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  2. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    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]. Now through application of elementary row operations, find the reduced echelon form of this n × 2n matrix.

  3. 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 .

  4. Low-rank approximation - Wikipedia

    en.wikipedia.org/wiki/Low-rank_approximation

    In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank.

  5. Rank (linear algebra) - Wikipedia

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

    The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A. A fundamental result in linear algebra is that the column rank and the row rank are always equal. (Three proofs of this result are given in § Proofs that column rank = row rank, below.)

  6. Row and column spaces - Wikipedia

    en.wikipedia.org/wiki/Row_and_column_spaces

    The rank is equal to the number of pivots in the reduced row echelon form, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4 × 4 matrix in the example above has rank three.

  7. Row echelon form - Wikipedia

    en.wikipedia.org/wiki/Row_echelon_form

    A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it.

  8. When are the fourth College Football Playoff rankings ...

    www.aol.com/fourth-college-football-playoff...

    The College Football Playoff will soon release its fourth top 25 rankings of the season. Here’s what you need to know and how to watch.

  9. Rouché–Capelli theorem - Wikipedia

    en.wikipedia.org/wiki/Rouché–Capelli_theorem

    The theorem can be read almost directly on the reduced row echelon form as follows. The rank of a matrice is number of nonzero rows in its reduced row echelon form. If the ranks of the coefficient matrix and the augmented matrix are different, then the last non zero row has the form […], corresponding to the equation 0 = 1.