All questions with tag [math: linear-algebra]


How can I normalize a percent to a value while still deriving results from the percent?

My mathematics abilities are corroded (at ideal) and also I was asking yourself if I can select individuals is below minds on attempting to identify just how to approach what I'm doing. My trouble is a little bit domain name details so I located a proxy trouble that matches flawlessly and also with any luck is less complex to clarify (simply exc...
2022-07-25 20:47:17

Solving linear inequalities over rings

The concrete trouble: for any kind of offered $N\ge 1$ I have a system of $2^N-1$ straight inequalities over $\mathbb{Z}_6^N$ which resembles this: for every single nonempty $S\subseteq[N]$ there is some $b_S\in\mathbb{Z}_6$ and also the inequality $\sum_{i\in S}x_i\ne b_S$. I intend to locate a remedy to all the inequalities simultaneously, cer...
2022-07-25 20:46:55

Need little hint to prove a theorem .

I have an iterative method \begin{eqnarray} X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots \end{eqnarray} with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar lying between 0 to 1)converging to the moore-penrose inverse $X =A^+$. $\{X_{k}\}$ are sequence of approximations. Assume that after $sth$ iteration...
2022-07-25 20:46:47

power series expansion of the square root of a Hermitian matrix

Is there a power series development of the square origin of a Hermitian matrix, as a procedure to compute the square origin without taking the inverted or diagonalizing the matrix? I locate for scalar number $x$, $$\sqrt{x}=\sum_{k=0}^\infty \frac{(-1)^k \left((-1+x)^k \left(-\frac12\right)_k\right)}{k!}\qquad\text{for }|-1+x|<1$$, under ...
2022-07-25 20:46:47

Completeness of normed spaces

As earlier, I have actually obtained a solution from this website that Bolzano Weierstrass' theory holds true for limited dimensional normed rooms, yet except boundless dimensional rooms. This, specifically = > all limited dim. normed rooms are full (in the feeling that every Cauchy series merges (w.r.t. standard) ). Nonetheless, is it real t...
2022-07-25 20:46:40

Finding the dot product.

Finding the dot item of $(-2w) \cdot w$ where $w=(0,-2,-2)$ $$ \text{dot product} = \frac{v . u}{\| v \| . \| u \|} $$ So $$-2 (0,-2,-2)=(0,4,4) \\ (0+4+4) = 8 \\ (0,-2,-2)=-4 \\ 8 \times -4=-32$$ $$\| v \|= \sqrt{0^2+(-2)^2+(-2)^2}=2 \sqrt 2 \\ \|-2w \|= \sqrt{0^2+4^2+4^2}= 4 \sqrt 2\\ \frac{-32}{2 \sqrt 2 \times 4 \sqrt 2} = -64$$ Is this...
2022-07-25 20:46:29

Why is a matrix $A\in \operatorname{SL}(2,\mathbb{R})$ with $|\operatorname{tr}(A)|<2$ conjugate to a matrix of the following form?

The trace $\operatorname{tr}(A)$ of a matrix $A$ is the sum of its diagonal entries. Apparently if $A\in \operatorname{SL}(2,\mathbb{R})$ and $|\operatorname{tr}(A)|&lt;2$, then $A$ is conjugate in $\operatorname{SL}(2,\mathbb{R})$ to a matrix of the form $$\left(\begin{array}{cc} \cos\theta &amp; \sin\theta\\ -\sin\theta &amp; \cos...
2022-07-25 20:46:00

Systems of linear equations

I am I exactly on the following: A 2X2 system of formula without remedy will resemble this: x+3y=3 2x+6y =-8 1/2 + 3/6 ≠ 3/8 A 2X2 system of formulas with in limited remedies will resemble this: x +2y=4 2x+4y=8 1/2 + 2/4 =4/8 A 2X2 system of formulas with one-of-a-kind remedy will resemble this x -y= 1 3x +2y=12 This is due to the fact that...
2022-07-25 20:46:00

Find the equation of a line which is perpendicular to a given vector and passing through a known point

There is offered a vector $2 \vec i + \vec j - 3 \vec k$ and also currently I intend to find the formula of a line that is vertical to the offered vector and also travelling through a well-known factor $(1,1,1)$. Just how can I address this?
2022-07-25 20:45:52

order of elements in a partition using Maple

I established this entire dividing yet I simply intend to have the finer the dividing for example: I have this M[{{1, 2, 3, 4, 5}}]+M[{{1}, {2, 3, 4, 5}}]+M[{{2}, {1, 3, 4, 5}}]+M[{{5}, {1, 2, 3, 4}}]+M[{{3}, {1, 2, 4, 5}}]+M[{{4}, {1, 2, 3, 5}}]+M[{{1, 2}, {3, 4, 5}}]+M[{{1, 3}, {2, 4, 5}}]+M[{{1, 4}, {2, 3, 5}}]+M[{{1, 5}, {2, 3, 4}}]+M[{{2,...
2022-07-25 20:45:27

Eigenvalues of a matrix $A$ such that $ A^2=0$.

Suppose the matrix $A$ is a $2 \times 2$ non-zero matrix with entries in $\Bbb C$. Which of the following statements must be true? $PAP^{-1}$ is a diagonal matrix for some invertible matrix $P$ with entries in $\Bbb R$. $A$ has only one distinct eigenvalue in $\Bbb C$ with multiplicity $2$. $A$ has two distinct eigenvalues in $\Bbb C$. $Av = v...
2022-07-25 20:45:20

About the intertwiners of a real representation and its complex conjugate

i am currently trying to understand a proof in Trautman's "The Spinorial Chessboard", namely theorem 4.2 on page 48. It states the following: If $\rho:\mathcal{A}\to\operatorname{End}_\mathbb{C} S$ is a complex, faithful and irreducible representation of a finite-dimensional, central simple algebra over $\mathbb{R}$ over a finite-dimen...
2022-07-25 20:43:48

Characteristic polynomial of the unique automorphism of the zero module

Is there any convention which makes sense of the characteristic polynomial of the unique automorphism of the zero module? This might seem like an odd question but it matters to me. The background (for those who understand): I am studying the K-theory of the category of pairs $(P,f)$ where $P$ is some projective $R$-module and $f$ is an automorph...
2022-07-25 20:43:33

Is it possible to determine if this matrix is ill-conditioned?

I intend to much better recognize ill - conditioning for matrices. Claim we are offered any kind of matrix $A$, where some components are $10^6$ in size and also some are $10^{-7}$ in size. Does this warranty that this matrix has a problem number more than 100? More than 1000? Despite the fact that we have not defined which components of $A$ hav...
2022-07-25 20:43:18

Characterization of the interior of a convex set

My inquiry is the following: allow $K$ be a convex embeded in $\mathbb{R}^n$ and also $x$ a component of the inside of $K$. Can I attest that there exist $z_1,...,z_n \in K$ linearly independent and also $\lambda_1,....,\lambda_n &gt; 0$ such that:. \begin{equation} x = \displaystyle \sum_{p=1}^n \lambda_p z_p \quad ??? \end{equation} If thi...
2022-07-25 20:42:46

Sign restriction on the Lagrange multiplier? Why?

Say we are offered a straight program where the objective is to decrease $c^Tx$ with the restraints $Ax\ge b$. Why exists an indicator constraint on the Lagrange multiplier related to the energetic restraints at the remedy?
2022-07-25 20:42:27

Derivative of a matrix

I need to know if this by-product is proper. I have actually acquired this yet not exactly sure if this is proper. I assume it is yet simply to validate F= A-(B/C)*D where A,B,C and D are square matrices dF/dx(partial derivative) = d(A-(B/C)*D)/dx ----- deriving the final result will be ----------- dA/dx - [(dB/dx - B*inv(C)*dC/dx)/C]*D - (B/...
2022-07-25 20:42:09

Complex eigenvalues of real matrices

Given a matrix $$A = \begin{pmatrix} 40 &amp; -29 &amp; -11\\ \ -18 &amp; 30\ &amp; -12 \\\ \ 26 &amp;24 &amp; -50 \end{pmatrix}$$ has a particular intricate number $l\neq0$ as an eigenvalue. Which of the adhering to have to additionally be an eigenvalue of $A$: $$l+20, l-20, 20-l, -20-l?$$ It appears that facili...
2022-07-25 20:41:44

Rotation matrix from an inertia tensor

I have a set of weighted points in 3D space (in fact, a molecule) and I'm trying to align the principal axes of this set with the $x$, and $y$ and $z$ axes. To do so, I've first translated my points so their barycenter coincides with the origin. Then, I've calculated the $I$ and its eigenvalues ($\lambda_i$) and eigenvectors ($V_i$). Then, I ne...
2022-07-25 20:41:33

Finding value of a constant in Differential Equations

I have the adhering to ODE Where offered is $x(0)=1$: $$(t+3)dx=4x^2dt$$ After splitting up of variables I obtained this: $$\frac{-1}{x} = 4\ln(t+3)+C$$ I assume this streamlines extra as: $$x=\frac{-1}{\ln((t+3)^4)+C}$$ Please inform me if this is proper, I additionally have trouble searching for C in this instance, MapleTA does decline m...
2022-07-25 20:40:38