# All questions with tag [math: matrices]

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## 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 $x$, $$\sqrt{x}=\sum_{k=0}^\infty \frac{(-1)^k \left((-1+x)^k \left(-\frac12\right)_k\right)}{k!}\qquad\text{for }|-1+x|&lt;1$$, under ...
2022-07-25 20:46:47
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## 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
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## 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
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## 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
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## 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
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Consider the indispensable matrix $$R = \left(\begin{matrix} 2 &amp; 4 &amp; 6 &amp; -8 \\ 1 &amp; 3 &amp; 2 &amp; -1 \\ 1 &amp; 1 &amp; 4 &amp; -1 \\ 1 &amp; 1 &amp; 2 &amp; 5 \end{matrix}\right).$$ 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 &amp; -1 &amp; 0 \\ -1 &amp; 0 &amp; -1 \\ 0 &amp; 1 &amp; 1 \end{bmatrix} \quad\text{and}\quad B=\begin{bmatrix} \frac{2}{\sqrt{3}} &amp; 0 &amp; 0 \\ \frac{4}{\sqrt{2}} &amp; \frac{-1}{\sqrt{2}} &amp; \frac{1}{\sqrt{2}...
2022-07-25 20:22:47
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## Interesting Determinant

Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1&lt;x_2&lt;\cdots&lt;x_n$. Define \begin{equation*} A=% \begin{bmatrix} 0 &amp; x_{2}-x_{1} &amp; \cdots &amp; x_{n-1}-x_{1} &amp; x_{n}-x_{1} \\ x_{2}-x_{1} &amp; 0 &amp; \cdots &amp; x_{n-1}-x_{2} &amp; x_{n}-x_{2} \\ \vdots &amp; \...
2022-07-25 20:21:54
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Consider having vector $$v = \begin{pmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{pmatrix}$$ Consider the last result: $$V = \begin{pmatrix} v_1 &amp; 0 &amp; \dots &amp; 0\\ 0 &amp; v_2 &amp; \dots &amp; 0\\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ 0 &amp; 0 &amp; \dots &amp; v_n\\ \end{pmatr... 2022-07-25 20:20:44 1 ## endomorphism as sum of two endomorphisms (nilpotent and diagonalizable) V is a field over \mathbb{C}. Show that \phi: V \to V can be written as \phi = \psi + \sigma where \psi is diagonalizable and \sigma is nilpotent. I managed to show this first part (you can transform so that \phi is on jordan form, and then split this matrix in a diagonal and a nilpotent ...). But the next part doesn't work for me:... 2022-07-25 20:19:30 0 ## About P_{{L},{M}}, projection transformation onto subspace L along subspace M . I need help to study following theorem: For every idempotent matrix E\in\mathbb{C}^{n\times n}, R(E) and N(E) are complementary subspaces with E = P_{{R(E)},{N(E)}}. Conversely, if L and M are complementary subspaces, there is a unique idempotent P_{{L},{M}} such that R(P_{{L},{M}}) = L, N(P_{{L},{M}}) = M where R(E) and N... 2022-07-25 20:19:01 2 ## Determinant of symmetrical factorized matrix Given A, B \in \mathbb{R}^{n\times n}, t \in \mathbb{R}\setminus \{0\} with b_{ij} = t^{i-j}\cdot a_{ij}. Prove \det(A) = \det(B). I first thought of induction. I can easily prove this for n \le 2. My induction hypothesis: \det(A) = \det(B) with A, B \in \mathbb{R}^{n\times n} Induction step: \det(B) = \sum_{i=1}^{n+1} b_{ij} \cdot ... 2022-07-25 20:18:09 1 ## Given a matrix, find a linear transformation that uses it The matrix is:$$\begin{pmatrix} 3+l &amp; 8 &amp; 3 &amp; 3+l \\ 8 &amp; 9 &amp; 3 &amp; 7 \\ 3 &amp; 3 &amp; 7 &amp; 8 \\ 3+l &amp; 7 &amp; 8 &amp; 13 \end{pmatrix}$$I'm offered the above matrix, and also I'm asked to figure if it can be the matrix of a straight transformation, for an... 2022-07-25 20:17:54 3 ## Finding the rank of a matrix Let A be a 5\times 4 matrix with real entries such that the space of all solutions of the linear system AX^t = (1,2,3,4,5)^t is given by\{(1+2s, 2+3s, 3+4s, 4+5s)^t :s\in \mathbb{R}\} where t denotes the transpose of a matrix. Then what would be the rank of A? Here is my attempt Number of linearly independent solution of a non homog... 2022-07-25 20:01:57 2 ## Finding dimension of a vector space Let H_n be the space of all n\times n matrices A = (a_{i,j}) with entries in \mathbb{R} satisfying a_{i,j} = a_{r,s} whenever i+j = r+s (i, j , r , s = 1, 2, \ldots, n). What would be dimension of H_n as a vector space over \mathbb{R}? i have options for the dimension 1 - n^2 2- n^2-n+1 3 - 2n+1 4- 2n-1 I am finding ... 2022-07-25 19:58:19 1 ## Determinant of symmetric Matrix with non negative integer element Let \begin{equation*} M=% \begin{bmatrix} 0 &amp; 1 &amp; \cdots &amp; n-1 &amp; n \\ 1 &amp; 0 &amp; \cdots &amp; n-2 &amp; n-1 \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots &amp; \vdots \\ n-1 &amp; n-2 &amp; \cdots &amp; 0 &amp; 1 \\ n&amp; n-1 &amp; \cdots &am... 2022-07-25 19:57:24 3 ## Determinant of symmetric matrix with the main diagonal elements zero How to confirm that the component of a symmetrical matrix with the major angled components absolutely no and also all various other components favorable is not absolutely no (i.e., that the matrix is invertible)? EDIT: OP shows in a comment that the access over the diagonal are to be distinctive. 2022-07-25 19:56:47 1 ## Largest eigenvalue of a positive semi-definite matrix is less than or equal to sum of eigenvalues of its diagonal blocks This inquiry is really comparable to this one. Allow$$B = \begin{bmatrix} B_{11} &amp; B_{12} \\ B_{12}' &amp; 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