Matrix (mathematics)

A matrix is a rectangular array of numbers or other mathematical objects. These objects are called entries or elements. They are arranged in horizontal lines called rows and vertical lines called columns.

Matrices are essential tools in many fields of science and math. They allow us to organize large amounts of data into a single structure. This structure makes it easier to perform complex calculations. In linear algebra, matrices represent linear maps. In geometry, they describe how shapes change through transformations. They are also used in numerical analysis to solve massive computational problems. Many of these problems involve matrices with huge dimensions.

The size of a matrix is defined by its rows and columns. We describe this as the dimension of the matrix. For example, a matrix with two rows and three columns is a two-by-three matrix. There is no limit to how many rows or columns a matrix can have. If a matrix has the same number of rows and columns, it is a square matrix.

A matrix with only one row is called a row matrix or a row vector. A matrix with only one column is called a column matrix or a column vector. We also use the term infinite matrix for grids with infinite rows or columns. In computer science, an empty matrix may have no rows or columns at all.

We can perform several basic operations on matrices. Matrix addition and subtraction require the matrices to have the same size. You calculate the result by performing the operation on each corresponding entry.

Scalar multiplication involves multiplying a single number, called a scalar, by every entry in the matrix. Another operation is the transpose. To find the transpose, you turn all the rows into columns and all the columns into rows. Matrix multiplication is more complex. It is only possible if the number of columns in the first matrix matches the number of rows in the second. This operation corresponds to the composition of linear transformations.

Unlike regular numbers, matrix multiplication is not commutative. This means the order of the matrices matters, and $AB$ does not always equal $BA$.

Matrices are powerful tools for describing geometric transformations. A matrix can act as a linear map to change a shape's position or appearance. For example, a matrix can represent a rotation.

It can also represent a scaling, which stretches or shrinks an object.

Other transformations include shearing or flipping a shape.

These changes are called geometric transformations. A matrix can even change the coordinate system used to describe a space. These transformations are fundamental to understanding how objects move and change in geometry.

Square matrices have unique properties that are central to matrix theory. One important value is the determinant. The determinant is a single number associated with a square matrix. It is fundamental for studying the matrix's properties. For instance, a square matrix is invertible only if its determinant is not zero. An invertible matrix allows you to undo a transformation. Another important concept is the eigenvalue. The eigenvalues of a square matrix are the roots of its characteristic polynomial. These values help describe how the matrix affects certain vectors.

Matrices are also used to manage systems of linear equations. Instead of writing out every equation separately, we can write them compactly as a single matrix equation. This makes solving complex systems much more efficient. We can use row operations to solve these equations. These include adding one row to another, multiplying a row by a non-zero constant, or switching two rows. These techniques are used in methods like Gauss-Jordan elimination. By using matrices, mathematicians can handle many variables at the same time. This is much faster than traditional algebraic methods.

Matrix theory has grown into a massive branch of mathematics. It began as a sub-branch of linear algebra. Over time, it expanded to include graph theory, algebra, combinatorics, and statistics.

It is used in almost every scientific field. In computer programming, matrices are often represented as arrays of arrays. While mathematicians usually start counting rows and columns at one, some computer languages start at zero. This difference is just a matter of convention. Regardless of the notation, matrices remain a core way to study patterns and solve problems in the modern world.