Questions tagged [matrix-norms]

This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

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Prove that $\kappa (A) = \sup\Big\{ \frac{||Ax||}{||Ay||},\ ||x|| = ||y||\Big\}$.

I am trying to prove this for my numerical analysis class. This is from chapter 4.4 of Kincaid and Cheney's book. So far I haven´t got any good idea. I have tried $$ \|A\| \|A^{-1}\| = \sup \|A\frac{u}...
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1 answer
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Upper bounds of $x^\top ABAx$

Given a unit vector $x=[1~~0~~0~\cdots~0]^\top\in\mathbb{R}^n$ and positive definite matrices $A, B\in\mathbb{R}^{n\times n}$. My question is How could we find the upper bound of $x^\top ABA x$ in ...
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What does the $\|^2_F$ represent in the notation $\operatorname{argmin}\left\|\text{some expression}\right\|_F^2$?

What does the $\|^2_F$ represent in the notation below $$\operatorname{argmin}\left\|\text{some expression}\right\|_F^2$$ I understand that $\operatorname{argmin}$ is the min of this expression. ...
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1 answer
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Upper bound on $\|A^{-1}\|_{\infty}$ for $A$ with unit length columns.

Consider a non-singular $A \in \mathbb{R}^{n\times n}$ with $\|\mathrm{col}_i(A)\| = 1$ ($i \in \{1,\ldots,n\}$), where $\mathrm{col}_i(A)$ is the $i$th column of $A$ and $\|\cdot\|$ is the Euclidean ...
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Does $\|A^k\| \leq a k^{n-1} \rho^k(A)$ hold?

Given a non-singular matrix $A \in \mathbb{R}^{n\times n}$, let $\rho(A)$ be its spectral radius. Then, $\forall k \in \mathbb{N_0}$, can we find a constant $a > 0$ such that the following ...
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2 votes
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Prove the convergence of the following sequences. [closed]

Let $(x_k)_{k∈\mathbb{N}}$ ⊂ $\mathbb{R^d}$ and $y$ ∈ $\mathbb{R^d}$. Show that $$(1)\ \ \ x_k → y ⇐⇒x_k-y→ (0,...,0) $$ And $$(2)\ \ \ x_k → y ⇐⇒||x_k-y||_2→ 0$$ $$$$ For (1) I started like this: Let ...
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1 vote
1 answer
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Show that the following function is continuous on R^2

Consider the function f : $\displaystyle\mathbb{R^2}→ \mathbb{R}$ with $$f(x_1,x_2)=\begin{cases} \frac{sin(|x_2|)cot(|x_1|)+cos(|x_2|)}{sin(|x_2|)-cot(|x_1|)cos(|x_2|)}(|x_1|+|x_2|)^{-1}, & \...
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1 answer
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$\lVert P \rVert_2 =1$ iff $P$ is an orthogonal projector - Proof

I need help with understanding a step in a proof for the following exercise: Let $P\in\mathbb{C}^{m\times n}$ be a non-zero projector. Show that $\lVert P \rVert_2 =1$ iff $P$ is an orthogonal ...
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The positive semidefinite(psd) of the 2 norm for vector

Suppose $q_n=(x_n,y_n)$ is a vector, $\Vert \cdot \Vert_2$ is the standard 2-norm like $\Vert q_n \Vert_2=\sqrt{x_n^2+y_n^2}$. Is $\sum_{n=0}^{N}\Vert q_{n+1}-q_{n} \Vert_2$ positive semidefinite(psd)...
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1 vote
1 answer
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Show that the set of extreme points of $S$ is its boundary

Let $S:=\{x:x^tx\leq 1\}$. Show that the set of extreme points of $S$ is its boundary. An extreme point, in mathematics, is a point in a convex set which does not lie in any open line segment joining ...
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1 vote
1 answer
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How to denote $L_p$ norm of a matrix $X$ in either row or column direction?

How to denote $L_p$ norm of a matrix $X$ in either row or column direction? The result of such an operation will be a column or row vector. Representing $\| X \|_p$ is ambiguous, isn't it? Do you have ...
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subgradient - subdifferential of spectral norm for a complex matrix

I found the following definition in this answer $$ \partial f(x) := \{x^* \in X^* \mid f(x') \ge f(x) + \langle x^*, x'-x\rangle\;\forall x' \in X\} $$ Can I define this $$ \langle A, B\rangle\ = Re(...
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2 answers
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Matrix-vector norm inequalities in tridiagonal matrix

Let's consider the linear system $A\vec{x}=\vec{b}$, with $A$ being a diagonally dominant by columns and tridiagonal matrix, that is: $$A=\begin{pmatrix} d_1&& c_2 && &&...
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Lower bound on the off-diagonal elements of a PSD matrix

Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, ...
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2 votes
1 answer
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Upper Bounds for Operator Norm of Block Diagonal matrix

Consider the real positive definite block matrix $$ X = \begin{bmatrix} A &B \\ B^T &C \end{bmatrix} $$ with dimensions: $A$ is $d \times d$, $C$ is $k \times k$, and $B$ is $d \times k$, so $...
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2 votes
0 answers
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Bounding a $\Vert X\Vert_{op}^2$ term with probability $1$ with $X$ being i.i.d. and $\mathbb P(X_{11} =1)=\mathbb P(X_{11} =-1)=\frac{1}{2}$.

Let $X$ be a $N \times N$ random matrix with $i.i.d.$ random entries, and $$\mathbb P(X_{11} =1)=\mathbb P(X_{11} =-1)=\frac{1}{2}$$ Define $$\Vert X\Vert_{op}=\sup_{\mathbf{v}\in\mathbb C^n:\Vert \...
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2 votes
2 answers
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Calculating the gradient of Log-Euclidean distance between SPD matrices on Riemannian manifold

In the paper Log-Euclidean metrics for fast and simple calculus on diffusion tensors, the geodesic distance between SPD matrices $A,B$ is defined as $$d(A,B)=||\log A- \log B||_F,$$ where $F$ is the ...
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1 vote
1 answer
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invertible element of finite dimension non-commutative algebra and the norm

Let $R$ be a commutative ring, $D$ be a finite dimensional associative $R$-algebra with unit. Then for $\alpha\in D$, left multiplication of $\alpha$, $L_{\alpha}: D\to D$ is $R$-linear map, and we ...
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  • 1,010
1 vote
1 answer
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Name for sum of diagonals for Hilbert Schmidt normalized matrix.

Suppose that $A$ is an $n \times n$ matrix. Then the Hilbert-Schmidt norm of $A$ is given by $$ \vert \vert A \vert \vert_{HS}^2= \sum_{i,j=1}^n \vert{A_{i,j} \vert}^2. $$ Now, if I insist that $ \...
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Prove that $\frac{1}{2}N_1(A_1,A_2)\leq N_2(A_1,A_2) \leq N_1(A_1,A_2)$ where $N_1$ and $N_2$ are two norms

Let $A_1$ and $A_2$ two bounded linear operators on a complex Hilbert space $E$. I want to prove that $$\frac{1}{2}N_1(A_1,A_2)\leq N_2(A_1,A_2) \leq N_1(A_1,A_2),$$ where $$N_1(A_1,A_2)=\...
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3 votes
1 answer
58 views

Subordinated matrix norm of diagonalizable matrix is its spectral radius

I want to solve the following problem Let $A$ be a matrix that admits a basis of eigenvectors (i.e. a diagonaliazable matrix). Find a norm $|||\cdot |||$ subordinated to a vector norm $|||\cdot |||$ ...
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  • 565
1 vote
1 answer
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Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$

Calculate $\frac{\partial}{\partial \mathbf{A}}\lVert \mathbf{A}^{\top}\mathbf{AX}-\mathbf{X} \rVert _{F}^{2}$ with $\mathbf{A}\in\mathbb{R}^{M\times N}$ and $\mathbf{X}\in\mathbb{R}^{N\times D}$, and ...
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  • 354
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1 answer
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Minimizing 2-norm of a matrix

Suppose I want to minimize the following matrix norm: $$\min_{\alpha\in\mathbb{R}, \beta\in\mathbb{R}^{n\times 1}} ||A-\alpha B-c*\beta'||_2, $$ where $A, B \in \mathbb{R}^{m\times n}$ and $c \in \...
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  • 259
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0 answers
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Matrix norm ordering upon multiplication by positive definite matrix

Assume I know the following to be true \begin{equation} ||A||>||B|| \end{equation} where $A$ and $B$ are $n\times n$ matrices and the norm can be any norm. If I define a positive definite matrix $...
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  • 282
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1 answer
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Two related matrices with same Hilbert Schmidt norm

We have $BGx$ where $B$ is a fixed m×N matrix, and $G$ is an N×n (random) matrix, $x$ is a fixed vector in $R^n$. By concatenating the rows of $G$ , we can view $G$ as a long vector in $R^{Nn}$. ...
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3 answers
91 views

Are there any solutions to $Ax=b$ satisfying $\Vert x \rVert=1$? [closed]

Let $$A = \begin{pmatrix} 1 & 1 & -3 \\ -3 & -2 & -2 \\ -7 & -5 & 1 \end{pmatrix}$$ be a $3$ by $3$ matrix and $$b = \begin{pmatrix} 4 \\ 6 \\ 8 \end{pmatrix}$$ be a column ...
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4 votes
1 answer
46 views

When does the following frobenius norm equality hold?

When we consider the square matrix ${\bf A}\in\mathbb{C}^{N\times N}$, then following ineqaulity always holds: $$ \left\|{\bf A^{\sf H}}{\bf A}\right\|_F = \left\|{\bf A}{\bf A^{\sf H}}\right\|_F\le\...
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1 vote
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Conditions for Weighted Norm

I'm constructing a weighted norm of the form $||N||_A = \sqrt{tr(N^T A N)}=\sqrt{N_{ij}A_{jk}N_{kl}\delta_{il}}$, where $\{A,N\} \in \mathbb{R}^{3x3}$. In order to keep the argument of the square root ...
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Bounding the operator norm of the inverse matrix of a continuous derivative

Given a continuously differentiable mapping $f: \overline{\Delta} \subset ℝ^n → ℝ^n$, $\overline{\Delta}$ being a closed cell, and that $\det f′(x) ≠ 0$ for all $x ∈ \overline{∆}$, show that the ...
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0 answers
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Upper bound on normwise relative error

Suppose I have two vectors $y_1, y_2 \in \mathbb{R}^{n\times 1}$ which are both transformations of the same vector $x$ by two different matrices $T_1, T_2 \in \mathbb{R}^{m\times n}$, i.e.: \begin{...
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1 vote
1 answer
40 views

Checking a possible corollary of Gelfand's formula

I have recently come across Gelfand's spectral formula for matrices, which Wikipedia gives as follows For any square matrix $A$ and matrix norm, we have $\lim_{k \to \infty} \lVert A^k \rVert ^{\frac{...
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0 answers
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Inequalities that relates the norms of $AB^{-1}$ and $A-B$?

Let $A,B$ be two $d\times d$ invertible real matrices. Are there any famous matrix inequalities relating the norms (any norm since all norms over finite dimeonsion spaces are equivalent) of $AB^{-1}$ ...
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1 vote
1 answer
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Sampling from equal-norm tight frame (Vershynin exercise 5.6.6)

I am struggling with Exercise 5.6.6 from Vershynin's "High-Dimensional Probability": Consider an equal-norm tight frame $(u_i)_{i=1}^{N}$ in $\mathbb{R}^n$. State and prove a result that ...
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1 vote
0 answers
33 views

Bounding norms of expressions involving hadamard products

Given square matrices $A, B, C$ and a vector $x$, I want to find an upper bound for expressions of the form $$ \| (A \circ B) C x\|_{2}. $$ Ideally, I want a bound that 'incorporates' the effect of $A$...
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0 votes
0 answers
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How to establish the identity $\Vert A B \Vert_F \leq \Vert A \Vert_F \Vert B \Vert_F$ for the Frobenius matrix norm? [duplicate]

The Frobenius matrix norm is defined for $m \times n$ real matrices as $$ \Vert A \Vert_F = + \sqrt{\mbox{Trace}(A^T A)} \tag{1} $$ For matrices $A$ and $B$ of sizes $(m \times n)$ and $(n \times k)$ ...
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  • 2,482
2 votes
0 answers
76 views

p-norm preserving matrix

I am reading Scott Aaronson, Is Quantum Mechanics An Island In Theoryspace? In Section 2 Other p-Norms, he tries to prove for $p>2$, the p-norm preserving matrix is generalized diagonal. Namely, ...
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  • 8,990
1 vote
0 answers
35 views

Inequality on the uniform norm of a conformal differentiable transformation

I'm currently working on a question on chaotic dynamics on fractals. Particularly if we have an iterated function system of conformal differentiable contraction mappings {$f_1$,...,$f_n$} we denote ...
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1 answer
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Bounding $\|A\|_2$ with $\max_i \|A_i \|_2$?

Given a Matrix $A=(A_1, \dots, A_n)$ with columns $A_i \in \mathbb{R}^n$. I am trying to understand the solution of my homework assignment where in one line we state the inequality $\|A(x-y)\|_2\leq \...
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  • 588
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0 answers
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Rewriting the one norm into a set of linear inequalities.

I am trying to solve this exercise in linear programming: Express $\|Ax - b\|_1$ As a linear program of the form: $$\text{minimize } v^T y$$ Subject to $$Bx \leq b_1$$ $$Cx = b_2$$ $$l \leq y \leq u$$...
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  • 2,301
0 votes
1 answer
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Proving that $||v^T||_2=||v||_2$ for $v\in \mathbb{R}^n$.

I'm trying to prove that $||v^T||_2=||v||_2$ for $v\in \mathbb{R}^n$. The first norm is the matrix norm, and the second norm is the usual euclidian norm. I already prove that: $\bullet ||v||_2\leq ||v|...
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  • 2,496
1 vote
0 answers
44 views

Converting a non-convex optimization problem into a convex one

I have this optimization problem to solve $$\begin{array}{ll} \underset{{\bf m},x}{\text{minimize}} & \| {\bf m} \|^2 \\ \text{subject to} & {\bf h}_k^\ast {\bf m} = x_k, \quad \forall k \\ &...
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1 vote
0 answers
34 views

How perturbing matrices $A,B$ perturbs words made of these matrices.

Let $A,B$ be $2\times 2$ matrices, with say operator norm (whichever one) $\|A\|<1.3,\|B\|<1.3$. If We assume we have perturbations $A',B'$ s.t $\|A-A'\|\leq \epsilon$ and $\|B-B' \|<\epsilon$...
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  • 1,610
0 votes
1 answer
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Are all the entries of the matirx exponential function non-negative? How to compute the infinity-norm of a matrix function?

The exponential functions are essential ingredients of Exponential Time Difference (ETD) integrators. Let us introduce following functions \begin{equation*} \varphi_k(z) = \int_0^1 \mathrm{e}^{(1 - s) ...
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  • 73
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1 answer
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Cond(PA)=Cond(A)

I want to proof the following equation. A is a nxn regular matrix and P a permutation Matrix $$cond_2(PA)=cond_2(A)$$ So the spectral norm of $||P||=1$ so $cond(PA)=||PA|| \cdot ||(PA)^{-1}||$ I tried ...
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1 vote
1 answer
39 views

Tridiagonal matrix with main diagonal equal to 1, inverse 1-norm upper bound

Description Suppose we have the following non singular tridiagonal matrix $$ B = \begin{bmatrix} 1 & a_1 \\ b_2 & 1 & a_2 \\ & \ddots & \ddots & \ddots \\ && b_{n-...
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  • 406
1 vote
1 answer
29 views

How to prove that the matrix norm $||A||_{1,2} = \max_{j} \, \left( \sum_{i=1}^p a_{i,j}^2 \right)^{\frac{1}{2}}$?

$A \in \mathbb{C}^{p\times q}$, $||A||_{1,2} = \max_{x \in \mathbb{C}^q } \big\{ ||Ax||_2 \; \big| \; ||x||_1 = 1 \big\}$. I tried using Holder's Inequality and extremum of quadratic function but did ...
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  • 133
0 votes
1 answer
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Proving a norm inequality for general matrices

I am currently doing exercises from Trefthen and Bau's numerical linear algebra. This problem has two parts. For the first part, I have managed to prove that if a set $S$ of complex numbers is ...
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  • 3,835
1 vote
1 answer
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Proving a norm limit based on Schur factorization

I am attempting to do exercises from Trefethen's numerical linear algebra. I am stuck on this one Using Schur decomposition, we are asked to prove that for an arbitrary square matrix $A$ and any ...
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1 vote
0 answers
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Decomposition of positive definite matrix.

Let $A$ be a positive definite matrix. Following are the decomposition's of matrix $A$: $A = PP$ $A = MM^{T}$ Empirically we found that $||P||_{F}^2 = ||M||_{F}^2$ but how to prove this ...
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  • 49
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1 answer
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If an invertible matrix $A$ satisfies $c||x|\leq ||Ax|| \leq C||x||$ for all $x$, can one say the same about its inverse?

For every $z$ with norm 1, we have that $$ 1 = ||z|| = ||AA^{-1}z|| \leq ||A||\cdot ||A^{-1}z|| \leq C||A^{-1}z||$$ and thus, this give us that $||A^{-1}z|| \geq \frac{1}{C}$. I thought that using the ...
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