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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 ...
(This is just a consequence of the fact that the inverse of an N×M transpose is an M×N transpose, although it is also easy to show explicitly that P −1 composed with P gives the identity.) As proved by Cate & Twigg (1977), the number of fixed points (cycles of length 1) of the permutation is precisely 1 + gcd( N −1, M −1) , where gcd is ...
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A T (among other notations). [1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [2]
To use column-major order in a row-major environment, or vice versa, for whatever reason, one workaround is to assign non-conventional roles to the indexes (using the first index for the column and the second index for the row), and another is to bypass language syntax by explicitly computing positions in a one-dimensional array.
MySQL uses a .frm file to store the definition of the table, but this file is not a part of the MyISAM engine; instead it is a part of the server. The data file has a .MYD (MYData) extension. The index file has a .MYI (MYIndex) extension. The index file, if lost, can always be recreated by recreating indexes.
In most cases, an index is used to quickly locate the data records from which the required data is read. In other words, the index is only used to locate data records in the table and not to return data. A covering index is a special case where the index itself contains the required data fields and can answer the required data.
Let H be a Hadamard matrix of order n.The transpose of H is closely related to its inverse.In fact: = where I n is the n × n identity matrix and H T is the transpose of H.To see that this is true, notice that the rows of H are all orthogonal vectors over the field of real numbers and each have length .
The transpose A T is an invertible matrix. A is row-equivalent to the n-by-n identity matrix I n. A is column-equivalent to the n-by-n identity matrix I n. A has n pivot positions. A has full rank: rank A = n. A has a trivial kernel: ker(A) = {0}. The linear transformation mapping x to Ax is bijective; that is, the equation Ax = b has exactly ...