Questions tagged [matrix-analysis]
For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.
258
questions
0
votes
1
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46
views
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 ...
- 25
0
votes
0
answers
20
views
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 ...
0
votes
0
answers
29
views
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 ...
- 369
0
votes
0
answers
34
views
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 ...
- 53
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votes
0
answers
14
views
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 ...
- 392
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votes
0
answers
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 ...
- 51
0
votes
1
answer
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 $\|M^2\|_\sharp>\|M\|_\sharp^2$?
- 1,282
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0
answers
32
views
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 ...
- 392
1
vote
1
answer
66
views
The spectral radius analysis
This is a question from numerical linear algebra. It originates from iteration method: Suppose $Ax = b$, we split $A = A_1+A_2$, then $A_1x = -A_2x+b$, if $A_1$ is invertible, then $x = -A_{1}^{-1}A_2 ...
- 271
3
votes
1
answer
82
views
Dimension of the manifold of symmetric rank $r$ $n\times n$ matrices
I'm currently reading through the paper "Low-rank matrix completion by Riemannian
optimization—extended version" by Vandereycken, and in this paper the author states that the set
$\mathcal{M}...
- 53
2
votes
1
answer
44
views
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....
- 66
0
votes
0
answers
47
views
Unitary matrix of polar decomposition
I know unitary matrix of polar decomposition ($A=UP$; $U$ is unitary and $P$ is positive semi-definite) cannot be unique (but $P$ is!) if $A$ is not invertible.
Can you give some examples that $A=UP=...
Semin Kwak
1
vote
0
answers
67
views
Trace of square of a rank-$1$ Hermitian matrix ${\bf A} = {\bf a}{\bf a}^H$
Given matrix ${\bf A} = {\bf a}{\bf a}^H$, where ${\bf a}$ is a complex column vector. Are the following correct?
${\rm Tr}({\bf A}) = {\rm Tr} \left( {\bf a}{\bf a}^H \right) = {\rm Tr}({\bf a}^H{\...
- 162
1
vote
0
answers
58
views
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
1
answer
125
views
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 ...
- 133
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votes
0
answers
20
views
What will the distribution of $u^HHH^Hu$ when elements of H follows the complex Gaussian independently and u is the eigenvector such that $\|u\|^2=1$?
I tried to solve this problem by directly expanding it. But I need some opinions on my solution. Here is the problem definition:
$\lambda=\frac{1}{|| \mathbf{H}^H\mathbf{u}||^2}\mathbf{u}^H\mathbf{H}\...
- 1
1
vote
1
answer
73
views
Maximal eigenvalue of Hermitian matrix lies on the diagonal of $A$
Let $A$ be Hermitian. Assume that the eigenvalues of $A$ are increasing ordered and that $a_{ii} = \lambda_n$. We have that $$\sum_{j=1}^{m}a_{ii} \geq \sum_{j=1}^{m}\lambda_j$$ and $$\sum_{j=1}^{m}a_{...
- 103
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votes
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96
views
How to find the Fréchet derivative of a matrix exponential?
Let $f : \Bbb R^{n\times n} \to \Bbb R^{n \times n}$ be the matrix exponential $$ f(A) = \sum_{k = 0}^\infty \frac{A^k}{k!} $$ Let $B$ be a matrix that commutes with $A$. Show that the value of ...
- 1
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votes
0
answers
34
views
Roots of monic polynomials with real coefficients as continuous functions of coefficients
Consider a monic polynomial
$$p(\lambda,a)=\lambda^n+a_1\lambda^{n-1}+\dots+a_{n-1}\lambda^{n-1}+a_n$$
with $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. Suppose that $p(\hat{a},\hat{\lambda}_i)=0$, $i=1,\dots,...
- 205
1
vote
1
answer
39
views
Relationship between matrix 2-norm of internal direct sum and orthogonal vector spaces of traceless matrices
Let A, B be vector spaces of traceless matrices such that A is orthogonal to B (in Frobenius inner product sense) and $S = \{ a + b\ |\ a \in A, b \in B \}$ is an internal direct sum.
Is there ...
1
vote
0
answers
33
views
Condition for 3*3 block matrix to be stable
Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times ...
- 113
0
votes
1
answer
104
views
How to interpret $A A' = \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$
(I edited the question to be more relevant and informative/specific. Hope it's better). I'm unsure of how to interpret
$$A A' = \begin{pmatrix}
0 & 0\\
0 & 0
\end{pmatrix}$$
where A and A' are ...
- 142
2
votes
2
answers
97
views
Convexity of the norm of a matrix exponential
I would like to know the convexity of the function below.
Denote the space of $ n \times n $ real symmetric matrices as $\mathcal{S}^n$, then define $f:\mathcal{S}^n \rightarrow \mathbb{R}$ as
$$
f(X) ...
- 165
0
votes
0
answers
28
views
Jacobian, vectorization and the Kroenecker product
Suppose
$f(U) = U^{T} A U$ with its derivative $d_{f}$ [U] $(H) = H^{T} A U + U^{T} A H$, then its Jacobian is given by
$J_f(vec U) = ((AU)^{T} \oplus I) \Pi + I \oplus U^{T} A$, (*)
where $\Pi = \Pi^{...
- 69
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answers
58
views
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 ...
- 4,953
2
votes
1
answer
231
views
Can we show the block matrix of the inverse of $A$ is greater than the inverse of the block matrix of $A$?
I know we have the relation $\left( A_{ii} \right) ^{-1}\le \left( A^{-1} \right) _{ii}$ where $A$ is a positive definite matrix. My problem is, can we extend the result to block matrices with $C\...
- 97
2
votes
1
answer
82
views
Bound for generalised Matrix $(p, k)$-norm
Generalised Matrix $(p, k)$-norm[1] generalizes Schatten and Ky-Fan norms as it is defined as follows:
$$
N_{p, k}(A) = (\sum_{i=1}^{k}\sigma_i^{p}(A))^{\frac{1}{p}}
$$
where $\sigma_i$ is i-th ...
0
votes
1
answer
33
views
Lipschitz continuity of diagonalizing orthogonal matrices
Let $A$, $B$ and $E$ be three symmetric $n \times n$ matrices where $B = A+E$.
Let $\mathcal{Q}$ be the set of orthonormal matrices that diagonalize $A$, i.e. the set of orthogonal matrices $Q$ such ...
1
vote
2
answers
36
views
Is the derivative of $\pmb x^\top (I+GB)^{-1} (I+GB)^{-\top}\pmb x$ with respect to $B$ a 4th-order tensor?
Is the derivative of $\pmb x^\top (I+GB)^{-\top} (I+GB)^{-1}\pmb x$ with respect to $B$ a 4th-order tensor?
Where $\pmb x$ is a vector, and $B$ is a matrix.
I followed the procedures in What is the ...
1
vote
1
answer
92
views
ij-element of (A^T)A matrix
I know that given an orthogonal matrix $A\in\mathbb{R}^{n\times n}$, $A^\top A=AA^\top=I_n$.
I saw that the $ij$-element of $A^\top A$ can be expressed as
$$(A^\top A)_{ij} = (A^\top a_j)_i=(\text{row ...
- 13
1
vote
0
answers
46
views
Orthogonal similarity of block real symmetric matrix
Can a block real symmetric matrix whose sum of diagonal blocks is an identity matrix
$$\left[
\begin{array}{cccc}
H^{T}_{1}H_{1} & H^{T}_{2}H_{2} & \cdots & H^{T}_{1}H_{n+1} \\
H^...
- 21
0
votes
1
answer
229
views
Differentiation under the integral sign in higher dimensions
Let's say I have a function $f : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$. Are there conditions under which the following holds
$$ \frac{\partial}{\partial\mathbf{x}^T}\int_\Omega f(\mathbf{x},...
- 1
0
votes
0
answers
23
views
Orthonormal Eigenvectors of $C C^{*}$.
I'm reading a paper, and somewhere in this paper, he considered a set of orthonormal vectors of the matrix $C C^{*}$ corresponding to $1$, where $C$ here is a companion matrix. I tried searching but I ...
- 431
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votes
2
answers
85
views
If $A^*= A$ and $A^m= 0$, then $A= 0$.
If $A^*= A$ and $A^m= 0$, then $A= 0$.
My attempt:
If $m = 2$, then $\text{tr}(A^*A)= 0 \Rightarrow A=0$
and then the result follows for $m = \{2,4,8,16,32,\ldots\}$.
But I don't know how it works in ...
-2
votes
1
answer
92
views
For any matrices $A$ and $B$ of the same size, show that $\mathrm{Im}(A ,B) = \mathrm{Im}(A) + \mathrm{Im}(B)$.
I just want to be more familiar with block matrices and while I am reading Fuhzen Zhang book, I found this problem
$\mathrm{Im}( A,B)$ means the range of the block matrix
$$\begin{bmatrix} A & B\...
1
vote
1
answer
65
views
Is Polar Decomposition Unique?
For an arbitrary complex matrix $A$, it is known that $A$ has either a decomposition $P U$ or $V Q$, where $P = (A A^{*})^{1 / 2}$, $Q = (A^{*} A)^{1 / 2}$, and $U, V$ are isometry. Question: Is it ...
- 431
2
votes
1
answer
53
views
Eigenvalues of Hessian of Lagrange Function
I am interested to know the eigenvalues of the following block matrix, which I obtained from the Lagrange function (optimization). We have $H_{n\times n}$ here positive definite (i.e., all the ...
- 133
0
votes
0
answers
29
views
preservation of positive semidefiniteness
if P is positive semidefinite, Q is invertible, P and Q are of the same dimension, then is $Q^TPQ$ still positive semidefinite? If so why?
edit: I am aware that sylvester's law of inertia might be ...
- 419
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votes
1
answer
67
views
Unitary equivalence of two matrices
Let $B\in M_n(\mathbb{C})$ with $BB^*+B^*B=I$. Is $
\begin{pmatrix}
0 & B \\
B & 0 \\
\end{pmatrix}
$ is unitarily similar to $
\begin{pmatrix}
0 & \vert B\vert \\
\vert B^*\vert & 0 \\...
- 404
7
votes
1
answer
343
views
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 ...
- 466
0
votes
1
answer
35
views
Finding a real matrix satisfying certain norm requirements
Suppose $M_{100}(\mathbb{R})$ denote the space of $100\times 100$ with real entries matrices with a natural norm.
I am trying to show that there exists some $\delta>0$ such that if $M\in M_{100}(\...
- 500
3
votes
1
answer
397
views
How to prove that generalized Vandermonde matrix is invertible?
Given $$A = \left( z_i^{\lambda_k}\right)_{i,j = 1,\ldots, n} =
\begin{pmatrix}
z_1^{\lambda_1} & z_1^{\lambda_2} & \cdots & z_1^{\lambda_n} \\
z_2^{\lambda_1} & z_2^{\lambda_2} & ...
- 99
3
votes
3
answers
199
views
Interpreting "geometric mean" $\exp(\frac{1}{2} \log(P) + \frac{1}{2} \log(Q))$ for singular symmetric matrices $P, Q$
Let $\mathcal S_+^d$ be set of real $d \times d$ symmetric positive semidefinite matrices and $\mathcal S_{++}^d$ be set of real $d \times d$ symmetric positive definite matrices
Following section 1....
- 4,755
1
vote
2
answers
133
views
What is the derivative of a matrix-valued composite function?
Preliminaries: Jordan algebra
In the sense of Jordan algebra, the following arrow matrix is often used to express the Jordan product $\circ$ which will be defined later: For a vector $a\in\Bbb R^m$,
$$...
- 135
0
votes
1
answer
54
views
Write linear constraints into linear transformation
Suppose we have the following block matrix
$$
X=\begin{bmatrix}
X_{11} & X_{12} & X_{13}\\
X_{21} & X_{22} & X_{23}\\
X_{31} &X_{32} & X_{33}
\end{bmatrix}
$$
where $X_{ij} \in ...
- 47
2
votes
0
answers
109
views
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 ...
- 51
0
votes
1
answer
38
views
The minimal polynomial of a matrix
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+...
- 1,988
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}^{...
- 618
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. ...
- 497
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 ...
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