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The nullity of a matrix is the dimension of the null space, and is equal to the number of columns in the reduced row echelon form that do not have pivots. [7] The rank and nullity of a matrix A with n columns are related by the equation: + =.
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 number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of the kernel of f). [1] [2] [3] [4]
A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0. a ij = δ ij: Lehmer matrix: a ij = min(i, j) ÷ max(i, j). A positive symmetric matrix. Matrix of ones: A matrix with all entries equal to one. a ij = 1. Pascal matrix: A matrix containing the entries of Pascal's triangle. Pauli matrices
The signature of a metric tensor is defined as the signature of the corresponding quadratic form. [2] It is the number (v, p, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities.
For , a pseudoinverse of A is defined as a matrix + satisfying all of the following four criteria, known as the Moore–Penrose conditions: [4] [5] A A + {\displaystyle AA^{+}} need not be the general identity matrix, but it maps all column vectors of A to themselves: A A + A = A . {\displaystyle AA^{+}A=\;A.}
It follows that Ax 1, Ax 2, …, Ax r are linearly independent. Now, each Ax i is obviously a vector in the column space of A. So, Ax 1, Ax 2, …, Ax r is a set of r linearly independent vectors in the column space of A and, hence, the dimension of the column space of A (i.e., the column rank of A) must be at least as big as r.
Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space. The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form.