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Matrix (mathematics)

Matrix (mathematics)

rectangular array of numbers, symbols, or expressions, arranged in rows and columns From Wikipedia, the free encyclopedia

Matrix (mathematics)
Definitions and notationsExampleOperationsAdditionMultiplication of two matricesSpecial matricesSquare matrixIdentityInverse matrixOne column matrixDeterminantsProperties of determinantsRelated pagesReferencesOther websites

In mathematics, a matrix (plural: matrices) is a rectangle of numbers, arranged in rows and columns. The rows are each left-to-right (horizontal) lines, and the columns go top-to-bottom (vertical). The top-left cell is at row 1, column 1 (see diagram at right).

Thumb
Specific entries of a matrix are often referenced by using pairs of subscripts, for the numbers at each of the rows & columns.

Matrices are often represented by capital roman letters such as A {\displaystyle A} {\displaystyle A}, B {\displaystyle B} {\displaystyle B} and C {\displaystyle C} {\displaystyle C},[1] and there are rules for adding, subtracting and "multiplying" matrices together, but the rules are different than for numbers.[2] As an example, the product A B {\displaystyle AB} {\displaystyle AB} does not always give the same result as B A {\displaystyle BA} {\displaystyle BA}, which is the case for the multiplication of ordinary numbers.[3] A matrix can have more than 2 dimensions, such as a 3D matrix. Also, a matrix can be one-dimensional, as a single row or a single column.

Many natural sciences use matrices quite a lot. In many universities, courses about matrices (usually called linear algebra) are taught very early, sometimes even in the first year of studies. Matrices are also very common in computer science, engineering, physics, economics, and statistics.[4]

The horizontal lines in a matrix are called rows. The vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (or m×n matrix). m and n are called its dimensions.

The places in the matrix where the numbers are, are called entries.[2] The entry of a matrix called "A" that is in the row number i and column number j is called the i,j entry of A. This is written as A[i,j] or ai,j.

A := ( a i j ) m × n {\displaystyle A:=(a_{ij})_{m\times n}} {\displaystyle A:=(a_{ij})_{m\times n}} defines a m × n matrix called A. Each entry in this matrix is called ai,j when "i" is between 1 and m, and when j is between 1 and n.

Example

The matrix

[ 1 2 3 1 2 7 4 9 2 6 1 5 ] {\displaystyle {\begin{bmatrix}1&2&3\\1&2&7\\4&9&2\\6&1&5\end{bmatrix}}} {\displaystyle {\begin{bmatrix}1&2&3\\1&2&7\\4&9&2\\6&1&5\end{bmatrix}}}

is a 4×3 matrix. This matrix has m=4 rows, and n=3 columns.

The element A[2,3] or a2,3 is 7.

Addition

The sum of two matrices is the matrix, which (i,j)-th entry is equal to the sum of the (i,j)-th entries of two matrices:

[ 1 3 2 1 0 0 1 2 2 ] + [ 0 0 5 7 5 0 2 1 1 ] = [ 1 + 0 3 + 0 2 + 5 1 + 7 0 + 5 0 + 0 1 + 2 2 + 1 2 + 1 ] = [ 1 3 7 8 5 0 3 3 3 ] {\displaystyle {\begin{bmatrix}1&3&2\\1&0&0\\1&2&2\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\\2&1&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0&2+5\\1+7&0+5&0+0\\1+2&2+1&2+1\end{bmatrix}}={\begin{bmatrix}1&3&7\\8&5&0\\3&3&3\end{bmatrix}}} {\displaystyle {\begin{bmatrix}1&3&2\\1&0&0\\1&2&2\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\\2&1&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0&2+5\\1+7&0+5&0+0\\1+2&2+1&2+1\end{bmatrix}}={\begin{bmatrix}1&3&7\\8&5&0\\3&3&3\end{bmatrix}}}

The two matrices have the same dimensions. Here, A + B = B + A {\displaystyle A+B=B+A} {\displaystyle A+B=B+A} is true (and is true in general for matrices of equal dimensions).

Multiplication of two matrices

The multiplication of two matrices is a bit more complicated. First, find the dimensions of each matrix as m × n {\displaystyle m\times n} {\displaystyle m\times n}, where m {\displaystyle m} {\displaystyle m} is the number of rows, and n {\displaystyle n} {\displaystyle n} is the number of columns. If A {\displaystyle A} {\displaystyle A} has dimensions m 1 × n 1 {\displaystyle m_{1}\times n_{1}} {\displaystyle m_{1}\times n_{1}} and B {\displaystyle B} {\displaystyle B} has dimensions m 2 × n 2 {\displaystyle m_{2}\times n_{2}} {\displaystyle m_{2}\times n_{2}}, then the matrix A B {\displaystyle AB} {\displaystyle AB} will have dimensions m 1 × n 2 {\displaystyle m_{1}\times n_{2}} {\displaystyle m_{1}\times n_{2}}. The values of n 1 {\displaystyle n_{1}} {\displaystyle n_{1}}and m 2 {\displaystyle m_{2}} {\displaystyle m_{2}} must be equal, or else the multiplication of the two matrices is impossible.[3]

Multiply each row by each column. In the following example, to get the top left value of the resulting matrix, get the first row of the first matrix, [ a 1 , a 2 ] {\displaystyle [a_{1},a_{2}]} {\displaystyle [a_{1},a_{2}]}, and the first column of the second matrix, [ b 1 , b 3 ] {\displaystyle [b_{1},b_{3}]} {\displaystyle [b_{1},b_{3}]}. Then sum the products of each corresponding index: ( a 1 × b 1 + a 2 × b 3 ) {\displaystyle (a_{1}\times b_{1}+a_{2}\times b_{3})} {\displaystyle (a_{1}\times b_{1}+a_{2}\times b_{3})}.

[ a 1 a 2 a 3 a 4 ] ⋅ [ b 1 b 2 b 3 b 4 ] = [ ( a 1 ⋅ b 1 + a 2 ⋅ b 3 ) ( a 1 ⋅ b 2 + a 2 ⋅ b 4 ) ( a 3 ⋅ b 1 + a 4 ⋅ b 3 ) ( a 3 ⋅ b 2 + a 4 ⋅ b 4 ) ] {\displaystyle {\begin{bmatrix}a_{1}&a_{2}\\a_{3}&a_{4}\\\end{bmatrix}}\cdot {\begin{bmatrix}b_{1}&b_{2}\\b_{3}&b_{4}\\\end{bmatrix}}={\begin{bmatrix}(a_{1}\cdot b_{1}+a_{2}\cdot b_{3})&(a_{1}\cdot b_{2}+a_{2}\cdot b_{4})\\(a_{3}\cdot b_{1}+a_{4}\cdot b_{3})&(a_{3}\cdot b_{2}+a_{4}\cdot b_{4})\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}a_{1}&a_{2}\\a_{3}&a_{4}\\\end{bmatrix}}\cdot {\begin{bmatrix}b_{1}&b_{2}\\b_{3}&b_{4}\\\end{bmatrix}}={\begin{bmatrix}(a_{1}\cdot b_{1}+a_{2}\cdot b_{3})&(a_{1}\cdot b_{2}+a_{2}\cdot b_{4})\\(a_{3}\cdot b_{1}+a_{4}\cdot b_{3})&(a_{3}\cdot b_{2}+a_{4}\cdot b_{4})\\\end{bmatrix}}}

This is an example with numbers:

[ 3 5 1 4 ] ⋅ [ 2 3 5 0 ] = [ ( 3 ⋅ 2 + 5 ⋅ 5 ) ( 3 ⋅ 3 + 5 ⋅ 0 ) ( 1 ⋅ 2 + 4 ⋅ 5 ) ( 1 ⋅ 3 + 4 ⋅ 0 ) ] = [ 31 9 22 3 ] {\displaystyle {\begin{bmatrix}3&5\\1&4\\\end{bmatrix}}\cdot {\begin{bmatrix}2&3\\5&0\\\end{bmatrix}}={\begin{bmatrix}(3\cdot 2+5\cdot 5)&(3\cdot 3+5\cdot 0)\\(1\cdot 2+4\cdot 5)&(1\cdot 3+4\cdot 0)\\\end{bmatrix}}={\begin{bmatrix}31&9\\22&3\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}3&5\\1&4\\\end{bmatrix}}\cdot {\begin{bmatrix}2&3\\5&0\\\end{bmatrix}}={\begin{bmatrix}(3\cdot 2+5\cdot 5)&(3\cdot 3+5\cdot 0)\\(1\cdot 2+4\cdot 5)&(1\cdot 3+4\cdot 0)\\\end{bmatrix}}={\begin{bmatrix}31&9\\22&3\\\end{bmatrix}}}
  • The result of the multiplication is called the product. The product is another matrix. It has the same number of rows as the first matrix and the same number of columns as the second matrix.
  • The multiplication of matrices is not commutative, which means that in general, A B ≠ B A {\displaystyle AB\neq BA} {\displaystyle AB\neq BA}.[4]
  • The multiplication of matrices is associative, which means that ( A B ) C = A ( B C ) {\displaystyle (AB)C=A(BC)} {\displaystyle (AB)C=A(BC)}.[4]

There are some matrices that are special.

Square matrix

See the main article: Square matrix

A square matrix has the same number of rows as columns, so m=n.[5]

An example of a square matrix is

[ 5 − 2 4 0 9 1 − 7 6 8 ] {\displaystyle {\begin{bmatrix}5&-2&4\\0&9&1\\-7&6&8\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}5&-2&4\\0&9&1\\-7&6&8\\\end{bmatrix}}}

This matrix has 3 rows and 3 columns: m=n=3.

Identity

See the main article: Identity matrix

Every square dimension set of a matrix has a special counterpart called the "identity matrix", represented by the symbol I {\displaystyle I} {\displaystyle I}.[1] The identity matrix has nothing but zeroes except on the main diagonal, where there are all ones. For example:

[ 1 0 0 0 1 0 0 0 1 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\\end{bmatrix}}}

is an identity matrix. There is exactly one identity matrix for each square dimension set. An identity matrix is special because when multiplying any matrix by the identity matrix, the result is always the original matrix with no change.

Inverse matrix

See the main article: Invertible matrix

An inverse matrix is a matrix that, when multiplied by another matrix, equals the identity matrix. For example:

[ 7 8 6 7 ] ⋅ [ 7 − 8 − 6 7 ] = [ 1 0 0 1 ] {\displaystyle {\begin{bmatrix}7&8\\6&7\\\end{bmatrix}}\cdot {\begin{bmatrix}7&-8\\-6&7\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}7&8\\6&7\\\end{bmatrix}}\cdot {\begin{bmatrix}7&-8\\-6&7\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}

[ 7 − 8 − 6 7 ] {\displaystyle {\begin{bmatrix}7&-8\\-6&7\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}7&-8\\-6&7\\\end{bmatrix}}} is the inverse of [ 7 8 6 7 ] {\displaystyle {\begin{bmatrix}7&8\\6&7\\\end{bmatrix}}} {\displaystyle {\begin{bmatrix}7&8\\6&7\\\end{bmatrix}}}.

The formula for the inverse of a 2x2 matrix, A = [ x y z v ] {\displaystyle A={\begin{bmatrix}x&y\\z&v\end{bmatrix}}} {\displaystyle A={\begin{bmatrix}x&y\\z&v\end{bmatrix}}} is:

1 det ( A ) [ v − y − z x ] {\displaystyle {\frac {1}{\det(A)}}{\begin{bmatrix}v&-y\\-z&x\end{bmatrix}}} {\displaystyle {\frac {1}{\det(A)}}{\begin{bmatrix}v&-y\\-z&x\end{bmatrix}}}

Where det {\displaystyle \det } {\displaystyle \det } is the determinant of the matrix. In a 2x2 matrix, the determinant is equal to:

x v − y z {\displaystyle {xv-yz}} {\displaystyle {xv-yz}}

If the determinant is 0, then it is referred to as 'singular', and has no inverse.[6]

One column matrix

A matrix, that has many rows, but only one column, is called a column vector.

See the main article: Determinant

The determinant takes a square matrix and calculates a simple number, a scalar. To understand what this number means, take each column of the matrix and draw it as a vector. The parallelogram drawn by those vectors has an area, which is the determinant. For all 2x2 matrices, the formula is very simple: det ( [ a b c d ] ) = a d − b c {\displaystyle \det \left({\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}\right)=ad-bc} {\displaystyle \det \left({\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}\right)=ad-bc}

For 3x3 matrices the formula is more complicated: det ( [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] ) = a 1 ( b 2 c 3 − c 2 b 3 ) − a 2 ( b 1 c 3 − c 1 b 3 ) + a 3 ( b 1 c 2 − c 1 b 2 ) {\displaystyle \det \left({\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\\\end{bmatrix}}\right)=a_{1}(b_{2}c_{3}-c_{2}b_{3})-a_{2}(b_{1}c_{3}-c_{1}b_{3})+a_{3}(b_{1}c_{2}-c_{1}b_{2})} {\displaystyle \det \left({\begin{bmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\\\end{bmatrix}}\right)=a_{1}(b_{2}c_{3}-c_{2}b_{3})-a_{2}(b_{1}c_{3}-c_{1}b_{3})+a_{3}(b_{1}c_{2}-c_{1}b_{2})}

There are no simple formulas for the determinants of larger matrices, and many computer programmers study how to get computers to quickly find large determinants.

Properties of determinants

There are three rules that all determinants follow. These are:

  • The determinant of an identity matrix is 1
  • If two rows or two columns of the matrix are exchanged, then the determinant is multiplied by -1. Mathematicians call this alternating.
  • If all the numbers in one row or column are multiplied by another number n, then the determinant is multiplied by n. Also, if a matrix M has a column v that is the sum of two column matrices v 1 {\displaystyle v_{1}} {\displaystyle v_{1}} and v 2 {\displaystyle v_{2}} {\displaystyle v_{2}}, then the determinant of M is the sum of the determinants of M with v 1 {\displaystyle v_{1}} {\displaystyle v_{1}} in place of v and M with v 2 {\displaystyle v_{2}} {\displaystyle v_{2}} in place of v. These two conditions are called multi-linearity.
  • Determinant
  • Eigenvalues and eigenvectors
  • Matrix analysis
  • Matrix function
  • Numerical linear algebra
  • System of linear equations
  • Transpose
  1. [1]
    "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-19.
  2. [2]
    "Matrices". www.mathsisfun.com. Retrieved 2020-08-19.
  3. [3]
    "How to Multiply Matrices". www.mathsisfun.com. Retrieved 2020-08-19.
  4. [4]
    "Matrix | mathematics". Encyclopedia Britannica. Retrieved 2020-08-19.
  5. [5]
    Weisstein, Eric W. "Matrix". mathworld.wolfram.com. Retrieved 2020-08-19.
  6. [6]
    "Singular Matrix - Definition, Properties, Examples, Meaning". Cuemath. Retrieved 2024-09-20.
The English Wikibook Linear Algebra has more information on:
Matrices
Wikiversity has more on: Matrices at
Linear algebra#Matrices
Wikimedia Commons has media related to Matrices.
History
  • MacTutor: Matrices and determinants Archived 2015-03-08 at the Wayback Machine
  • Matrices and Linear Algebra on the Earliest Uses Pages
  • Earliest Uses of Symbols for Matrices and Vectors
Online books
  • Kaw, Autar K. (September 2008), Introduction to Matrix Algebra, ISBN 978-0-615-25126-4
  • Brookes, M. (2005), The Matrix Reference Manual, London: Imperial College, retrieved 2008-12-10
  • Online Calculus Courses Archived 2018-07-08 at the Wayback Machine
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Definitions and notations

Operations

Special matrices

Determinants

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References

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