Search results
Results from the WOW.Com Content Network
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:
The vector space of matrices over is denoted by . For A ∈ K m × n {\displaystyle A\in \mathbb {K} ^{m\times n}} , the transpose is denoted A T {\displaystyle A^{\mathsf {T}}} and the Hermitian transpose (also called conjugate transpose ) is denoted A ∗ {\displaystyle A^{*}} .
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]
Such an belongs to 's null space and is sometimes called a (right) null vector of . The vector x {\displaystyle \mathbf {x} } can be characterized as a right-singular vector corresponding to a singular value of A {\displaystyle \mathbf {A} } that is zero.
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.
Consider the linear space of polynomials of a bounded degree. The derivative operator is a linear map. We know that applying the derivative to a polynomial decreases its degree by one, so when applying it iteratively, we will eventually obtain zero. Therefore, on such a space, the derivative is representable by a nilpotent matrix.