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In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; ... which multiply matrices efficiently, ...
The identity matrices (which are the square matrices whose entries are zero outside of the main diagonal and 1 on the main diagonal) are identity elements of the matrix product. It follows that the n × n matrices over a ring form a ring, which is noncommutative except if n = 1 and the ground ring is commutative.
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV ∗, where U and V are unitary matrices and D is a diagonal matrix. An example of a matrix in Jordan ...
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 ...
Comment: The diagonal elements of D are called the singular values of A. Comment: Like the eigendecomposition above, the singular value decomposition involves finding basis directions along which matrix multiplication is equivalent to scalar multiplication, but it has greater generality since the matrix under consideration need not be square.
One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily composed and inverted. Composition is accomplished by matrix multiplication. Row and column vectors are operated upon by matrices, rows on the left and columns on the right. Since text reads from left to right, column ...
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.