User:Dcljr/Matrix

The terms below are used in the branch of mathematics called matrix theory, which is often considered a subfield of linear algebra. For specific types of matrices, see the List of matrices. For some matrix operations, see Matrix.

Matrix

A rectangular array of objects which are usually members of a ring.

The definitions below will assume the following matrix:

${\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &&\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}=(a_{ij})_{i=1,\ldots ,m \atop j=1,\ldots ,n}}$

Element

One of the objects in a matrix.

a_ij for a specific choice of i and j.

Size or dimensions

The number of rows and columns, respectively, of a matrix; usually expressed in the form m × n, read "m by n".

i-th row of matrix A

${\displaystyle {\begin{bmatrix}a_{i1}&a_{i2}&\cdots &a_{in}\end{bmatrix}}}$

j-th column of matrix A

${\displaystyle {\begin{bmatrix}a_{1j}\\a_{2j}\\\vdots \\a_{mj}\end{bmatrix}}}$

Main diagonal

The elements whose row and column number match.

${\displaystyle \{a_{kk}:\,k=1,\ldots ,\min(m,n)\}}$

Transpose

An operation resulting in a new matrix whose rows are the columns of the original matrix and whose columns are the rows of the original matrix, or the resulting matrix itself.

${\displaystyle A^{T}=(a_{ji})_{i=1,\ldots ,m \atop j=1,\ldots ,n}}$

Trace

The sum of the elements on the main diagonal.

${\displaystyle {\mbox{tr}}\,A=a_{11}+a_{22}+\ldots +a_{kk},{\mbox{ where }}k=\min(m,n)}$

...

Minor

The determinant of the matrix obtained by deleting a given row and column from the original matrix.

${\displaystyle M_{ij}={\begin{vmatrix}a_{11}&\cdots &a_{1,j-1}&a_{1,j+1}&\cdots &a_{1n}\\\vdots &&\vdots &\vdots &&\vdots \\a_{i-1,1}&\cdots &a_{i-1,j-1}&a_{i-1,j+1}&\cdots &a_{i-1,n}\\a_{i+1,1}&\cdots &a_{i+1,j-1}&a_{i+1,j+1}&\cdots &a_{i+1,n}\\\vdots &&\vdots &\vdots &&\vdots \\a_{m1}&\cdots &a_{m,j-1}&a_{m,j+1}&\cdots &a_{mn}\end{vmatrix}}}$

Note that the i-th row and j-th column are missing in the above determinant.

...

Vector

A matrix with one row (a row vector) or one column (column vector).

See Vector for more information.

Linear transformation

The function that results from multiplying a given matrix by an appropriately sized vector of variables.

${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},{\mbox{ where }}f(x)=Ax,\ x\in \mathbb {R} ^{n}.}$

Rank

The dimension of the space generated by the rows of a given matrix.

The dimension of the image of the linear transformation represented by the matrix.