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If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
The left null space of A is the set of all vectors x such that x T A = 0 T. It is the same as the null space of the transpose of A. The product of the matrix A T and the vector x can be written in terms of the dot product of vectors:
A graph and its transpose. In the mathematical and algorithmic study of graph theory, the converse, [1] transpose [2] or reverse [3] of a directed graph G is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of the corresponding edges in G.
Transposition, producing the transpose of a matrix A T, which is computed by swapping columns for rows in the matrix A; Transpose of a linear map; Transposition (logic), a rule of replacement in philosophical logic; Transpose relation, another name for converse relation
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map.
(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 ...
Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA T will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.