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The left null space, or cokernel, of a matrix A consists of all column vectors x such that x T A = 0 T, where T denotes the transpose of a matrix. The left null space of A is the same as the kernel of A T. The left null space of A is the orthogonal complement to the column space of A, and is dual to the cokernel of the
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:
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]
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.
Let and be Banach spaces, : a closed linear operator whose domain () is dense in , and ′ the transpose of . The theorem asserts that the following conditions are equivalent: The theorem asserts that the following conditions are equivalent:
Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering , and statistics , such as signal processing , least squares fitting of data, and process control .
The ginv function calculates a pseudoinverse using the singular value decomposition provided by the svd function in the base R package. An alternative is to employ the pinv function available in the pracma package. The Octave programming language provides a pseudoinverse through the standard package function pinv and the pseudo_inverse() method.
A square matrix which is equal to the negative of its conjugate transpose, A * = −A. Skew-symmetric matrix: A matrix which is equal to the negative of its transpose, A T = −A. Skyline matrix: A rearrangement of the entries of a banded matrix which requires less space. Sparse matrix: A matrix with relatively few non-zero elements.