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A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
The answer to the decision problem for the existential theory of the reals, given this sentence as input, is the Boolean value true. The inequality of arithmetic and geometric means states that, for every two non-negative numbers x {\displaystyle x} and y {\displaystyle y} , the following inequality holds: x + y 2 ≥ x y . {\displaystyle ...
Quantifier elimination is a term used in mathematical logic to explain that, in some theories, every formula is equivalent to a formula without quantifier. This is the case of the theory of polynomials over an algebraically closed field , where elimination theory may be viewed as the theory of the methods to make quantifier elimination ...
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns ...
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement " ∃ x {\displaystyle \exists x} such that … {\displaystyle \ldots } " can be viewed as a question "When is there an x {\displaystyle x} such that … {\displaystyle \ldots ...
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions.
In three-dimensional Euclidean space, these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as:
Second, we solve the equation = for x. In both cases we are dealing with triangular matrices (L and U), which can be solved directly by forward and backward substitution without using the Gaussian elimination process (however we do need this process or equivalent to compute the LU decomposition itself).