port from mathematics-physics notes
This commit is contained in:
parent
a4e106ce02
commit
c009ea53f0
124 changed files with 13224 additions and 0 deletions
|
|
@ -0,0 +1,69 @@
|
|||
# Elementary matrices
|
||||
|
||||
> *Definition*: an *elementary* matrix is defined as an identity matrix with exactly one elementary row operation undergone.
|
||||
>
|
||||
> 1. An elementary matrix of type 1 $E_1$ is obtained by changing two rows $I$.
|
||||
> 2. An elementary matrix of type 2 $E_2$ is obtained by multiplying a row of $I$ by a nonzero constant.
|
||||
> 3. An elementary matrix of type 3 $E_3$ is obtained from $I$ by adding a multiple of one row to another row.
|
||||
|
||||
For example the elementary matrices could be given by
|
||||
|
||||
$$
|
||||
E_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}, \qquad E_2 = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 3\end{pmatrix}, \qquad E_3 = \begin{pmatrix}1 & 0 & 3\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}.
|
||||
$$
|
||||
|
||||
> *Theorem*: if $E$ is an elementary matrix, then $E$ is nonsingular and $E^{-1}$ is an elementary matrix of the same type.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
If $E$ is the elementary matrix of type 1 formed from $I$ by interchanging the $i$th and $j$th rows, then $E$ can be transfomred back into $I$ by interchanging these same rows again. Therefore, $EE = I$ and hence $E$ is its own inverse.
|
||||
|
||||
IF $E$ is the elementray matrix of type 2 formed by multiplying the $i$th row of $I$ by a nonzero scalar $\alpha$ then $E$ can be transformed into the identity matrix by multiplying either its $i$th row or its $i$th column by $1/\alpha$.
|
||||
|
||||
If $E$ is the elemtary matrix of type 3 formed from $I$ by adding $m$ times the $i$th row to the $j$th row then $E$ can be transformed back into $I$ either by subtracting $m$ times the $i$th row from the $j$th row or by subtracting $m$ times the $j$th column from the $i$th column.
|
||||
|
||||
> *Definition*: a matrix $B$ is **row equivalent** to a matrix $A$ if there exists a finite sequence $E_1, E_2, \dots, E_K$ of elementary matrices with $k \in \mathbb{N}$ such that
|
||||
>
|
||||
> $$
|
||||
> B = E_k E_{k-1} \cdots E_1 A.
|
||||
> $$
|
||||
|
||||
It may be observed that row equivalence is a reflexive, symmetric and transitive relation.
|
||||
|
||||
> *Theorem*: let $A$ be an $n \times n$ matrix, the following are equivalent
|
||||
>
|
||||
> 1. $A$ is nonsingular,
|
||||
> 2. $A\mathbf{x} = \mathbf{0}$ has only the trivial solution $\mathbf{0}$,
|
||||
> 3. $A$ is row equivalent to $I$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Let $A$ be a nonsingular $n \times n$ matrix and $\mathbf{\hat x}$ is a solution of $A \mathbf{x} = \mathbf{0}$ then
|
||||
|
||||
$$
|
||||
\mathbf{\hat x} = I \mathbf{\hat x} = (A^{-1} A)\mathbf{\hat x} = A^{-1} (A \mathbf{\hat x}) = A^{-1} \mathbf{0} = \mathbf{0}.
|
||||
$$
|
||||
|
||||
Let $U$ be the row echelon form of $A$. If one of the diagonal elements of $U$ were 0, the last row of $U$ would consist entirely of zeros. But then $A \mathbf{x} = \mathbf{0}$ would have a nontrivial solution. Thus $U$ must be a strictly triangular matrix with diagonal elements all equal to 1. It then follows that $I$ is the reduced row echelon form of $A$ and hence $A$ is row equivalent to $I$.
|
||||
|
||||
If $A$ is row equivalent to $I$ there exists elementary matrices $E_1, E_2, \dots, E_k$ with $k \in \mathbb{N}$ such that
|
||||
|
||||
$$
|
||||
A = E_k E_{k-1} \cdots E_1 I = E_k E_{k-1} \cdots E_1.
|
||||
$$
|
||||
|
||||
Since $E_i$ is invertible for $i \in \{1, \dots, k\}$ the product $E_k E_{k-1} \cdots E_1$ is also invertible, hence $A$ is nonsingular.
|
||||
|
||||
If $A$ is nonsingular then $A$ is row equivalent to $I$ and hence there exists elemtary matrices $E_1, \dots, E_k$ such that
|
||||
|
||||
$$
|
||||
E_k E_{k-1} \cdots E_1 A = I,
|
||||
$$
|
||||
|
||||
multiplyting both sides on the right by $A^{-1}$ obtains
|
||||
|
||||
$$
|
||||
E_k E_{k-1} \cdots E_1 = A^{-1}
|
||||
$$
|
||||
|
||||
a method for computing $A^{-1}$.
|
||||
94
docs/mathematics/linear-algebra/matrices/matrix-algebra.md
Normal file
94
docs/mathematics/linear-algebra/matrices/matrix-algebra.md
Normal file
|
|
@ -0,0 +1,94 @@
|
|||
# Matrix algebra
|
||||
|
||||
> *Theorem*: let $A, B$ and $C$ be matrices and $\alpha$ and $\beta$ be scalars. Each of the following statements is valid
|
||||
>
|
||||
> 1. $A + B = B + A$,
|
||||
> 2. $(A + B) + C = A + (B + C)$,
|
||||
> 3. $(AB)C = A(BC)$,
|
||||
> 4. $A(B + C) = AB + AC$,
|
||||
> 5. $(A + B)C = AC + BC$,
|
||||
> 6. $(\alpha \beta) A = \alpha(\beta A)$,
|
||||
> 7. $\alpha (AB) = (\alpha A)B = A (\alpha) B$,
|
||||
> 8. $(\alpha + \beta)A = \alpha A + \beta A$,
|
||||
> 9. $\alpha (A + B) = \alpha A + \alpha B$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
In the case where an $n \times n$ matrix $A$ is multiplied by itself $k$ times it is convenient to use exponential notation: $AA \cdots A = A^k$.
|
||||
|
||||
> *Definition*: the $n \times n$ **identity matrix** is the matrix $I = (\delta_{ij})$, where
|
||||
>
|
||||
> $$
|
||||
> \delta_{ij} = \begin{cases} 1 &\text{ if } i = j, \\ 0 &\text{ if } i \neq j.\end{cases}
|
||||
> $$
|
||||
|
||||
Obtaining for the multiplication of a $n \times n$ matrix $A$ with the identitiy matrix; $A I = A$.
|
||||
|
||||
> *Definition*: an $n \times n$ matrix $A$ is said to be **nonsingular** or **invertible** if there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$. The matrix $A^{-1}$ is said to be a **multiplicative inverse** of $A$.
|
||||
|
||||
If $B$ and $C$ are both multiplicative inverses of $A$ then
|
||||
|
||||
$$
|
||||
B = BI = B(AC) = (BA)C = IC = C,
|
||||
$$
|
||||
|
||||
thus a matrix can have at most one multiplicative inverse.
|
||||
|
||||
> *Definition*: an $n \times n$ matrix is said to be **singular** if it does not have a multiplicative inverse.
|
||||
|
||||
Or similarly, an $n \times n$ matrix $A$ is singular if $A \mathbf{x} = \mathbf{0}$ for some non trivial $\mathbf{x} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. For a nonsingular matrix $A$, $\mathbf{x} = \mathbf{0}$ is the only solution to $A \mathbf{x} = \mathbf{0}$.
|
||||
|
||||
> *Theorem*: if $A$ and $B$ are nonsingular $n \times n$ matrices, then $AB$ is also nonsingular and
|
||||
>
|
||||
> $$
|
||||
> (AB)^{-1} = B^{-1} A^{-1}.
|
||||
> $$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Let $A$ and $B$ be nonsingular $n \times n$ matrices. If we suppose $AB$ is nonsingular and $(AB)^{-1} = B^{-1} A^{-1}$ we have
|
||||
|
||||
$$
|
||||
(AB)^{-1}AB = (B^{-1} A^{-1})AB = B^{-1} (A^{-1} A) B = B^{-1} B = I, \\
|
||||
AB(AB)^{-1} = AB(B^{-1} A^{-1}) = A (B B^{-1}) A^{-1} = A A^{-1} = I.
|
||||
$$
|
||||
|
||||
> *Theorem*: let $A$ be a nonsingular $n \times n$ matrix, the inverse of $A$ given by $A^{-1}$ is nonsingular.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Let $A$ be a nonsingular $n \times n$ matrix, $A^{-1}$ its inverse and $\mathbf{x} \in \mathbb{R}^n$ a vector. Suppose $A^{-1} \mathbf{x} = \mathbf{0}$ then
|
||||
|
||||
$$
|
||||
\mathbf{x} = I \mathbf{x} = (A A^{-1}) \mathbf{x} = A(A^{-1} \mathbf{x}) = \mathbf{0}.
|
||||
$$
|
||||
|
||||
> *Theorem*: let $A$ be a nonsingular $n \times n$ matrix then the solution of the system $A\mathbf{x} = \mathbf{b}$ is $\mathbf{x} = A^{-1} \mathbf{b}$ with $\mathbf{x}, \mathbf{b} \in \mathbb{R}^n$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Let $A$ be a nonsingular $n \times n$ matrix, $A^{-1}$ its inverse and $\mathbf{x}, \mathbf{b} \in \mathbb{R}^n$ vectors. Suppose $\mathbf{x} = A^{-1} \mathbf{b}$ then we have
|
||||
|
||||
$$
|
||||
A \mathbf{x} = A (A^{-1} \mathbf{b}) = (A A^{-1}) \mathbf{b} = \mathbf{b}.
|
||||
$$
|
||||
|
||||
> *Corollary*: the system $A \mathbf{x} = \mathbf{b}$ of $n$ linear equations in $n$ unknowns has a unique solution if and only if $A$ is nonsingular.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
The proof follows from the above theorem.
|
||||
|
||||
> *Theorem*: let $A$ and $B$ be matrices and $\alpha$ and $\beta$ be scalars. Each of the following statements valid
|
||||
>
|
||||
> 1. $(A^T)^T = A$,
|
||||
> 2. $(\alpha A)^T = \alpha A^T$,
|
||||
> 3. $(A + B)^T = A^T + B^T$,
|
||||
> 4. $(AB)^T = B^T A^T$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
105
docs/mathematics/linear-algebra/matrices/matrix-arithmetic.md
Normal file
105
docs/mathematics/linear-algebra/matrices/matrix-arithmetic.md
Normal file
|
|
@ -0,0 +1,105 @@
|
|||
# Matrix arithmetic
|
||||
|
||||
## Definitions
|
||||
|
||||
> *Definition*: let $A$ be a $m \times n$ *matrix* given by
|
||||
>
|
||||
> $$
|
||||
> A = \begin{pmatrix} a_{11} & a_{12}& \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}
|
||||
> $$
|
||||
>
|
||||
> with $a_{ij}$ referred to as the entries of $A$ or scalars in general, with $(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$. For real entries in $A$ we may denote $A \in \mathbb{R}^{m \times n}$.
|
||||
|
||||
This matrix may be denoted in a shorter way by $A = (a_{ij})$.
|
||||
|
||||
> *Definition*: let $\mathbf{x}$ be a $1 \times n$ matrix, referred to as *row vector* given by
|
||||
>
|
||||
> $$
|
||||
> \mathbf{x} = \begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{pmatrix}
|
||||
> $$
|
||||
>
|
||||
> with $x_i$ referred to as the entries of $\mathbf{x}$, with $i \in \{1, \dots, n\}$. For real entries we may denote $\mathbf{x} \in \mathbb{R}^n$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: let $\mathbf{x}$ be a $n \times 1$ matrix, referred to as *column vector* given by
|
||||
>
|
||||
> $$
|
||||
> \mathbf{x} = (x_1, x_2, \dots, x_n)
|
||||
> $$
|
||||
>
|
||||
> with $x_i$ referred to as the entries of $\mathbf{x}$, with $i \in \{1, \dots, n\}$. Also for the column vector we have for real entries $\mathbf{x} \in \mathbb{R}^n$.
|
||||
|
||||
From these two definitions it may be observed that row and column vectors may be used interchangebly, however using both it is important to state the difference. Best practice is to always work with row vectors and take the transpose if necessary.
|
||||
|
||||
## Matrix operations
|
||||
|
||||
> *Definition*: two $m \times n$ matrices $A$ and $B$ are said to be **equal** if $a_{ij} = b_{ij}$ for each $i(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: if $A$ is an $m \times n$ matrix and $\alpha$ is a scalar, then $\alpha A$ is the $m \times n$ matrix whose $(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$ entry is $\alpha a_{ij}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: if $A = (a_{ij})$ and $B = (b_{ij})$ are both $m \times n$ matrices, then the sum $A + B$ is the $m \times n$ matrix whose $(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$ entry is $a_{ij} + b_{ij}$ for each ordered pair $(i,j)$.
|
||||
|
||||
If $A$ is an $m \times n$ matrix and $\mathbf{x}$ is a vector in $\mathbb{R}^n$, then
|
||||
|
||||
$$
|
||||
A \mathbf{x} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n
|
||||
$$
|
||||
|
||||
with $A = (\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n})$.
|
||||
|
||||
> *Definition*: if $\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}$ are vectors in $\mathbb{R}^m$ and $x_1, x_2 \dots, x_n$ are scalars, then a sum of the form
|
||||
>
|
||||
> $$
|
||||
> x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n
|
||||
> $$
|
||||
>
|
||||
> is said to be a **linear combination** of the vectors $\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Theorem*: a linear system $A \mathbf{x} = \mathbf{b}$ is consistent if and only if $\mathbf{b}$ can be written as a linear combination of the column vectors $A$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
## Transpose matrix
|
||||
|
||||
> *Definition*: the **transpose** of an $m \times n$ matrix A is the $n \times m$ matrix $B$ defined by
|
||||
>
|
||||
> $$
|
||||
> b_{ji} = a_{ij},
|
||||
> $$
|
||||
>
|
||||
> for $j \in \{1, \dots, n\}$ and $i \in \{1, \dots m\}$. The transpose of $A$ is denoted by $A^T$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: an $n \times n$ matrix $A$ is said to be **symmetric** if $A^T = A$.
|
||||
|
||||
## Hermitian matrix
|
||||
|
||||
> *Definition*: the **conjugate transpose** of an $m \times n$ matrix A is the $n \times m$ matrix $B$ defined by
|
||||
>
|
||||
> $$
|
||||
> b_{ji} = \bar a_{ij},
|
||||
> $$
|
||||
>
|
||||
> for $j \in \{1, \dots, n\}$ and $i \in \{1, \dots m\}$. The **conjugate transpose** of $A$ is denoted by $A^H$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: an $n \times n$ matrix $A$ is said to be **Hermitian** if $A^H = A$.
|
||||
|
||||
## Matrix multiplication
|
||||
|
||||
> *Definition*: if $A = (a_{ij})$ is an $m \times n$ matrix and $B = (b_{ij})$ is an $n \times r$ matrix, then the product $A B = C = (c_{ij})$ is the $m \times r$ matrix whose entries are defined by
|
||||
>
|
||||
> $$
|
||||
> c_{ij} = \mathbf{a}_i \mathbf{b}_j = \sum_{k=1}^n a_{ik} b_{kj}
|
||||
> $$
|
||||
Loading…
Add table
Add a link
Reference in a new issue