Questions tagged [gaussian-elimination]
For questions on or related to the technique of Gaussian elimination, used in solving systems of linear equations.
695 questions
1 vote
1 answer
145 views
I thought Gaussian elimination preserved the solution to a set of equations
I have following matrix (8 rows, 10 columns) which I have augmented with an 11th constant term to represent 8 equations over 10 variables. ...
- 131
2 votes
1 answer
67 views
Semifield / Division Ring and Gaussian elimination.
Discrete logarithm is difficult because, unlike integer division where magnitudes exist, and calculation can proceed iteratively by test-comparing and subtracting, the concept of magnitude doesn't ...
- 355
0 votes
0 answers
33 views
If a matrix can be written as a product of atomic upper/lower triangular matrices, is its inverse calculated as any atomic triangular matrix?
With gauss elimination, the inverse of the matrix $M_{n-1}\dots M_2M_1=M$ is just $$ M^{-1}=\begin{bmatrix} -\vec{μ_1}\quad -\vec{μ_2} \quad \dots \quad -\vec{μ}_n\end{bmatrix} + I $$ I get that this ...
- 61
2 votes
1 answer
70 views
Efficiently finding out which vectors from some set are linearly independent of the other vectors in the set
I have a set $S = \{v_1, v_2, \cdots, v_N\}$ with $N$ vectors, each of which have $d$ dimensions. I would now like to find a set $S' = \{n | v_n \in \mathrm{span(}S \setminus \{v_n \}) \}$, i.e. the ...
- 45
-1 votes
1 answer
107 views
Gaussian elimination modulo $4$ [closed]
Does anyone know how to perform Gaussian elimination modulo $4$? Are there any ready-to-use code snippets or relevant websites available? I find that there are almost no existing code implementations ...
- 103
0 votes
3 answers
104 views
Finding the inverse of a 3 by 3 matrix using matrix row operations
Problem: Find the inverse of the following matrix: $$ \begin{bmatrix} 1 & 3 & 2 \\ -1 & 2 & 1 \\ 1 & 1 & 0 \end{bmatrix} $$ Answer: First we check the determinate to see if the ...
- 4,632
1 vote
1 answer
63 views
Provability of methodological algorithms, including Gauss elimination
I know there are algorithms for Gauss elimination, such as: Swapping two rows, Multiplying a row by a nonzero number, Adding a multiple of one row to another row. Looking at these three algorithms, ...
- 867
0 votes
1 answer
54 views
Why is my Gauss-Jordan inverse calculation incorrect despite consistent row operations?
I attempted to compute the inverse of the matrix $$ \begin{bmatrix} 1 & 1 & 1 \\ 5 & 0 & 2 \\ 5 & 2 & 9 \end{bmatrix} $$ using elementary row transformations. I combined it ...
0 votes
0 answers
45 views
What type of matrix inverse do I need here?
I have non-square binary matrices $A_1, A_2, B_2, B_3$ and binary vectors $v_1, v_2, v_3$ such that $$A_1v_1 = A_2v_2$$ $$B_2v_2 = B_3v_3$$ Given $v_1$ and $A_1, A_2, B_2, B_3$, I'd like to solve for $...
- 125
2 votes
0 answers
107 views
Mathematical advantage to pivoting before starting gaussian elimination
In one of my computational classes we discussed using gaussian elimination on the following matrix: $$ \begin{bmatrix} 10^{-17} & 2 \\ 1 & 3 \end{bmatrix} $$ The first step of gaussian ...
- 99
1 vote
2 answers
213 views
How can I transform this block matrix by applying elementary row operations?
I have a matrix with a sort-of "block-identity" structure, that I would like to transform (via elementary row operations) into a matrix of the following form: each column (except for the ...
- 121
0 votes
1 answer
78 views
LU factorization with full column of zeros
I've been practicing LU factorization for matrices. All is well, as long as the matrix is square and there are no zeros on the main diagonal, in which case I resort to $PA=LU$. So far so good. But ...
0 votes
2 answers
123 views
Intuition behind why row operations are allowed in Gaussian elimination [closed]
I am currently studying Computer Science and I have some problems understanding some concepts in my Linear Algebra class. I've been working with Gaussian elimination and solve systems of equations ...
1 vote
2 answers
168 views
Adding columns (rows) of totally unimodular matrices preserving total unimodularity
I recently found the paper titled Total unimodularity and decomposition method for large-scale air traffic cell transmission model stating: Assume a $m\times n$ matrix $A$ is totally unimodular and of ...
- 33
4 votes
1 answer
121 views
Can we think of the solution of a linear system of eqns as the new Origin for that particular system?
I couldn't understand why Gauss Elimination works. Algebraically, it is straightforward. For eg: x + y = 5 x - y = 1 If above two statements are true, then adding them result in a true statement. But, ...
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