All questions with tag [math: matrices]

0

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
0

Show that the set of matrices such that $\det A \neq 0$ is open

Possible Duplicate: ¢ Like the title claims just how would certainly you show that the set of matrices such that $\det A \neq 0$ is open ? I can not also see where to start! As I can not imagine just how I would certainly locate a matrix 'round' of size $\epsilon$ for every single component of the set?
2022-07-25 20:46:40
0

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
0

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
1

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
1

Determining the Smith Normal Form

Consider the indispensable matrix $$R = \left(\begin{matrix} 2 & 4 & 6 & -8 \\ 1 & 3 & 2 & -1 \\ 1 & 1 & 4 & -1 \\ 1 & 1 & 2 & 5 \end{matrix}\right).$$ Determine the framework of the abelian team offered by generators and also relationships. $$...
2022-07-25 20:43:11
0

Proper way to create Goppa code check matrix?

I'm trying to figure out what is the correct way to compute a check matrix for a binary Goppa code. So far (searching through the publications) I've found more than one possibility to do that, and I'm not sure how all those are related. Suppose I have a Goppa code with irreducible Goppa polynomial $g(x)$ of degree $t$ over $GF(2^m) \simeq F[x]/F...
2022-07-25 20:40:56
0

I need help to understand meaning of certain terms in a theorem

There are certain terms in the following theorem where I am finding difficulty to figure out. I need help. Theorem. Let $\mathbb{C}_{r}^{m\times n}$ denote the set of all complex $m\times n$ matrices of rank $r$. Let $A\in\mathbb{C}_{r}^{m\times n}$ , let $T$ be a subspace of $\mathbb{C^n}$ of dimension $s\leq r$, and let $S$ be a subspace of ...
2022-07-25 20:39:14
2

Finding Transition Matrix

Problem: Find the change matrix P such that $P^{-1}AP=B$ where: $$A=\begin{bmatrix} 3 & -1 & 0 \\ -1 & 0 & -1 \\ 0 & 1 & 1 \end{bmatrix} \quad\text{and}\quad B=\begin{bmatrix} \frac{2}{\sqrt{3}} & 0 & 0 \\ \frac{4}{\sqrt{2}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}...
2022-07-25 20:22:47
2

Interesting Determinant

Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$. Define \begin{equation*} A=% \begin{bmatrix} 0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & x_{n}-x_{1} \\ x_{2}-x_{1} & 0 & \cdots & x_{n-1}-x_{2} & x_{n}-x_{2} \\ \vdots & \...
2022-07-25 20:21:54
0

A matricial process to assign different values to elements of a diagonal matrix

Consider having vector $$v = \begin{pmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{pmatrix}$$ Consider the last result: $$ V = \begin{pmatrix} v_1 & 0 & \dots & 0\\ 0 & v_2 & \dots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \dots & v_n\\ \end{pmatr...
2022-07-25 20:20:44
1

endomorphism as sum of two endomorphisms (nilpotent and diagonalizable)

$V$ is a field over $\mathbb{C}$. Show that $\phi: V \to V$ can be written as $\phi = \psi + \sigma$ where $\psi$ is diagonalizable and $\sigma$ is nilpotent. I managed to show this first part (you can transform so that $\phi$ is on jordan form, and then split this matrix in a diagonal and a nilpotent ...). But the next part doesn't work for me:...
2022-07-25 20:19:30
0

About $P_{{L},{M}}$, projection transformation onto subspace $L$ along subspace $M$ .

I need help to study following theorem: For every idempotent matrix $E\in\mathbb{C}^{n\times n}$, $R(E)$ and $N(E)$ are complementary subspaces with $E = P_{{R(E)},{N(E)}}$. Conversely, if $L$ and $M$ are complementary subspaces, there is a unique idempotent $P_{{L},{M}}$ such that $R(P_{{L},{M}}) = L$, $N(P_{{L},{M}}) = M$ where $R(E)$ and $N...
2022-07-25 20:19:01
2

Determinant of symmetrical factorized matrix

Given $A, B \in \mathbb{R}^{n\times n}, t \in \mathbb{R}\setminus \{0\}$ with $b_{ij} = t^{i-j}\cdot a_{ij}$. Prove $\det(A) = \det(B)$. I first thought of induction. I can easily prove this for $n \le 2$. My induction hypothesis: $\det(A) = \det(B)$ with $A, B \in \mathbb{R}^{n\times n}$ Induction step: $\det(B) = \sum_{i=1}^{n+1} b_{ij} \cdot ...
2022-07-25 20:18:09
1

Given a matrix, find a linear transformation that uses it

The matrix is: $$\begin{pmatrix} 3+l & 8 & 3 & 3+l \\ 8 & 9 & 3 & 7 \\ 3 & 3 & 7 & 8 \\ 3+l & 7 & 8 & 13 \end{pmatrix}$$ I'm offered the above matrix, and also I'm asked to figure if it can be the matrix of a straight transformation, for an...
2022-07-25 20:17:54
3

Finding the rank of a matrix

Let $A$ be a $5\times 4$ matrix with real entries such that the space of all solutions of the linear system $AX^t = (1,2,3,4,5)^t$ is given by$\{(1+2s, 2+3s, 3+4s, 4+5s)^t :s\in \mathbb{R}\}$ where $t$ denotes the transpose of a matrix. Then what would be the rank of $A$? Here is my attempt Number of linearly independent solution of a non homog...
2022-07-25 20:01:57
2

Finding dimension of a vector space

Let $H_n$ be the space of all $n\times n$ matrices $A = (a_{i,j})$ with entries in $\mathbb{R}$ satisfying $a_{i,j} = a_{r,s}$ whenever $i+j = r+s$ $(i, j , r , s = 1, 2, \ldots, n)$. What would be dimension of $H_n$ as a vector space over $\mathbb{R}$? i have options for the dimension 1 - $n^2$ 2- $n^2-n+1$ 3 - $2n+1$ 4- $2n-1$ I am finding ...
2022-07-25 19:58:19
1

Determinant of symmetric Matrix with non negative integer element

Let \begin{equation*} M=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & n-2 & \cdots & 0 & 1 \\ n& n-1 & \cdots &am...
2022-07-25 19:57:24
3

Determinant of symmetric matrix with the main diagonal elements zero

How to confirm that the component of a symmetrical matrix with the major angled components absolutely no and also all various other components favorable is not absolutely no (i.e., that the matrix is invertible)? EDIT: OP shows in a comment that the access over the diagonal are to be distinctive.
2022-07-25 19:56:47
1

Largest eigenvalue of a positive semi-definite matrix is less than or equal to sum of eigenvalues of its diagonal blocks

This inquiry is really comparable to this one. Allow $$B = \begin{bmatrix} B_{11} & B_{12} \\ B_{12}' & B_{22} \end{bmatrix}$$ be a positive semidefinite matrix, where block $B_{11}$ is $p \times p$ . After that. $$\lambda_1(B) \le \lambda_1(B_{11}) + \lambda_1(B_{22})$$ where $\lambda_1$ is the biggest eigenvalue of ...
2022-07-25 19:55:57