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Concept# Square matrix

Summary

In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order . Any two square matrices of the same order can be added and multiplied.
Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if is a square matrix representing a rotation (rotation matrix) and is a column vector describing the position of a point in space, the product yields another column vector describing the position of that point after that rotation. If is a row vector, the same transformation can be obtained using , where is the transpose of .
Main diagonal
The entries (i = 1, ..., n) form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a11 = 9, a22 = 11, a33 = 4, a44 = 10.
The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal or counterdiagonal.
If all entries outside the main diagonal are zero, is called a diagonal matrix. If only all entries above (or below) the main diagonal are zero, is called an upper (or lower) triangular matrix.
The identity matrix of size is the matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
It is a square matrix of order , and also a special kind of diagonal matrix. It is called identity matrix because multiplication with it leaves a matrix unchanged:
for any m×n matrix .
A square matrix is called invertible or non-singular if there exists a matrix such that
If exists, it is unique and is called the inverse matrix of , denoted .
A square matrix that is equal to its transpose, i.e., , is a symmetric matrix. If instead , then is called a skew-symmetric matrix.
For a complex square matrix , often the appropriate analogue of the transpose is the conjugate transpose , defined as the transpose of the complex conjugate of .

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Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

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