# Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

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### Relating two seemingly different definitions of Riemannian metric.

I've been studying Riemannian geometry from John M. Lee's book on the same. My aim is to understand the geometry of Hermitian positive semidefinite matrices when viewed as a Riemannian manifold. In ...
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### Eigendecomposition of a block matrix of the form $\begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & 0 \end{pmatrix}$

Consider a matrix $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & 0 \end{pmatrix}$ where $A_{11}$ is symmetric. The eigendecomposition of $A_{11}$ is $A_{11} = U_1S_1U_1^T$ and the one of ...
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### Backward error of solving $Ax=b$ with perturbed matrix $A$

Let $A=\begin{bmatrix} 1 &-0.55 \\ -2 & 1.06 \end{bmatrix}, b=\begin{bmatrix} 1\\ -1 \end{bmatrix}$. How to compute the backward error of the algorithm of solving $Ax=b$, with respect to ...
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### Algorithms to reduce matrix bandwidth

I'm currently working on a variant of a non-convex low-rank matrix completion algorithm, whereby we take a uniform sample of entries in a (symmetric) matrix and look to complete said matrix. For ...
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### What is the second derivative of the spectral norm of a symmetric matrix?

It is well known that the derivative of a matrix $A$'s 2-norm with respect to $A$ is $$\frac{\partial \sigma_{\max}(A)}{\partial A}=u_1v_1^T,$$ where $u_1,v_1$ is the are the first column/row in the ...
42 views

### Finding eigenvalues of the sum of two matrices.

If the eigenvalues of two matrices are known is there any (generalized) way to tell the eigenvalues of the sum of the two matrices? I have come to hear that no such theorem exists that can handle the ...
31 views

### vector but not matrix norm

Is there a norm $\|\cdot\|_\sharp$ on $\mathbb R^{n\times n}$ for which $\| I\|_\sharp=1$ and there is a matrix $M$ such that $\|M^2\|_\sharp>\|M\|_\sharp^2$?
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### Why limit of a matrix's power exits will implies this?

I was reading this paper "Fast Linear Iterations for Distributed Averaging" and on page 3 it reads that: $\lim_{t \to \infty} W^t$ exists (i.e., $W$ is semi-convergent) if and only if there ...
1 vote
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### Derivatives of Matrix Functions of Different Dimensions

Notation: For matrices $A,B\in\mathbb R^{n\times m}$, we define the inner product $\langle A,B\rangle=\sum_{i,j}A_{ij}B_{ij}$ The basis vector $e_i$ is equal to $1$ at position $i$ and $0$ otherwise....
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1 vote
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### Upper bound of biggest singular value of Kronecker Sum via singular values and traces

I am investigating the biggest singular value of Kronecker Sum. I'd like to understand if my argument is correct. Let $A$, $B$ be square matrices of size $n$. Let $I$ be an identity matrix of size $n$....
1 vote
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### Chain rule for vector by vector derivative

I think it's clear that: \begin{equation} \frac{d(\mathbf{A} \mathbf{x})}{d \mathbf{x}}=\mathbf{A}, \quad \text{ where $\mathbf{A}$ is a matrix and $\mathbf{x}$ is a vector}. \end{equation} but if we ...
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### When does $A_1 A_2 A_3\ldots \xrightarrow{a.s}0$ for IID random matrices $A_i$?

Suppose $A_1,A_2,A_3,\ldots$ is an infinite sequence of $d\times d$ matrices sampled IID from some distribution. Under which conditions does the product converge to zero almost surely? The hard case ...
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### Is $l_2$ on $\mathbb{R}^n$ the only norm for which it is equal to its dual norm?

Given any norm $\|.\|$ on $\mathbb{R}^n$, its dual norm $\|.\|^D$ is defined as the following: $\|v\|^D = \sup_{\|x\|\leq 1} |(v,x)|$, where $(,)$ is the standard Euclidean Inner product. Under that ...
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### The necessary and sufficient constraint of $A$ such that $\text{tr}(A^n)\geq0$?
I'm looking for a sufficient and necessary constraint for the complex square matrix $A$ such that $\text{tr}(A^n)\geq0$ for $n=1,2,3...$ Up to now, it seems that the eigenvalues of $A$ must come in ...
Let $A=(a_{kl})$ be the matrix in $M_n(\mathbb{C})$ given by $a_{kk}=0$ and $a_{kl}=\frac12$ if $k\neq l$. Let $D$ be the diagonal matrix in $M_n(\mathbb{C})$ with $D=\operatorname{diag}(n,\frac{2n+... 0 votes 2 answers 733 views ### Projection onto the Stiefel manifold and the orthogonal Procrustes problem Let$m$and$n$be positive integers such that$m \ge n$, the case with$m >n$being particularly interesting to us. The Stiefel manifold is \begin{equation} \mathbb{S}^{m, n} = \{X \in \mathbb{R}^{... 3 votes 0 answers 54 views ### Bhatia, Ex 7.1.12: Principal angles between subspaces I am unable to solve this problem in Matrix Analysis by Rajendra Bhatia: I can't see how to prove that$\theta_i$are decreasing. I also have no intuition for why the angles are defined this way. ... 0 votes 1 answer 54 views ### Property regarding two positive definite matrices Suppose$A$and$B$are two$n \times n$symmetric positive definite real matrices and let$M$be an$n \times m$real matrix of full rank. Suppose that$A - B\$ is positive definite. Then show that ... 