A Matrix
- (now rare) The womb.
- (biology) The material or tissue in which more specialized structures are embedded.
- (biology) An extracellular matrix, the material or tissue between the cells of animals or plants.
- (biology) Part of the mitochondrion.
- (biology) The medium in which bacteria are cultured.
- (mathematics) A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory.
- (computing) A two-dimensional array.
- (electronics) A grid-like arrangement of electronic components, especially one intended for information coding, decoding or storage.
- A table of data.
- (geology) A geological matrix.
- (archaeology and paleontology) The sediment surrounding and including the artifacts, features, and other materials at a site.
- (analytical chemistry) The environment from which a given sample is taken.
- (printing, historical) In hot metal typesetting, a mold for casting a letter.
- (printing, historical) In printmaking, the plate or block used, with ink, to hold the image that makes up the print.
Synonym: printing form
- The cavity or mold in which anything is formed
- (dyeing) The five simple colours (black, white, blue, red, and yellow) from which all the others are formed.
- (material science) A binding agent of composite materials, e.g. resin in fibreglass.
| Synonyms | Edit |
(mathematics): array, table
(table of data): array, grid, spreadsheet, table
(computing): array
Mathematics
A matrix is a rectangular arrangement of numbers, symbols, or expressions organized into rows and columns.
\[A_{mn}=\begin{bmatrix}a_{11}&......&a_{1n}\\.....&a_{ij}&.....\\a_{m1}&.....&a_{mn}\end{bmatrix}\]
\[A_{mn}=\{a_{ij}\ (j,i); i,j \in Z ; 0\le i\le m, 0\le j\le n \}\]
- Element / Entry: An individual number or expression inside a matrix. \(a_{ij}\)
- Row: A horizontal line of elements in a matrix. Row c or \(R_c=\) \(\{a_{ij}\in A_{mn}; i=c, 1\le j\le n\}\)
- Column: A vertical line of elements in a matrix. Column c or \(C_c=\)\(\{a_{ij}\in A_{mn}; j=c, 1\le i\le m\}\)
- Dimension / Order: \((m× n)\)The ordered product of the number of rows (m) and number of columns (n).
- Main Diagonal: The set of elements in the diagonal running from the top-left to the bottom-right. D=\\(D=\{a_{ij}\in A_{mn}; i=j; 0\le i\le m, 0\le j\le n \}\)
- Anti-diagonal: a diagonal line running in the opposite direction to the usual main diagonal of a matrix. \( D'=\{a_{ij}\in A_{mn}; i+j=n+1; 0\le i\le m, 0\le j\le n \} \)
Operation on Matrices
Equality
Two matrices are equal if they have the same dimensions and every corresponding element has the same value.
\(A_{mn}=[a_{ij}]\), \(B_{uv}=[b_{kl}]\) (I)
\[A=B\leftrightarrow \{m=u, n=v , a_{ij}=b_{kl}; i,j=k,l \}\]
Addition
Additive identity
\(A=B\leftrightarrow \ a_{ij}=b_{ij}\) (I)
So there's a matrix \(C_{mn}\in M_{mn}; C=[c_{ij}] \) which satisfying \(A+C=B\) (II)
(I,II): \(A+C=A\) (III)
Thus \(C_{nm}=0\) but this zero is not the real zero.
Sum of Matrices
If \(A+C=B\)
So:
\[A+C=D_{mn}\]
Thus \(d_{ij}=b_{ij}\)
Although \(a_{ij}+c_{ij}=b_{ij}\)
If (I) still fulfilled so: \(c_{ij}=0\)
Thus
\[C_{mn}=0_{mn}\leftrightarrow c_{ij}=0\]
Generally: \(A+B=C\longleftrightarrow c_{ij}=a_{ij}+b_{ij}\) (IV)
Since matrix addition is defined element by element using addition in the real numbers, it inherits the corresponding algebraic properties of real-number addition (commutativity, associativity, additive identity, and additive inverse).
Additive Inverse
\[A,B \in M_{mn}; A+B=0 \longleftrightarrow B=-A, -A=[-a_{ij}]_{mn}\]
\[A+B=0\]
\[[a_{ij}+b_{ij}]=[0_{ij}]\]
\[a_{ij}+b_{ij}=0_{ij} \]
\[b_{ij} =0_{ij}-a_{ij}\]
\[b_{ij} =0-a_{ij}\]
\[b_{ij} =-a_{ij}\]
Thus \(B=-A=[-a_{ij}]_{mn}\)
Scala Multiplication
If \(\sum A=k.A \)
\[[\sum a_{ij}]=k.A\]
\[k.A=[k.a_{ij}]\]
Properties of scalar multiplication
\[\forall A \in M_{mn} \ , k\in \mathbb {R} \iff k.A \in M_{mn}\]
- Distributive over matrix addition
\[\forall A,B \in M_{mn} \ , k\in \mathbb {R} \iff k(A+B)=k.A+k.B \]
- Distributive over scalar addition
\[\forall A \in M_{mn} \ , k,l\in \mathbb {R} \iff (k+l).A=k.A+l.A \]
- Associative with scalar multiplication
\[\forall A \in M_{mn} \ , k,l\in \mathbb {R} \iff k.(l.A)=(k.l).A\]
- Identity scalar
\[\forall A \in M_{mn} \implies 1.A=A\]
- Zero scalar
\[\forall A \in M_{mn} \implies 0.A=0_{mn}\]
Matrix Transposition
The transpose of a matrix is the matrix obtained by interchanging the rows and columns of the matrix.
\[\forall A \in M_{mn} \ \iff (A )^T=B_{nm}=A^T; b_{ij}=a_{ji}\]
Properties of Transposition
- Transpose of a transpose
\[\forall A\in M_{mn} \iff (A^T)^T=A\]
- Transpose of a sum
\[\forall A, B\in M_{mn} \iff (A+B)^T=A^T+B^T\]
- Transpose of scalar multiplication
\[\forall A\in M_{mn},k\in \mathbf R \iff (k.A)^T=k.A^T\]
Multiplication of Matrices
The multiplication is defined basically by multiplication of an one-row matrix with its transpose as followed:
\[\forall A\in M_{1×n}\iff A.(A^T)=[\sum_{j=1}^{n} (a_{1j})^2]\]
The product of a one-row matrix representing a vector and its transpose is a one-element matrix whose element is equal to the square of the norm of the represented vector.
\[\forall A\in M_{1×n}\iff A.(A^T)=[\|\vec{A}\|^2]; \vec A=\sum_{j=1}^na_{1j}.\vec e_j\]
- Theorem
\[B=\lambda. A^T\iff \vec B\perp \vec A \Lambda A.B=[\|\vec A\| .\|\vec B\| ]\]
\[\forall A\in M_{1×n}\iff A.(A^T)=[\|\vec{A}\|^2]; \vec A=\sum_{j=1}^na_{1j}.\vec e_j\]
\[\vec A=\sum_{j=1}^na_{1j}.\vec e_j \implies \lambda. \vec A=\sum_{j=1}^n\lambda a_{1j}.\vec e_j\]
\[\vec A(\lambda. \vec A)= (\sum_{j=1}^na_{1j}.\vec e_j)( \sum_{j=1}^n\lambda a_{1j}.\vec e_j) \\ =\lambda (\sum_{j=1}^na_{1j}.\vec e_j)( \sum_{j=1}^n a_{1j}.\vec e_j)\\ =\lambda (\sum_{j=1}^na_{1j}.\vec e_j)^2\\ =\lambda \sum_{j=1}^n(a_{1j})^2\\\]