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Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne ... 1answer 80 views The density of diagonalizable matrices of M_n(\mathbb{C}) problem. For any matrix A = (a_{ij})_{1\leq i,j\leq n} \in M_n(\mathbb{C}), we pose ||A|| = \max_{1\leq i,j\leq n} |a_{ij}|. 1. Show that ||.|| define a norm on M_n(\mathbb{C}) and that \forall A, ... 1answer 267 views Norm of Matrix transpose I have a problem below: Let \|\cdot\| denotes the norm matrix $$\|A\|=\max \frac {\|Ax\|}{\|x\|},$$ for every A. Now suppose that H: \mathbb{R}^k \rightarrow ... 1answer 92 views Determinant of Schur Complement If I have an n \times n real-valued non-symmetric matrix \mathbf{M}, which has determinant |\mathbf{M}| > 0, what can I say about the determinant of the matrix \mathbf{Q}^T \mathbf{M}^{-1} ... 1answer 39 views W(A)=\{x^HAx : x^Hx=1,{x\in \mathbb{C}}\}, {A\in \mathbb{R}}^{n\cdot n} How do I show that set is symmetrical set regard to real axis? I need help to solve this task, so I would accept any suggestion: If {A\in \mathbb{R}}^{n\cdot n}, show that set W(A)=\{x^HAx : x^Hx=1,\,{x\in \mathbb{C^n}}\}, is a symmetrical set with respect ... 2answers 131 views Picard iterations of a matrix I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2} defined ... 1answer 54 views How to show A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\} is open in \mathcal{L}(\mathbb R^m, \mathbb R^n) How can we show A=\{T\in \mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{T is onto}\} is open in \mathcal{L}(\mathbb R^m, \mathbb R^n)? Here \mathcal{L}(\mathbb R^m, \mathbb R^n) is the set of ... 1answer 134 views Convergence in norm independent of the choice of the norm. When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing \left\Vert X_n - Y_n\right\Vert\rightarrow 0 as n\rightarrow 0 ... 1answer 293 views The openness of the set of positive definite square matrices Let \mathbb{R}^{n\times n} be the vector space of square matrices with real entries. For each A\in \mathbb{R}^{n\times n} we consider the norms given by:$$ \displaystyle\|A\|_1=\max_{1\leq j\leq ...
Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is \left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...