0
votes
2answers
60 views

Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
0
votes
4answers
64 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
0
votes
1answer
17 views

Find subset of rows whose entries sum to an even number in each column

I am trying to implement Fermat factorization with factor bases. The textbook suggests using row-reduction to find a linearly dependent set of rows. How does one go about finding such a linearly ...
1
vote
2answers
131 views

Solving inequalities with fractions with unfactorable polynomials

So I've been cracking my head open trying to solve this inequality: $$\frac {x+1}{2-x} \le \frac {x}{3+x}$$ I've been taught you have to put all factors to one side of the inequality (leaving zero ...
2
votes
2answers
139 views

Solve the equations $z^2 + (2 - 2i)z + 2i = 0 $ by completing the square

I tried solving this thing by completing the square and I always end up with something like this $(z^2 + (2 - 2i)z - 2i) + 2i + 2i = 0 $ and it doesn't seem like to me that you can factor the part in ...
0
votes
1answer
39 views

Inversion in factor rings

I have this polynomials: $f = x^{4} + 3x^{3} + x^2 + 3 \in \mathbb{Z}_{5}[x]$, $g = x + 2 \in \mathbb{Z}_{5}[x]$ Does g + (f) have inversion in ring $(\mathbb{Z}_{5}[x]/(f),+,.)$ ? I should found ...
0
votes
1answer
60 views

Frobenius matrix norm vs. 2-norm

From this article about the singular value decomposition: Let $A$ be an $n \times d$ matrix and think of the rows of $A$ as $n$ points in $d$-dimensional space. The Frobenius norm of $A$ is the ...
0
votes
0answers
38 views

Polynomial factorization over an infinite field - is there an algorithm

In my previous questioned I asked how do I factor a polynomial, and I gave an easy example of a polynomial of degree 2. But now I have another question I need to solve. I need to factor ...
3
votes
1answer
79 views

Cholesky decomposition: any theoretical value?

Just read the Wikipedia article on Cholesky decomposition. All the applications listed there were numerical. Are there theoretical arguments to which it is important? For instance, here there is an ...
5
votes
2answers
277 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
3
votes
2answers
73 views

Factorization of a linear combination of matrices

I'm trying to understand the determinant from Axler Sheldon's paper and there is one point in the very beginning that I don't understand :S (Link below to the paper) ...
1
vote
1answer
92 views

how to interpret theorem about polynomial factorization over modulo ring?

polynomial $X^n+a_1X^{n-1}+...+a_n \in \Bbb Z_2[X]$ doesn't have linear factors $\iff a_n(1+\sum a_i) \neq 0$. How then $f(X)=X+1$ can has no linear factors? Doesn't the condition expands to ...
1
vote
1answer
323 views

Cholesky/LU decomposition from matrix and its inverse?

Usually, we have a matrix $A$ and want to calculate the $LU$ (or sometimes Cholesky, depending on $A$'s properties) decomposition. This is often the hard part. Now, if we have the $LU$ decomposition ...
2
votes
1answer
48 views

(CHECK) Cardinality of Terms in the Expansion of a Product of Multinomials

QUESTION: How many terms are there in the expansion of $$(x+y)(a+b+c)(e+f+g)(h+i)$$ I'd like some help with this one, but I'd also like to discuss a method of generalization on the problem, ...
0
votes
1answer
27 views

Relationship between 2 Dimensional Quadratic systems and roots

Given four points $(x_1, y_1) (x_2, y_2) (x_3, y_3) (x_4, y_4)$ How does one construct a system of two equations: $a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$ $b_1x + b_2x^2 + b_3y + b_4y^2 + ...
1
vote
0answers
111 views

What is the condition for a polynomial to be factorizable in linear real factors?

I have a polynomial $p_a(x,y)= x^2F(a)+y^2G(a)-xH(a)-I(a)$ where $F(a)$, $G(a)$, $H(a)$ and $I(a)$ some real fuctions of $a$ are. Which conditions must satisfy $a$ so that I can factorize the ...
0
votes
1answer
229 views

Zeroes in a 3x3 Matrix Determinant

My professor found the cubic roots of a 3x3 matrix by doing the following. I don't understand how step 2 came about and why he applied the same for step 4 on row 1 instead of row 2. Step 1: ...
0
votes
1answer
115 views

Suitable Loss function for Order preserving Factoring of a matrix?

(Old-Question) Given a $n\times n$ symmetric matrix $X$, I would like to factor it using a vector $c$ of size $n \times 1$ such that: $\sum_{i,j} [X_{ij} \cdot c_i\cdot c_j]$ is minimum. How can I ...
3
votes
1answer
688 views

Factorize a Symmetric matrix as an 'Approximation' with an outer product.

(deprecated-taken back based on discussion(OLD)) What is a good way to factor a symmetric matrix $X$ as an outer product of two vectors $u$ and $v$. i.e, Find two vectors $u$ and $v$ such that ...
1
vote
1answer
1k views

General method for factorizing matrix determinants

I'm learning how to factorize determinants of a square matrix in school, but we haven't learnt a general method to do that, besides 'creating zeros'. So I thought maybe I'll ask here if someone does ...