Questions tagged [matrix-analysis]

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

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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 ...
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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 \...
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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
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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(...
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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}}...
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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, ...
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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, ...
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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
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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\...
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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 ...
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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 ...
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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
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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
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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 ...
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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 ...
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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{...
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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
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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 $$ \lVert A\rVert=\sup_{\lVert x\rVert=1}\lVert Ax\rVert. $$ ...
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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\...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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\...
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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
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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
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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))...
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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 ...
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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 ...
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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
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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*...
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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
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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
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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
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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
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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
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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 ...
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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} \...
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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
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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
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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
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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 $...