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The set of Toeplitz matrices is a subspace of the vector space of matrices (under matrix addition and scalar multiplication). Two Toeplitz matrices may be added in O ( n ) {\displaystyle O(n)} time (by storing only one value of each diagonal) and multiplied in O ( n 2 ) {\displaystyle O(n^{2})} time.
A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix. The following matrix is square diagonal matrix: [] If the entries are real numbers or complex numbers, then it is a normal matrix as well.
In the trivial case, S = I, where I is the identity matrix, gives regular OFDM without spreading. The received signal can also be expressed as: r = F −1 Λ H FF −1 (Λ C F)b, where S = Λ C F, and C is a circulant matrix defined by C = F −1 Λ C F, where Λ C is the circulant's diagonal matrix.
The identity matrix under Hadamard multiplication of two m × n matrices is an m × n matrix where all elements are equal to 1. This is different from the identity matrix under regular matrix multiplication, where only the elements of the main diagonal are equal to 1. Furthermore, a matrix has an inverse under Hadamard multiplication if and ...
The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix .
Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix: = + + + + = (), where is given by the companion matrix = []. The set of n × n {\displaystyle n\times n} circulant matrices forms an n {\displaystyle n} - dimensional vector space with respect to addition and scalar multiplication.
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop: