Questions tagged [matrix-analysis]
For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.
123
questions
0
votes
1answer
53 views
When $f\left(\|Ax\|\right)\leq \|f(A)x\|$ is true? [closed]
When is the following relation true?
$$f\left(\|Ax\|\right)\leq \|f(A)x\|$$ where $A$ is any matrix and $\|\cdot\|$ is the spectral norm.
Edit note :
The function $𝑓$ is any real concave ...
0
votes
2answers
51 views
Taylor Expansion of Logarithm of Determinant near Identity for Non-Diagonalizable Matrix
I have been working on a problem where I need to Taylor expand an expression of the form $\log \det(I-A)$ in terms of traces of the matrices $A^m$ for $m \in \mathbb N$, where $A$ is a general $n \...
0
votes
1answer
27 views
Problem about inequality with symetric matrices and inner product
Let $A$ and $B$ be two matrices of order $n$ with entries in $\mathbb{R}$.
$\newcommand{\lg}{\langle}$ $\newcommand{\rg}{\rangle}$
a) If $A$ and $B$ are symmetric then
$$ \lg(A^{2} + B^{2})x, x \rg ...
4
votes
2answers
102 views
Spectral norm of projected matrix
Let $M_{n,m}$ be the set of real matrices of $n\times m$, and let $T:M_{n,m}\to M_{n,m}$ be a orthogonal projection operator, i.e., $T$ is such that for any $A,B\in M_{n,m}$
$$T(A+B)=T(A)+T(B),$$
$$T(...
0
votes
0answers
29 views
Convex functions defined on eigenvalues
Let $f:\mathbb R^n\to \mathbb R$ be a symmetric convex $C^2$ function defined by $(\lambda_1,\cdots,\lambda_n)\mapsto f(\lambda_1,\cdots,\lambda_n)$.
Define $F:Sym(n)\cong\mathbb R^{\frac{n(n+1)}{2}}...
2
votes
3answers
75 views
Invertibility of infinite-dimensional matrix
I have a matrix $M \in \mathbb{R}^{n \times n}$ whose columns are linearly independent. Hence, $M$ is invertible.
How to extend this conclusion to the case where $n$ is infinite?
Specifically, ...
0
votes
0answers
10 views
Determinant of matrix exponential
It easy to see that the statement hold for every diagonlizable matrices:
$X=CDC^{-1}, D=\text{diag}(\lambda_{1},\cdots,\lambda_{n})$, because $$e^{X}=Ce^{D}C^{-1}=C\text{diag}(e^{\lambda_{1}},\cdots, ...
-1
votes
1answer
30 views
Why is the computation of the general $p$-norm hard?
Consider a $2 \times 2$ matrix. Can we find a general polynomial for computing the operator norm induced by a $p$-norm?
1
vote
1answer
34 views
About positive definiteness of analytic matrix-valued function
Let $A(\theta):\mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ be a analytic matrix-valued function such that
$A(\theta)=A^T(\theta)\geq 0,\quad \forall \theta\in\mathbb{R}$
For each fixed $x\in\...
0
votes
0answers
26 views
Logarithmic derivative of matrix function
in my research I ended up with a term of the following form:
$$ C(x)^\prime:C^{-1}(x)^\prime $$
In my case the matrix function $C(x)\in\mathbb{R}^{3\times 3}$ and is always s.p.d. so we can rewrite ...
1
vote
1answer
34 views
Prove that $A$ is semiprimitive [closed]
Matrix $A \ge 0$ is a semi primitive matrix, if for some $k$ matrix $A^{k}$ contains a positive column.
Prove, if characteristic polynomial of stochastic matrix $A$ equal to $t^{(n-1)}(t-1)$, then A ...
1
vote
1answer
51 views
Real Hadamard powers of matrices
Let $A$ be an entrywise nonnegative matrix and for any $r>0$, $A^{\circ r} = [a_{ij}^{r}]$. My question is what properties of matrix $A$ are preserved for all $r>0$ or for all $r$ in some ...
1
vote
1answer
51 views
$A$ is a $12\times12$ matrix such that $A^{10 }=I$, and $\operatorname{rank}(A-I)=5$. How can we show $\operatorname{rank}(A^2 +A+I)\leq 7$?
If $A$ is a $12\times12$ matrix such that $A^{10 }=I$, and $\operatorname{rank}(A-I)=5$, how can we show that $\operatorname{rank}(A^2 +A+I)\leq 7$?
4
votes
2answers
74 views
Equality in trace duality
For $A,B\in\mathbb{R}^{n\times m}$ we have the trace duality property
$$|\langle A, B \rangle|\leq \|A\|_1 \|B\|_{\infty}$$
where $\|A\|_p$ is the Schatten $p$-norm (i.e. $\|\cdot \|_1$ is the ...
1
vote
0answers
29 views
About trace duality and other two bounds…
I've been reading this paper of Koltchinskii, Tsybakov and Lounici about nuclear norm penalization. Let me give you the context for the next three questions regarding Theorem 1 (I don't think that ...
0
votes
0answers
28 views
Invariance of numerical radius
Let $X=\begin{pmatrix} r & x\\ \vert y\vert e^{i\theta} & r\end{pmatrix}$ where $r,x>0, y\in\mathbb{C}$ and $\theta\in [0,2\pi)$ and $Y=\begin{pmatrix} r & x\\ \vert y\vert & r\end{...
0
votes
1answer
37 views
Sufficient condition for total nonnegativity
If $A$ is any $m \times n$ entrywise non negative matrix, is it true that, if all initial minors of $A$ are nonnegative then $A$ is totally nonnegative (TN)?
I know the analogous result is true for ...
0
votes
1answer
109 views
Matrix exponential is continuous
I want to prove that the function $ \exp\colon M_n(\mathbb{C})\to \mathrm{GL}_n(\mathbb{C}) $ is continuous under standard matrix norm
$$ \lVert A\rVert=\sup_{\lVert x\rVert=1}\lVert Ax\rVert. $$
...
0
votes
1answer
140 views
Numerical radius of $2 \times 2$ complex matrix
Let $$X=\begin{pmatrix} r & x\\ y & r\end{pmatrix} \in M_n(\mathbb{C}).$$ Here $r> 0$, $x, y \in \mathbb{C}$. Calculate the numerical radius $w(X)$ of $X$ where $$w(X) := \sup\limits_{\Vert\...
2
votes
2answers
140 views
Do eigenvalues depend smoothly on the matrix elements of a diagonalizable matrix?
Suppose I have a matrix $M(t)$ whose matrix elements depend smoothly on a real parameter $t$. I also know that this matrix is diagonalizable for the $t$s I'm interested in. Can I say that its ...
0
votes
1answer
45 views
Trace of a Summation of Positive Definite Matrices
Given that $$tr \Bigl( \sum_{i=1}^k (\alpha A_{i} + B_{i})^2 \Bigr) \geqslant 0 \forall \alpha \in \mathbb{R} $$
with $A_{i}$ and $B_{i}$ being positive definite $n\times n$ complex matrices $\forall ...
1
vote
1answer
200 views
Index of a Nilpotent matrix
I was wondering why there can't be a nilpotent matrix of index greater than its no. of rows.
Like why there does not exist a nilpotent matrix of index 3 in $M_{2×2}(F)$
When I look up on the ...
0
votes
1answer
21 views
Trace of a Summation of Positive Definite Matrices with a Real Coefficient
Show that $$\mbox{tr} \left( \sum_{i=1}^k (\alpha A_{i} + B_{i})^2 \right) \geqslant 0, \qquad \forall \alpha \in \mathbb{R}$$ given that $A_{i}$ and $B_{i}$ are positive definite $n \times n$ complex ...
1
vote
1answer
68 views
Which of the statements is/are TRUE?
Let $K = \big [k_{i,j} \big ]_{i,j = 1}^{\infty}$ be an infinite matrix over $\Bbb C$ (the set of all complex numbers) such that
$(\text {i})$ for each $i \in \Bbb N$ (the set of all natural ...
0
votes
0answers
54 views
example of Stahl's theorem
I'm reading about Stahl's theorem
So the theorem says that : If $A$ and $B$ are $n \text{ by } n$ Hermitian matrices and $B$ is positive semi-definite, define function $f(t)=tr(exp(A-tB)$ for $t>=0$...
3
votes
0answers
236 views
“Almost Normal” Matrix and Gap between Spectral Radius/Norm
Let's denote
$$\Vert{A}\Vert := \max_{x\neq0}\frac{x^* Ax}{x^*x}$$
and let $\rho(A)$ denote the largest absolute value of the eigenvalues of matrix $A$. From basic linear algebra, one could ...
0
votes
1answer
30 views
Inequality regarding diagonal entries and eigenvalues of a PSD matrix
I want to show the following inequality regarding schur's majorization theorem $$\overset{n}{\underset{i = 1}{\sum}}a_i\lambda_i\leq \overset{k}{\underset{i = 1}{\sum}}\lambda_i,$$ where $\lambda_1\...
0
votes
1answer
25 views
Two results of monotone operators of matrices
Let $A$ and $B$ be two $p\times p$ real symmetric matrices such that $A\succeq B$, meaning that $A-B$ is non-negative definite. I want to prove these two results.
First result: $\lambda_j(A)\geq \...
1
vote
1answer
99 views
Show that the smallest eigenvalue is strictly positive.
I have a $2\times 2$ real, symmetric and positive definite matrix $B_{x,n}$ which depends on a point $x\in[0,1]$ and $n\in\mathbb{R}$. I want to show that for sufficiently large $n$, the smallest ...
1
vote
1answer
39 views
Inclusion of numerical range and translation and rotation invariance
Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Then prove that following two are equivalent:
(i) $W(B\otimes X)\subseteq W(B\otimes T)$, for all $B\in M_n$
(ii) $W(C\otimes (aX+bI_n))...
0
votes
0answers
34 views
Show that if $g(x) = f(Ax + b)$, then $\delta g(x) = A^T \delta f(Ax + b)$
Looking for a simplier expaination for the following:
Show the following for sub-gradients:
(a) If $g(x) = f(Ax + b)$, then $\delta g(x) = A^T \delta f(Ax + b)$.
I've found the trivial ...
0
votes
1answer
58 views
Approximation of selfadjoint matrix in $M_n(\mathbb{C})$
We first observe that by spectral theorem of a selfadjoint matrix, any selfadjoint matrix $A\in M_n(\mathbb{C})$ can be written as (up to unitarily equivalence) $A=A_{+}-A_{-}$ where $A_{+}, A_{-}\geq ...
1
vote
0answers
32 views
Perturbation of a matrix and eigen values
Let $T\in\mathscr{B(\mathcal{H})}$ and $A,B\in M_n(\mathbb{C})$ where $A$ is a hermitian matrix s.t. $$\lambda_1(A\otimes I_n+B\otimes X+B^*\otimes X^*)\leq \lambda_1(A\otimes I+B\otimes T+B^*\otimes ...
1
vote
1answer
42 views
Ordering of positive semidefinite matrices
Suppose $A \in \mathbb R^{n \times r}$ with $r < n$ and $M, X \in \mathbb R^{n \times n}$ are symmetric and positive definite. I am wondering whether the following relation is correct:
\begin{align*...
0
votes
0answers
13 views
cp-rank $A \leq \lim_{n \to \infty} \inf$ cp-rank $A_n$.
Let $A$ be a $n \times n$ completely positive matrix and cp-rank is the minimal number of summands in a rank $1$ representation of $A$, $A = \sum_{i=1}^{k}b_ib_i^T, b_i \geq 0$, where $b_i \geq 0$ ...
0
votes
0answers
35 views
Min-sum-max norm optimization with orthogonality constraint and matrix regularization
Let $S = \{s_1,\cdots,s_N\}\subset\mathbb{R}^n$ be a finite set of points, and $A\in\mathbb{R}^{n\times n}$ an invertible matrix. Then I would like to solve the following optimization problem:
$$
\...
0
votes
0answers
37 views
Dilation of a contraction in the connection with numerical range.
Let $T\in\mathcal{B(\mathcal{H})}$ be a contraction and $X\in M_n$ with $\Vert X\Vert\leq 1$ s.t. $W(X)\subseteq \overline{W(T)}$ where $W(T)=\{\langle Tx,x\rangle :\Vert x\Vert=1\}$ is the numerical ...
1
vote
1answer
62 views
Subgradient of function $f_p(\mathbf{A})$ that has as output the $p^{th}$ largest singular value
Suppose I have a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$. Let $\{\sigma_1,\sigma_2,.....,\sigma_n \}$ be its $n$ singular values. Calculating a sub-gradient of its operator norm (largest ...
1
vote
1answer
37 views
Let $Q,P$ be square row stochastic matrices with nonnegative real eigenvalues. Is $ || (2Q-I)(2P-I)||_\infty \leq 3 $?
Here $\lvert \lvert P \rvert \rvert_\infty$ is the maximum absolute row sum of the matrix $P$, which is $1$ when $P$ is row stochastic.
While the hypothesis in the title seems to hold when tested, I ...
1
vote
2answers
31 views
The relation between the trace of a positive definite matrix and the trace of the inverse of this matrix
How to prove this inequality, that:
$$Tr(\mathbf{X})\geq \frac{N^2}{Tr(\mathbf{X}^{-1})}$$
where $\mathbf{X}\in \mathbb{R}^{N\times N}$ is an arbitrary positive definite matrix.
Horn R A, Johnson C ...
0
votes
1answer
122 views
How do I prove that $\sum_{i,j=1}^na_{ij}b_{ij} \geq 0$ if $A=(a_{ij})$ and $B=(b_{ij})$ are both positive semidefinite $n \times n$ matrices? [closed]
This is an exercise in linear algebra:
Let $A=(a_{ij})$ and $B=(b_{ij})$
both be positive semidefinite $n \times n$ (real) matrices.
Prove that
$$
\sum_{i=1}^n \sum_{j=1}^n a_{ij}b_{ij} \...
0
votes
0answers
73 views
Intuition behind factorisation in Schur Complement Lemma proof
The theorem states:
Let $T \in \mathbb{S}^m$ be positive definite and $X \in \mathbb{S}^n$ and $U \in \mathbb{R}^{n \times m}$. Then $M$ is positive semidefinite if and only if $X - UT^{-1}U^T$ is, ...
0
votes
1answer
31 views
How obtain the partial derivative of quadratic function $f(z)=\frac{1}{2} x^T(z)Q(z)x(z)$?
Consider $z\in \mathbb{R}$, $x(z):\mathbb{R} \to \mathbb{R}^{n\times 1}$ and $Q(z):\mathbb{R} \to \mathbb{R}^{n\times n}$ such that $Q(z)=Q^T(z)>0$. If we define the function $f(z):\mathbb{R} \to \...
7
votes
1answer
124 views
Conjecture about $(0,1)$-matrices
Let $A$ be an $m$ by $n$ $(0,1)$-matrix. For $1\leq i \leq m$ and $1\leq j \leq n$, let $f(A,i,j)$ be the number of entries in $A$ not in row $i$, not in column $j$, and not equal to $a_{ij}$.
I ...
0
votes
1answer
179 views
Second order Taylor expansion of Frobenius norm
I have the following function
$||{\bf A} - {\bf BC}||^2_F$,
where ${\bf A} \in \mathbb{C}^{m \times n}$, ${\bf B} \in \mathbb{C}^{m \times k}$, and ${\bf C} \in \mathbb{C}^{k \times n}$, which is a ...
2
votes
1answer
73 views
Singular value of matrices whose entries are on the unit circle
I found that
\begin{align*}
\max_{|z_{mn}|=1} \sigma_2\left(
\begin{bmatrix}
z_{11} & z_{12} & z_{13}\\
z_{21} & z_{22} & z_{23}\\
z_{31} & z_{32} & z_{33}
\end{bmatrix}\right)=...
2
votes
1answer
66 views
Is it possbile to upper bound ${\bf tr}(ABC)$ by ${\bf tr}(AB)$ provided $A,B,C$ are all real, symmetric and positive definite?
Suppose $A,B,C$ are all real symmetric and positive definite matrices. Is it possible to lower bound ${\bf tr}(ABC)$ in terms of ${\bf tr}(AB)$.
I tried to follow Von Neumann's trace inequality: ...
1
vote
0answers
57 views
Lower bound ${\bf tr}(XY) - {\bf tr}(XZY)$ by ${\bf tr}(X)$ provided $X, Y, Z$ all real symmetric
Suppose $X, Y, Z$ are all real square, symmetric matrices and in particular $X, Y$ are positive definite. I am interested in a tighter lower bound of the quantity
\begin{align*}
{\bf tr}(XY) - {\bf tr}...
0
votes
1answer
29 views
Diophantine matrix equation - Solver or algorithm
I look for a solver or reference rather than an answer about how to solve the diophantine matrix equation. One states it below.
$\mathbf{X}\mathbf{N} + \mathbf{Y}\mathbf{D} = \mathbf{I}$
All entries ...
6
votes
1answer
154 views
Estimate parameter $a$ such that $tr \left[ A (B- (I-aC)B(I-aC) ) \right] > 0$.
Suppose $A, B, C$ are all real symmetric and positive definite matrices. Consider the function $f: \mathbb R \to \mathbb R$ given by
$$ a \mapsto {\bf tr}\left[ A (B- (I-aC)B(I-aC) ) \right],$$
where $...
