User:Dcljr/Matrix

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


 * Element
 * One of the objects in a matrix.
 * aij 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 &times; n, read "m by n".


 * i-th row of matrix A
 * $$\begin{bmatrix} a_{i1} & a_{i2} & \cdots & a_{in} \end{bmatrix}$$


 * j-th column of matrix A
 * $$\begin{bmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \end{bmatrix}$$


 * Main diagonal
 * The elements whose row and column number match.
 * $$\{a_{kk}:\,k = 1, \ldots, \min(m,n)\}$$


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


 * Trace
 * The sum of the elements on the main diagonal.
 * $$\mbox{tr}\,A = a_{11} + a_{22} + \ldots + a_{kk}, \mbox{ where } k=\min(m,n)$$

...


 * Minor
 * The determinant of the matrix obtained by deleting a given row and column from the original matrix.
 * $$M_{ij}=\begin{vmatrix} a_{11} & \cdots & a_{1,j-1} & a_{1,j+1} & \cdots & a_{1n} \\ \vdots & & \vdots & \vdots &  & \vdots \\ a_{i-1,1} & \cdots & a_{i-1,j-1} & a_{i-1,j+1} & \cdots & a_{i-1,n} \\ a_{i+1,1} & \cdots & a_{i+1,j-1} & a_{i+1,j+1} & \cdots & a_{i+1,n} \\ \vdots &  & \vdots & \vdots &  & \vdots \\ a_{m1} & \cdots & a_{m,j-1} & a_{m,j+1} & \cdots & a_{mn} \end{vmatrix}$$
 * 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.
 * $$f:\mathbb{R}^n\to\mathbb{R}^m,\mbox{ where }f(x)=Ax,\ x\in\mathbb{R}^n.$$


 * 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.