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a ij are 1 if i divides j or if j = 1; otherwise, a ij = 0. A (0, 1)-matrix. Shift matrix: A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. a ij = δ i+1,j or a ij = δ i−1,j: Multiplication by it shifts matrix elements by one position. Zero matrix: A matrix with all entries equal to zero. a ij = 0.
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R. [2] The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1. When multiplied by another matrix, it isolates a specific row or column in arbitrary position.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple λ of the identity matrix I. Its effect on a vector is scalar multiplication by λ . For example, a 3×3 scalar matrix has the form: [ λ 0 0 0 λ 0 0 0 λ ] ≡ λ I 3 {\displaystyle {\begin{bmatrix}\lambda &0&0\\0&\lambda &0\\0&0&\lambda \end{bmatrix ...
The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that: When multiplied by itself, the result is itself; All of its rows and columns are linearly independent. The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However ...
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. [ 1 ] [ 2 ] That is, the matrix A {\displaystyle A} is idempotent if and only if A 2 = A {\displaystyle A^{2}=A} .