Matrix - Revision history

Rongzhou: /* Identity matrix */

2026-10-17T12:07:29Z

Identity matrix

← Older revision		Revision as of 12:07, 17 October 2026
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	$$		$$

	~~Then~~ it is an '''identity matrix''' of dimension $m$, denoted $\mathbf{I}_m$.		then it is an '''identity matrix''' of dimension $m$, denoted $\mathbf{I}_m$.

	From the definition of [[#Matrix multiplication]], it can be inferred that		From the definition of [[#Matrix multiplication]], it can be inferred that

Rongzhou: /* Binary operations */

2026-10-17T12:07:04Z

Binary operations

← Older revision		Revision as of 12:07, 17 October 2026
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	\|}		\|}

	== Binary ~~operations~~ ==		== [[Binary operation]]s ==

	=== Single matrix operations ===		=== Single matrix operations ===

Rongzhou: /* Formal definition */

2026-10-17T12:05:17Z

Formal definition

← Older revision		Revision as of 12:05, 17 October 2026
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	$$		$$

	is a matrix of '''~~degree~~''' $m \times n$ over [[field (mathematical concept)\|field]] $\mathbb{F}$.		is a matrix of '''dimension''' $m \times n$ over [[field (mathematical concept)\|field]] $\mathbb{F}$.

	== Dimension ==		== Dimension ==

Rongzhou: /* Formal definition */

2026-10-17T12:04:58Z

Formal definition

← Older revision		Revision as of 12:04, 17 October 2026
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	$$		$$

	is a matrix of ~~dimension~~ $m \times n$ over [[field]] $\mathbb{F}$.		is a matrix of '''degree''' $m \times n$ over [[field (mathematical concept)\|field]] $\mathbb{F}$.

	== Dimension ==		== Dimension ==

Rongzhou at 14:06, 15 October 2026

2026-10-15T14:06:44Z

← Older revision		Revision as of 14:06, 15 October 2026
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			A '''matrix''' (plural: '''matrices''') is a 2-dimensional ordered list of numbers:

			$$
			\mathbf{M}_{m \times n} =
			\begin{bmatrix}
			a_{11} & a_{12} & \cdots & a_{1n} \\
			a_{21} & a_{22} & \cdots & a_{2n} \\
			\vdots & \vdots & \ddots & \vdots \\
			a_{m1} & a_{m2} & \cdots & a_{mn}
			\end{bmatrix},
			$$
			where element $a_{ij}$ is in row $i$, column $j$.

			It can be therefore written:

			$$
			\mathbf{M}_{m \times n} = (a_{ij}),i \in \{1,\dots,m\}, j \in \{1,\dots,n\}
			$$

			== Formal definition ==

			Any function satisfying

			$$
			\mathbf{M} : \{1,\dots,m\} \times \{1,\dots,n\} \to \mathbb{F}
			$$

			is a matrix of dimension $m \times n$ over [[field]] $\mathbb{F}$.

			== Dimension ==

			If a matrix $\mathbb{M}_{m \times n}$ has $m$ rows and $n$ columns, its '''dimension''' is $m \times n$.

			Special cases:

			{\| class="wikitable"
			\|-
			!
			! Known as
			\|-
			\| $m=n$
			\| '''Square matrix'''
			\|-
			\| $m=1$
			\| '''Row vector'''
			\|-
			\| $n=1$
			\| '''Column vector'''
			\|}

			== Binary operations ==

			=== Single matrix operations ===

			For matrice $\mathbf{A}=(a_{ij}) \in \mathbb{F}$:

			{\| class="wikitable"
			\|-
			! Name
			! Operation
			\|-
			\| '''scalar multiplication'''
			\| $c\mathbf{A} = (c\,a_{ij})$
			\|-
			\| '''transposition'''
			\| $\mathbf{A}^T = (a_{ji})$
			\|}

			=== Two matrices operations ===

			==== Matrix addition/subtraction ====

			It is defined for two matrices of '''identical sizes''' $\mathbf{A}_{m \times n}=(a_{ij}), \mathbf{B}_{m \times n}=(b_{ij})\in \mathbb{F}$:

			$$
			(\mathbf{A} \pm \mathbf{B})= (a_{ij} \pm b_{ij})
			$$

			==== Matrix multiplication ====

			It is defined for two matrices of '''compatible sizes''' $\mathbf{A}_{m \times n}=(a_{ij}), \mathbf{B}_{n \times p}=(b_{ij})\in \mathbb{F}$:

			$$
			\forall i \in \{1,\dots,m\}, j \in \{1,\dots,p\} \\
			(\mathbf{A}\mathbf{B})_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}.
			$$

			<blockquote>
			Notice that $\mathbf{AB}$ is a $m \times p$ matrix over $\mathbb{F}$
			</blockquote>

			== Identity matrix ==

			If '''square matrix''' $\mathbf{I}=(a_{ij})$ of dimension $m \times m$ satisfies

			$$
			\begin{cases}
			i=j: a_{ij}=1 \\
			i \neq j: a_{ij}=0
			\end{cases}
			$$

			Then it is an '''identity matrix''' of dimension $m$, denoted $\mathbf{I}_m$.

			From the definition of [[#Matrix multiplication]], it can be inferred that

			$$
			\mathbf{A}_{m \times n}\mathbf{I}_n = \mathbf{A}_{m \times n} = \mathbf{I}_m\mathbf{A}_{m \times n}
			$$

			== Determinant ==




	== Matrix operations in python ==		== Matrix operations in python ==

Rongzhou: /* Matrix operations in python */

2026-10-15T12:33:54Z

Matrix operations in python

← Older revision		Revision as of 12:33, 15 October 2026
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	product_A_B=np.dot(A,B) # A.B		product_A_B=np.dot(A,B) # A.B
	</syntaxhighlight>		</syntaxhighlight>

			<ref>{{#cite:Q2182}}</ref>

Rongzhou at 08:03, 15 October 2026

2026-10-15T08:03:18Z

← Older revision		Revision as of 08:03, 15 October 2026
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	=== Single matrix operations ===		=== Single matrix operations ===

			==== Transposition, determinant, comatrix, inverse ====

	<syntaxhighlight lang=python copy>		<syntaxhighlight lang=python copy>
	A_T=A.T # Transpose A: -> 2D nparray: since A is an nparray, nupmy understands what you mean!		A_T=A.T # Transpose A: -> 2D nparray: since A is an nparray, nupmy understands what you mean!
			det_A=np.linalg.det(A): -> float
			inverse_A=np.linalg.inv(A): -> 2D nparray
			</syntaxhighlight>

			==== Element-wise,column-wise, and row-wise sum ====

			<syntaxhighlight lang=python copy>
	sum_A=np.sum(A) # -> float: element-wise sum of A		sum_A=np.sum(A) # -> float: element-wise sum of A
	column_wise_sum=np.sum(A, axis=0): -> 1D nparray		column_wise_sum=np.sum(A, axis=0): -> 1D nparray

Rongzhou: /* Matrice operations in python */

2026-10-15T07:16:23Z

Matrice operations in python

← Older revision		Revision as of 07:16, 15 October 2026
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	== ~~Matrice~~ operations in python ==		== Matrix operations in python ==

	=== Initialisation ===		=== Initialisation ===

Rongzhou: /* Initialisation */

2026-10-15T07:15:58Z

Initialisation

← Older revision		Revision as of 07:15, 15 October 2026
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	<syntaxhighlight lang=python copy>		<syntaxhighlight lang=python copy>
	import numpy as np # import ~~numpy~~		import numpy as np # import dependencies: always the 1st step!
	</syntaxhighlight>		</syntaxhighlight>

Rongzhou: Created page with "== Matrice operations in python == === Initialisation === import numpy as np # import numpy The matrices are to be inputted as 2D numpy arrays, so: $$ \forall a_{ij} \in \mathbf{A}, \text{a[i][j]}=a_{ij} $$ For example, to initialise the identity matrice $\mathbf{I}_3$: $$ I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix} $$ I=np.array([[1,0,0],[0,1,0],[0,0,1]])
2026-10-15T07:14:41Z

Created page with "== Matrice operations in python == === Initialisation === <syntaxhighlight lang=python copy> import numpy as np # import numpy </syntaxhighlight> The matrices are to be inputted as 2D numpy arrays, so: $$ \forall a_{ij} \in \mathbf{A}, \text{a[i][j]}=a_{ij} $$ For example, to initialise the identity matrice $\mathbf{I}_3$: $$ I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix} $$ <syntaxhighlight lang=python copy> I=np.array([[1,0,0],[0,1,0],[0,0,1]]) </syntaxhighli..."

New page
== Matrice operations in python ==

=== Initialisation ===

<syntaxhighlight lang=python copy>
import numpy as np # import numpy
</syntaxhighlight>

The matrices are to be inputted as 2D numpy arrays, so:

$$
\forall a_{ij} \in \mathbf{A}, \text{a[i][j]}=a_{ij}
$$

For example, to initialise the identity matrice $\mathbf{I}_3$:

$$
I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}
$$

<syntaxhighlight lang=python copy>
I=np.array([[1,0,0],[0,1,0],[0,0,1]])
</syntaxhighlight>

For two matrices $\mathbf{A},\mathbf{B}$ initialised as 2D arrays:

=== Single matrix operations ===

<syntaxhighlight lang=python copy>
A_T=A.T # Transpose A: -> 2D nparray: since A is an nparray, nupmy understands what you mean!
sum_A=np.sum(A) # -> float: element-wise sum of A
column_wise_sum=np.sum(A, axis=0): -> 1D nparray
row_wise_sum=np.sum(A, axis=1): -> 1D nparray
</syntaxhighlight>

=== Two matrices operations ===

==== Element-wise operations ====

<syntaxhighlight lang=python copy>
sum_A_B=np.add(A,B) # A+B
difference_A_B=np.subtract(A,B) # A-B
element_wise_product_A_B=np.multiply(A,B) # A \circ B
quotient_A_B=np.divide(A,B) # A/B
</syntaxhighlight>

<blockquote>
Be extremely careful with <code>np.multiply</code>, which gives the '''element-wise product''', NOT the matrix product.
</blockquote>

==== Matrix operations ====

<syntaxhighlight lang=python copy>
product_A_B=np.dot(A,B) # A.B
</syntaxhighlight>