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Definition

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The statement of the theorem

A matrix is a rectangular array of numbers (or other mathematical objects), called the "entries" of the matrix. Matrices are subject to standard operations such as addition and multiplication. Most commonly, a matrix over a field FF is a rectangular array of elements of ⁠ FF ⁠. A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries are discussed below. For instance, this is a real matrix: A=[−1.30.620.45.59.7−6.2].\mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}. The numbers (or other objects) in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are respectively called rows and columns. The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the number of rows and columns that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with m rows and n columns is called an m × n matrix, or m-by-n matrix, where m and n are called its dimensions. For example, the matrix A{\mathbf {A} } above is a 3 × 2 matrix. Matrices with a single row are called row matrices or row vectors, and those with a single column are called column matrices or column vectors. A matrix with the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix. Overview of a matrix size Name Size Example Description Row matrix 1×n1\times n [372]{\begin{bmatrix}3&7&2\end{bmatrix}} A matrix with one row and more than one columns, sometimes used to represent a vector Column matrix n×1n\times 1 [418]{\begin{bmatrix}4\\1\\8\end{bmatrix}} A matrix with one column and more than one rows, sometimes used to represent a vector Square matrix n×nn\times n [91351117263]{\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\end{bmatrix}} A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing. - ^Lang (2002), Chapter XIII. - ^Fraleigh (1976), p. 209. - ^Nering (1970), p. 37. - ^ ^{a}^{b}Brown (1991), p. 1. - ^Golub & Van Loan (1996), p. 3. - ^Horn & Johnson (1985), p. 5. - ^Gbur (2011), p. 89. - ^Cite error: The named reference empty was invoked but never defined (see the help page).