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  • numerical methods - What does it mean to pivot (linear algebra . . .
    So the natural idea is to pick the largest of the remaining entries, call it the pivot (turning axis) and use that row as the basis for the elimination step To keep constructing the echelon form, rows are swapped or rotated (most efficiently using a row index array), adding permutation steps to the elementary row transformations
  • linear algebra - Question about finding pivot columns in a matrix . . .
    Here's a matrix with a pivot off the diagonal \begin{matrix} 2 3 6 \\ 0 0 1 \\ \end{matrix} Here both the 2 and the 1 are in pivot positions because they correspond to a leading 1 in the reduced echelon form of the matrix
  • Is it okay to determine pivot positions in a matrix in echelon form . . .
    The author haven't yet checked if the positions in 2 and -5 satisfy the definition of pivot position when the echelon matrix is reduced to a reduced echelon form I think if it becomes the reduced echelon form, the columns above the pivot positions, the author determined, can be nonzeros and the entries in pivot positions, author dermined, can
  • matrices - Intuitive way of knowing why pivot positions matter . . .
    However, if we had a pivot in each column where n = r, then we know for a fact that the only solution to our system is the trivial one Although this illustration may seem trivial, I think it will give you a better idea of why pivot columns are important and what they tell us about the solutions to a system of linear equations
  • Why do pivot columns indicate linear independence?
    What you can tell from the pivot columns is that the pivot columns are one of the subsets of linearly independent columns We know this because whenever you encounter a column that is a linear combination of the previous pivot columns, it does not become a pivot column
  • Finding number of pivots when matrix includes variables
    Instead of performing Gaussian elimination, we can use the fact that the number of pivots is equal to the rank of the matrix, which can be determined by examining its various minors: the rank of a matrix is equal to the maximum order of its non-vanishing minors
  • Pivots, determinant and eigenvalues - Mathematics Stack Exchange
    If the resulting matrix is upper-triangular, the determinant of the matrix is the product of the diagonal entries As for property (2); as the constant term in the characteristic polynomial the determinant of a matrix is always the product of its eigenvalues, with appropriate algebraic multiplicity
  • linear algebra - Pivot positions and reduced row echelon form . . .
    To answer this question, we look at if the rightmost column of the augmented matrix $[A\quad b]$ is a pivot column But this doesn't make any sense to me, because a pivot column is only going to have 1 non 0 variable in it, and the point of an augmented matrix is to solve the system of equations for a column (vector) of multiple variables
  • Pivoting on a matrix element - Mathematics Stack Exchange
    $\begingroup$ No, a "pivot" need not be equal to 1 From standard English, a "pivot" is "a person or thing on which something depends or turns; the central or crucial factor " The entry you are considering is the "pivot" because are using it as the central crucial factor to eliminate the other entries in the column (make them a $0$)
  • What does a having pivot in every row tell us? What about a pivot in . . .
    $\begingroup$ why does having a pivot in every row necessarily mean Ax=b has at least one solution? Even if there weren't pivots in every row, couldn't we still have solutions—for eg, if A = [4 5 6 ; 0 0 0] and b= [5 ; 0] then we have 4*x_1 + 5*x_2 + 6*x_3 = 5, which does give at least one solution (x_2 and x_3 in particular are free variables), but A doesn't have a pivot in every row





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