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For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K (m,n) is the nm × mn matrix which, for any m × n matrix A, transforms vec(A) into vec(A T): K (m,n) vec(A) = vec(A T) .
Matrix representation is a method used by a computer language to store column-vector matrices of more than one dimension in memory. Fortran and C use different schemes for their native arrays. Fortran uses "Column Major" ( AoS ), in which all the elements for a given column are stored contiguously in memory.
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m×n matrix and B is an n×p matrix, then their matrix product AB is the m×p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column ...
A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space. A linear map A from a vector space of dimension n into a vector space of dimension m maps a column vector
A vector specified as a row matrix is known as a row vector; one specified as a column matrix is known as a column vector. Again, an n -dimensional vector v {\displaystyle \mathbf {v} } can be specified in either of the following forms using matrices:
The row space of this matrix is the vector space spanned by the row vectors. The column vectors of a matrix. The column space of this matrix is the vector space spanned by the column vectors. In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column ...