} or #{|A|}.
+    It satisfies the following properties:
+    \ul{
+        \li{#{\det A^T = \det A} }
+        \li{#{\det AB = \det A \det B}}
+        \li{#{\det \lambda A = \lambda^n \det A}}
+    }
+}

trees/def/def-0045.tree

@@ -0,0 +1,9 @@
+\title{First Minor and Cofactor}
+\taxon{Definition}
+\p{
+    If #{A} is a square matrix, then the \strong{minor} of the entry in the i-th row and j-th 
+    column (also called the #{(i, j)} minor, or a first minor) is the \strong{determinant} of 
+    the sub-matrix formed by deleting the i-th row and j-th column.
+    The #{(i, j)} minor is denoted as #{M_{ij}}.
+    The \strong{Cofactor} is obtained by multiplying the minor by #{(-1)^{i+j}}.
+}

trees/def/def-0046.tree

@@ -0,0 +1,17 @@
+\title{Cofactor Matrix}
+\taxon{Definition}
+\p{
+    The matrix formed by all of the [cofactors](def-0045) of a square matrix #{A} is called the cofactor matrix,
+    or \strong{comatrix}:
+    ##{
+        C = \left[ 
+            \begin{array}{cccc}
+                C_{11} & C_{12} & \cdots & C_{1n} \\
+                C_{21} & C_{22} & \cdots & C_{2n} \\
+                \vdots & \vdots & \ddots & \vdots \\
+                C_{n1} & C_{n2} & \cdots & C_{nn}
+            \end{array}
+        \right]
+    }
+    The \strong{Adjugate matrix} of #{A} is the transpose of the cofactor matrix.
+}

trees/def/def-0047.tree

@@ -0,0 +1,5 @@
+\title{Singular Matrix}
+\taxon{Definition}
+\p{
+    A square matrix that is not \strong{invertible} is called \strong{singular} or degenerate
+}

trees/def/def-0048.tree

@@ -0,0 +1,10 @@
+\title{Matrix Polynomial}
+\taxon{Definition}
+\p{
+    A \strong{matrix polynomial} is a polynomial with square matrices as variables.
+    The general form of a matrix polynomial is:
+    ##{
+        P(A) = \sum_{i=0}^{n} a_i A^i
+    }
+    where #{A^0 = I} is the identity matrix.
+}

trees/def/def-0049.tree

@@ -0,0 +1,19 @@
+\title{Matrix Partitioning}
+\taxon{Definition}
+\import{macros}
+\p{
+    Let #{A \in \C^{m\times n} }. A \strong{partitioning} of #{A} is a representation of #{A} in the form
+    ##{
+        A = \begin{bmatrix}
+            A_{11} & A_{12} & \cdots & A_{1q} \\
+            A_{21} & A_{22} & \cdots & A_{2q} \\
+            \vdots & \vdots & \ddots & \vdots \\
+            A_{p1} & A_{p2} & \cdots & A_{pq}
+        \end{bmatrix}
+    }
+    where #{A_{ij} \in \C^{m_i \times n_j} } for #{1 \leq i \leq p} and #{1 \leq j \leq q} such that
+    ##{
+        \sum_{i=1}^p m_i = m \quad \text{and} \quad \sum_{j=1}^q n_j = n.
+    }
+    The partitioned matrix operations are similar to the operations on the normal matrix. 
+}

trees/math/math-0008.tree

@@ -0,0 +1,106 @@
+\title{Matrix}
+\taxon{Linear Algebra}
+\import{macros}
+\p{
+    This post shows operations and applications over matrix.
+}
+\transclude{def-0043}
+\p{
+    If the following condition satisfies:
+    ##{
+        a_{ij} = a_{ji} \quad \forall i,j
+    }
+    Then the matrix is called symmetric.
+}
+\transclude{def-001N}
+\transclude{def-0044}
+\p{
+    To compute the inverse of a matrix, we need \strong{Adjugate matrix}.
+}
+\transclude{def-0045}
+\transclude{def-0046}
+\p{
+    Then the inverse of #{A} is the transpose of the cofactor matrix times the reciprocal of the determinant of #{A}:
+    ##{
+        A^{-1} = \frac{1}{\det A} \cdot \text{adj} A = \frac{1}{\det A} \cdot C^T
+    }
+}
+\transclude{def-0047}
+\subtree{
+    \title{An important property of the inverse of a matrix}
+    \p{
+        ##{
+            A \cdot \text{adj} A = \text{adj} A \cdot A = \det A \cdot I
+        }
+    }
+    \proof{
+        Let #{A \cdot \text{adj} A = (b_{ij})} and we have
+        ##{
+            b_{ij} = a_{i1}A_{j1} + a_{i2}A_{j2} + \cdots + a_{in}A_{jn} = \delta_{ij} \cdot \det A
+        }
+        Hence we have #{A \cdot \text{adj} A = \det A \cdot I} 
+    }
+}
+\subtree{
+    \title{Matrix Polynomial and Computation}
+    \transclude{def-0048}
+    \p{
+        If #{A} is a diagonal matrix, then the polynomial of #{A} is the diagonal matrix of the polynomial of the diagonal elements of #{A}.
+        ##{
+            p(A) = \begin{bmatrix}
+                p(a_{11}) & 0 & \cdots & 0 \\
+                0 & p(a_{22}) & \cdots & 0 \\
+                \vdots & \vdots & \ddots & \vdots \\
+                0 & 0 & \cdots & p(a_{nn})
+            \end{bmatrix}
+        }
+    }
+    \p{
+        If #{A = P\Lambda P^{-1}}, then #{A^k = P \Lambda ^k P^{-1}} and hence
+        ##{
+            p(A) = a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n = P \Lambda P^{-1}
+        }
+    }
+}
+\subtree{
+    \title{Solving a Linear System}
+    \transclude{thm-0011}
+    \p{
+        Matrix partitioning is the process of dividing a matrix into smaller submatrices. 
+        This is often done to simplify the computation of matrix operations, such as matrix multiplication.
+    }
+    \transclude{def-0049}
+    \p{
+        If the partitioned matrix is formed as diagonal blocks, then we can compute the determinant of the matrix by the following formula:
+        ##{
+            \det A = \det A_1 \cdot \det A_2 \cdots \det A_n
+        }
+        And the inverse of the matrix is
+        ##{
+            A^{-1} = \begin{bmatrix}
+                A_1^{-1} & O & \cdots & O \\
+                O & A_2^{-1} & \cdots & O \\
+                \vdots & \vdots & \ddots & \vdots \\
+                O & O & \cdots & A_n^{-1}
+            \end{bmatrix}
+        }
+    }
+    \p{
+        The column partitioning of matrix is useful. 
+        If we have #{m\times s} matrix #{A = (a_{ij})} and #{s\times n} matrix #{B=(b_{ij})},
+        their product can be written:
+        ##{
+            AB = \begin{bmatrix} A_1 \\ A_2 \\ \vdots A_m \end{bmatrix}
+            \begin{bmatrix}
+                B_1 & B_2 & \cdots & B_n
+            \end{bmatrix} = 
+            \begin{bmatrix}
+                A_1B_1 & A_1B_2 & \cdots & A_1B_n \\
+                A_2B_1 & A_2B_2 & \cdots & A_2B_n \\
+                \vdots & \vdots & \ddots & \vdots \\
+                A_mB_1 & A_mB_2 & \cdots & A_mB_n
+            \end{bmatrix}
+        }
+        We can show that #{A=O\iff A^TA=O}.
+    }
+}

trees/notes.tree

@@ -15,4 +15,5 @@
 \transclude{cs-0005}
 \transclude{cs-0006}
 \transclude{cs-0007}
-\transclude{math-0007}
+\transclude{math-0007}
+\transclude{math-0008}

trees/thm/thm-0011.tree

@@ -0,0 +1,17 @@
+\title{Cramer's rule}
+\taxon{Theorem}
+\p{
+    Consider a system of #{n} linear equations for #{n} unknowns, represented in matrix multiplication form as follows:
+    ##{
+        A \cdot X = B
+    }
+    where #{A} is a square matrix of order #{n}, #{X} is a column matrix of order #{n} and #{B} is a column matrix of order #{n}.
+    ##{
+        X = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}
+    }
+    The Cramer's rule states that the solution to the system of equations is given by:
+    ##{
+        x_i = \frac{\text{det}(A_i)}{\text{det}(A)}
+    }
+    where #{A_i} is the matrix obtained by replacing the #{i}th column of #{A} by #{B}.
+}