I need to find the column reduction form of a matrix. Are there any easy methods, free online books or pdfs where I can get examples?
From a computational viewpoint there is not much difference between row reduction and column reduction. Instead of doing the operations on rows you are doing the operations on columns. In fact you can turn column reduction into row reduction.
Take the transpose of the matrix, do row reduction (this can be found in any linear algebra text) and at the end take the transpose again.
Let A be an n x n matrix and let I be the n x n identity matrix. Place A over I so that a 2n x n matrix results. Let's denote it by [A/I]. Using elementary column operations, reduce [A/I] to say [B/E]. For example, B could be in reduced column echelon form. Then, AE=B where E represents a product of elementary column operations.
Just as Jonathan Needleman says, you can also perform elementary row operation to get the same thing. Start by placing I to the right of A^t so that an n x 2n matrix results. Denote it by [A^t,I]. Using elementary row operations (the 'same' as the column operations that were used to determine B), [A^t,I] reduces to [B^t,E^t] and E^t*A^t = B^t, or equivalently AE = B.
I guess that's Reducers columns mean obtaining a matrix in which the elements of each column are zeros except one at most. An algorithm is easy to apply (no easy explanation in this context ...) is as follows:
1) Mark with a circle the first element of the first column (a11); This marked element is called "pivot".
2) Enter the row containing the pivot (invariant).
3) Enter 0 for other elements of the column that is the pivot.
4) The calculation of the remaining elements is done this way: imagine that the pivot and the element to calculate are the vertices of a diagonal of a rectangle. Visualize the rectangle to determine the other two vertices. Now calculate the difference between the product of the diagonal elements pivot and the product of the elements of the other diagonal; this is the value of the new item.
5) Repeat steps 1) -4), choosing now as pivot the second element of the second column.
6) The algorithm is complete when you have completed steps 1) -4) taking as pivots each of the elements of the main diagonal.
If you can not understand, give me an email and I'll try to explain in more detail in a PDF, with mathematical notation.
Greetings from Spain.
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