Add content for vectors-matrix-ops/ranks-determinants

Add actual content to the skeleton of the
`vectors-matrix-ops/ranks-determinants` section.

Signed-off-by: Eggert Karl Hafsteinsson <[email protected]>
Signed-off-by: Teodor Dutu <[email protected]>
Signed-off-by: Razvan Deaconescu <[email protected]>

chapters/vectors-matrix-ops/ranks-determinants/reading/README.md

@@ -1 +1,213 @@
 # Ranks and Determinants
+
+## The Rank of a Matrix
+
+The rank of an $n \times p$ matrix $A$, denoted by $\text{rank}(A)$, is the largest number of columns of $A$, which are not linearly dependent (i.e. the number of linearly independent columns).
+
+### Details
+
+Vectors $a_1, a_2, \ldots, a_n$ are said to be linearly dependent if there exist constants $k_1, \ldots, k_n$ that are not all zero, such that
+
+$$k_1 a_1 + k_2 a_2 + \ldots + k_n a_n = 0.$$
+
+Note that if such constants exist, then we can write one of the $a$ 's as a linear combination of the rest, e.g. if $k_1 \neq 0$ then
+
+$$a_1=\mathbf{c_1} = -\frac{k_2}{k_1} a_2 - \ldots - \frac{k_2}{k_1} a_n$$
+
+It can be shown that the rank of $A$, is the same as the rank of $A'$
+
+i.e. the maximum number of linearly independent rows of $A$.
+
+:::note Note
+
+Note that if $\text{rank}(A) = p$, then the columns are linearly independent.
+
+:::
+
+### Examples
+
+:::info Example
+
+If
+
+$$A= \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right]$$
+
+then $\text{rank}(A)$ = 2, since
+
+$$k_1 \left( \begin{array}{cc} 1 \\ 0 \\ \end{array} \right) + k_2 \left( \begin{array}{cc} 0 \\ 1 \\ \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \\ \end{array} \right)$$
+
+if and only if
+
+$$\left( \begin{array}{cc} k_1 \\ k_2 \\ \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \\ \end{array} \right)$$
+
+so the columns are linearly independent.
+
+:::
+
+:::info Example
+
+If
+
+$$A = \left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1\\ 0 & 0 & 0 \\ \end{array} \right]$$
+
+then $\text{rank}(A)$ = 2.
+
+:::
+
+:::info Example
+
+If
+
+$$A = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ \end{array} \right]$$
+
+then $\text{rank}(A)$ = 2. since
+
+$$1 \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ \end{array} \right) + 0 \left( \begin{array}{ccc} 0 \\ 1 \\ 1 \\ \end{array} \right) + (-1) \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ \end{array} \right) = 0$$
+
+(and hence the rank cannot be more than 2) but
+
+$$k_1 \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \\ \end{array} \right) + k_2 \left( \begin{array}{ccc} 0 \\ 1 \\ 1 \\ \end{array} \right)$$
+
+if and only if $k_1=k_2=0$ (and hence the rank must be at least 2).
+
+:::
+
+## The Determinant
+
+Recall that for a $2 \times 2$ matrix,
+
+$$A= \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$
+
+the inverse of $A$ is
+
+$$A^{-1}= \frac{1}{ad-bc} \begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix}$$
+
+### Details
+
+:::note Definition
+
+The number $ad-bc$ is called the **determinant** of the $2 \times 2$ matrix $A$.
+
+:::
+
+:::note Definition
+
+Now suppose $A$ is an $n \times n$ matrix.
+An **elementary product** from the matrix is a product of $n$ terms based on taking exactly one term from each column of row $x$.
+Each such term can be written in the form $a_{1j_1} \cdot a_{2j_2} \cdot a_{3j_3} \cdot \ldots \cdot a_{nj_n}$ where $j_1, \ldots, j_n$ is a permutation of the integers $1,2, \ldots, n$.
+Each permutation $\sigma$ of the integers $1,2,\ldots,n$ can be performed by repeatedly interchanging two numbers.
+
+:::
+
+:::note Definition
+
+A **signed elementary product** is an elementary product with a positive sign if the number of interchanges in the permutation is even but negative otherwise.
+
+:::
+
+The determinant of $A$, $\det(A)$ or $\vert A \vert$, is the sum of all signed elementary products.
+
+### Examples
+
+:::info Example
+
+$$A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$
+
+then
+
+$\vert A \vert = a_{1\underline{1}} a_{2\underline{2}} - a_{1\underline{2}}a_{2\underline{1}}$.
+
+:::
+
+:::info Example
+
+If
+
+$$A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix},$$
+
+Then
+
+$\vert A \vert$
+
+= $a_{11} a_{22} a_{33}$ This is the identity permutation and has positive sign
+
+$-a_{11} a_{23} a_{32}$ This is the permutation that only interchanges $2$ and $3$
+
+$-a_{12} a_{21} a_{33}$ Only one interchange
+
+$+a_{12} a_{23} a_{31}$ Two interchanges
+
+$+a_{13} a_{21} a_{32}$ Two interchanges
+
+$-a_{13} a_{22} a_{31}$ Three interchanges
+
+:::
+
+:::info Example
+
+$$A= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$$
+
+$\vert A \vert = -1$
+
+:::
+
+:::info Example
+
+$$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$
+
+$\vert A \vert = 1 \cdot 2 \cdot 3 = 6$
+
+:::
+
+:::info Example
+
+$$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 3 & 0 \end{bmatrix}$$
+
+$\vert A \vert = 0$
+
+:::
+
+:::info Example
+
+$$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 3 & 0 \end{bmatrix}$$
+
+$\vert A \vert = -6$
+
+:::
+
+:::info Example
+
+$$A= \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$$
+
+$\vert A \vert = 0$
+
+:::
+
+:::info Example
+
+$$A= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$
+
+$\vert A \vert = 0$
+
+:::
+
+## Ranks, Inverses and Determinants
+
+The following statements are true for an $n\times n$ matrix $A$ :
+
+- $\text{rank} (A)= n$
+
+- $\det(A)\neq 0$
+
+- $A$ has an inverse
+
+### Details
+
+Suppose $A$ is an $n\times n$ matrix.
+Then the following are truths:
+
+- $\text{rank} (A)= n$
+
+- $\det(A)\neq 0$
+
+- $A$ has an inverse