Search results
Results From The WOW.Com Content Network
There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending ...
When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]
The characteristic equation, also known as the determinantal equation, [1] [2] [3] is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory , the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix .
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, where a i j {\displaystyle a_{ij}} is the entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th ...
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 ...
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
For a function f of three or more variables, there is a generalization of the rule shown above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point.
The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant.