The identity matrix is the identity element in the ring of square matrices. The unit matrix or identity matrix I n, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MI n=M and I nN=N for any m-by- n matrix M and n-by- k matrix N. M( n, C), the ring of complex square matrices, is a complex associative algebra. M( n, R), the ring of real square matrices, is a real unitary associative algebra. Unless n = 1, this ring is not commutative. The set of all square n-by- n matrices, together with matrix addition and matrix multiplication is a ring. We have ( A + B) tr = A tr + B tr and ( AB) tr = B tr A tr.Ī square matrix is a matrix which has the same number of rows and columns. If A describes a linear map with respect to two bases, then the matrix A tr describes the transpose of the linear map with respect to the dual bases, see dual space. The transpose of an m-by- n matrix A is the n-by- m matrix A tr (also sometimes written as A T or t A) formed by turning rows into columns and columns into rows, i.e. The rank of a matrix A is the dimension of the image of the linear map represented by A this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A. More generally, a linear map from an n-dimensional vector space to an m-dimensional vector space is represented by an m-by- n matrix, provided that bases have been chosen for each. This follows from the above-mentioned associativity of matrix multiplication. Now if the k-by- m matrix B represents another linear map g : R m → R k, then the linear map g o f is represented by BA. We say that the matrix A "represents" the linear map f. For every linear map f : R n → R m there exists a unique m-by- n matrix A such that f( x) = Ax for all x in R n. Here and in the sequel we identify R n with the set of "columns" or n-by-1 matrices. This same property makes them powerful data structures in high-level programming languages. Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next. Linear transformations, ranks and transpose It is important to note that commutativity does not generally hold that is, given matrices A and B and their product defined, then generally AB ≠ BA. C( A + B) = CA + CB for all m-by- n matrices A and B and k-by- m matrices C ("left distributivity").( A + B) C = AC + BC for all m-by- n matrices A and B and n-by- k matrices C ("right distributivity").( AB) C = A( BC) for all k-by- m matrices A, m-by- n matrices B and n-by- p matrices C ("associativity").Matrix multiplication has the following properties: These two operations turn the set M( m, n, R) of all m-by- n matrices with real entries into a real vector space of dimension mn. If A is an m-by- n matrix and B is an n-by- p matrix, then their matrix product AB is the m-by- p matrix ( m rows, p columns) given by: Multiplication of two matrices is well-defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying the scalar c by every element of A (i.e. For example:Īnother, much less often used notion of matrix addition is the direct sum. Given m-by- n matrices A and B, their sum A + B is the m-by- n matrix computed by adding corresponding elements (i.e. Is a 1×9 matrix, or 9-element row vector. A 1 × n matrix (one row and n columns) is called a row vector, and an m × 1 matrix (one column and m rows) is called a column vector. However, the convention that the indices i and j start at 1 is not universal: some programming languages start at zero, in which case we have 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1.Ī matrix where one of the dimensions equals one is often called a vector, and interpreted as an element of real coordinate space. We often write to define an m × n matrix A with each entry in the matrix A called a ij for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. The row is always noted first, then the column. The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or ( i, j)-th entry of A. The dimensions of a matrix are always given with the number of rows first, then the number of columns. A matrix with m rows and n columns is called an m-by- n matrix (written m× n) and m and n are called its dimensions. The horizontal lines in a matrix are called rows and the vertical lines are called columns.
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