n×mmatriisi

An n×m matrix is a rectangular array of elements arranged in n rows and m columns, where n and m are positive integers. Matrices are fundamental objects in linear algebra and are used extensively in various fields of mathematics, physics, engineering, computer science, and economics. Each element in the matrix is typically denoted by a variable with two subscripts, such as aᵢⱼ, where i represents the row index and j represents the column index.

The dimensions of a matrix are specified by the number of rows and columns it contains. For example, a 2×3 matrix has two rows and three columns, meaning it contains six elements arranged in two horizontal rows and three vertical columns. Matrices can be square (where n equals m), rectangular (where n does not equal m), or degenerate (where either n or m is zero, though this is less common in standard usage).

Matrices can represent linear transformations, systems of linear equations, or data tables. Operations such as addition, subtraction, and scalar multiplication can be performed on matrices, provided their dimensions are compatible. Matrix multiplication is also possible, but it requires that the number of columns in the first matrix matches the number of rows in the second matrix. The result is a new matrix where each element is computed as the dot product of a row from the first matrix and a column from the second.

Applications of matrices include solving linear systems, performing geometric transformations (such as rotations or scaling), and modeling networks or graphs. They are also foundational in advanced topics like eigenvalues, eigenvectors, and matrix decompositions, which are crucial in data analysis, machine learning, and quantum mechanics.