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2. Rank (linear algebra) - Wikipedia

   en.wikipedia.org/wiki/Rank_(linear_algebra)

   In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4] Rank is thus a measure of the "nondegenerateness ...

3. Rank–nullity theorem - Wikipedia

   en.wikipedia.org/wiki/Rank–nullity_theorem

   For the rank theorem of multivariable calculus, see constant rank theorem. Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: 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 ...

4. Sherman–Morrison formula - Wikipedia

   en.wikipedia.org/wiki/Sherman–Morrison_formula

   In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a " rank -1 update" to a matrix whose inverse has previously been computed. [1][2][3] That is, given an invertible matrix and the outer product of vectors and the formula cheaply computes an updated matrix inverse.

5. Cauchy–Binet formula - Wikipedia

   en.wikipedia.org/wiki/Cauchy–Binet_formula

   Cauchy–Binet formula. In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that ...

6. Matrix decomposition - Wikipedia

   en.wikipedia.org/wiki/Matrix_decomposition

   Matrix decomposition. In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

7. Rank factorization - Wikipedia

   en.wikipedia.org/wiki/Rank_factorization

   In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .

8. Projection matrix - Wikipedia

   en.wikipedia.org/wiki/Projection_matrix

   Properties. The projection matrix has a number of useful algebraic properties. [5][6] In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix . [4] (. Note that is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized as follows: [4] and.

9. Jacobian matrix and determinant - Wikipedia

   en.wikipedia.org/wiki/Jacobian_matrix_and...

   The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x. When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f.