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The linear transformation mapping x to Ax is bijective; that is, the equation Ax = b has exactly one solution for each b in K n. (There, "bijective" can equivalently be replaced with "injective" or "surjective".) The columns of A form a basis of K n. (In this statement, "basis" can equivalently be replaced with either "linearly independent set ...
The Gershgorin circle theorem is useful in solving matrix equations of the form Ax = b for x where b is a vector and A is a matrix with a large condition number.
A square matrix A is called invertible or non-singular if there exists a matrix B such that [28] [29] = =, where I n is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A −1.
where is the matrix formed by replacing the i-th column of A by the column vector b. A more general version of Cramer's rule [ 10 ] considers the matrix equation A X = B {\displaystyle AX=B}
The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB. [1] Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, [2] to represent the composition of linear maps ...
The linear least squares problem is to find the x that minimizes ‖ Ax − b ‖, which is equivalent to projecting b to the subspace spanned by the columns of A. Assuming the columns of A (and hence R) are independent, the projection solution is found from A T Ax = A T b. Now A T A is square (n × n) and invertible, and also equal to R T R.
Input: initial guess x (0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion Output: solution when convergence is reached Comments: pseudocode based on the element-based formula above k = 0 while convergence not reached do for i := 1 step until n do σ = 0 for j := 1 step until n do if j ≠ i then ...
The vector equation is equivalent to a matrix equation of the form = where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries. [ 4 ]