linear algebra – Nullspace of reduced row low level form for integer matrices

I work with matrices that have integer entries (using exact arithmetic). I want the results of operations such as nullspace to also be vectors with strictly integer entries.

Until now, I have implemented a reduced rank form at low ranks (adapted from here). The use of the entire row reduction (by previous link) allows me to correctly get the row space and column space of a matrix with coefficients of & # 39; 39; integer also having integer coefficients. The connection of the null space to the r ref of a matrix is ​​also well known (for example here). The question is how can I generate the null space using the low integer rref that I have (which is similar, there are only diagonal elements that are not necessarily necessary) $ 1 $). The use of the algorithm to generate nullspace from rref with low rref fails for example for the following matrix:

$$ A = pmatrix {5 & 3 & 7 \ 10 & 6 & 14 \ 8 & 3 & 1} $$

the low rref calculated is:

$$ rref (A) = pmatrix {1 & -3 & -19 \ 0 & 9 & 51 \ 0 & 0 & 0} $$

Using the algorithm to generate the null space from the above r ref failed.

For example, the output is:

$ pmatrix {19 \ – 51 \ 1} $

How can I change the algorithm to generate zero space from low rref?

Thank you!

PS This is for a mine library for integer calculations that also supports integer matrices. Therefore, I implement exact and fractional algorithms.

linear algebra – Calculates dim $ S $, where $ S $ is a subspace of $ V_3 $

Let $ S $ denote the set of all vectors $ (x, y, z) $ in $ V_3 $ whose components satisfy $ x = y = z $. Determine if $ S $ is a subspace of $ V_3 $. Yes $ S $ is a subspace, calculates dim $ S $.

I found that $ S $ is a subspace of $ V_3 $. But I do not know how to calculate its dimension. I work on Apostol Vol 2, and it addresses many theoretical explanations on the dimension, but never shows how to find it. I know the answer is $ 1 $, but why?

linear algebra – Maximum determinant of matrix nxn with elements 0 and 1

Generalize https://math.stackexchange.com/questions/3265627/largest-value-of-a-third-order-determinant-whose-elements-are-0-or-1 I would like to propose two related issues

(a) find the maximum value of the determinant of a $ n times n $-matrix whose elements are $ 0 or $ 1 $ (Christianized a matrix 0-1 here).

(b) find all possible values ​​of the determinants of a matrix 0-1 of order $ n $.

I found the solutions for the little ones $ n = 2, …, $ 5 by "brute force" in Mathematica by simply listing the values ​​of the determinants as much as possible $ 2 ^ {n ^ 2} $ matrices.

On my PC, I could not go further $ n = $ 5 due to lack of memory.

I have two of them questions

1) Is anyone aware of an analytical solution?

2) Can you improve the Mathematica code that I will show soon in an automatic answer? (This part is intentionally delayed to allow the reader to find his own solutions).

linear algebra – At the limit of eigenvalues ​​for any matrix

I read this pre-print: https://arxiv.org/abs/1605.00531. The authors mention the following:

"We know that eigenvalues $ z_k $ of any matrix $ Z $ are in the rectangle $ text {Re} (z_k) in sigma_1 $, $ text {Im} (z_k) in sigma_2 $ or $ sigma_1 $ is the range of $ frac {1} {2} (Z + Z ^ dagger) $ and $ sigma_2 $ is the range of $ frac {1} {2} (Z-Z ^ dagger) $. "

I thought the range was a set of all the linear combinations of the columns of $ Z $? is $ sigma_1 $ the cardinality of this set?

More generally, how can I prove their statement or where can I find proof? The reference used by the author is inaccessible with my university.

Thank you so much!

linear algebra – Existence of hyperplane having no intersection with a given Z-module

Let $ x_1, …, x_n $ to be vectors in $ ℝ ^ d $, with typically $ n> d $. I am interested in knowing when it is possible to find a vector $ aεℝ ^ d $ such as $ (aᵀx_i) _ {1≤i≤n} $ are linearly independent on $ mathbb {Z} $ (that is to say more $ mathbb {Q} $).

I'm not very familiar with this stuff, but as I understand it all $ S = {_ {i = 1} ^ n α_ix_i: αε mathbb {Z} ^ n $ is what we call a $ mathbb {Z} $-module? So the property would be tantamount to finding $ aεℝ ^ n $ such as $ ∀αε mathbb {Z} ^ n: aᵀ (_ {i = 1} ^ n α_ix_i) = 0 implies α = 0 $that is, the search for a vector hyperplane that does not intersect with $ S _ { backslash {0 }} $?

I do not see if there is a simple argument to always answer yes / no, or if the condition $ (x_i) _ {1≤i≤n} $ is more difficult to characterize.

linear algebra – Pseudo metric on the orthogonal group

Let $ O (n) $ to be all of all $ n times n $ orthogonal matrices. Define an equivalence relation $ sim $ sure $ O (n) $ as following: $ U sim V $ if there is a permutation matrix $ Pi $ and a diagonal matrix $ Lambda $ where all the diagonal entries are 1 or -1, such as $ U = V Pi Lambda $. Let $ tilde {O} (n) $ to be the fixed quotient $ O (n) / sim $.

The question is: is there a psedo metric? $ rho $ sure $ O (n) $ so that there is a well-defined metric $ tilde { rho} $ sure $ tilde {O} (n) $ such as $ tilde { rho} ([U],[V]) = rho (U, V) $, or $[U],[V] in tilde {O} (n) $ are the equivalent classes with representative elements $ U, V in O (n) $ respectively.

Are Linear Scanning Automata Complete?

Linear terminal automata are only Turing machines with finished ribbon, instead of infinite ribbon.

But that prevents them from Turing Complete? Why?

Linear Transformations – Find all the vectors for …

"Given the linear transformation F: $ R ^ 2 $ at $ R ^ 2 $, $ F ((x; y)) = (2x; x) $ find all vectors $ v $ of $ R ^ 2 $ so that $ F (v) = 2v $"

Can you help me? If I think that the vector $ v $ as $ (v1; v2) $, the linear transformation tells me that the resulting vector would $ (2v1; v1) $but after what should I do?

The greatest ideal in delineated linear maps on Schatten- $ p $ class

Let $ 1 leq p < infty. $ Denote $ S_p ( ell_2) $ be the set of all compact operators $ x $ sure $ ell_2 $ such as $ Tr (| x | ^ p) < infty. $ To define $ | x | _ {S_p ( ell_2)}: = Tr (| x | ^ p) ^ { frac {1} {p}}. $ This is done $ S_p ( ell_2) $ a Banach space.
What is the largest closed bilateral ideal of Banach algebra with all of the linear maps delineated on $ S_p ( ell_2) $?

linear algebra – Multiple vectors to block the diagonal matrix

I was reading somewhere and I found this.

Given a set of 1 X M vectors of rows of dimensions aI, where 1 <= i <= N. Let B be a diagonal matrix per block given by,

B = diag {a1, a2….., aNOT }

which is of dimension N X NM.

Can any one tell me how this matrix B is built?