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Questions tagged [matrix-analysis]

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1
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1answer
22 views

Could we find a constant $a$ such that $\operatorname{Tr}(M^{\top}N + N^{\top}M) \ge a \operatorname{Tr}(M^{\top}M)$?

Suppose $\mathcal K_1, \mathcal K_2 \subset M_{m \times n}(\mathbb R)$ are two connected compact sets both containing $0$ and for every nonzero $M \in \mathcal K_1$ and $N \in \mathcal K_2$, we have $...
1
vote
2answers
60 views

If $\det\left(\begin{smallmatrix}a & 1 & 1\\1 & b & 1\\1 & 1 & c\end{smallmatrix}\right) > 0 $ then prove that $abc> -8$

If the value of the determinant $$\begin{vmatrix} a & 1 & 1\\ 1 & b & 1\\ 1 & 1 & c \end{vmatrix} > 0 $$ then prove that $abc> -8$ I have calculated its determinant and ...
0
votes
2answers
34 views

How to find Inverse-squared of a matrix

Given the matrix $$ A = \begin{bmatrix} 1 & 2 & 0 \\ 2 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$ Find $A^{-2}$ I have solved it using Cayley Hamilton theorem, but it is ...
0
votes
0answers
19 views

Question about convexity of least-squares problem and pseudoinverse

In the rank-deficient case, for A $\in \mathbb{C}^{m\times n}$, the solution of the least-squares problem, with b $\in \mathbb{C}^m $, $$min_{x \in \mathbb{C}}||Ax-b||_2$$ is not unique. (i) Prove ...
2
votes
0answers
57 views

How do eigenvalues change if we perturbe the diagonal entries of a matrix?

Suppose $A \in M_n(\mathbb R)$ is a stable matrix, i.e., all eigenvalues are on the left open half plane of $\mathbb C$. If in particular, all the Gershgorin disks $\Gamma_j$ corresponding to rows $j=...
0
votes
1answer
45 views

Eigendecomposition of $(a^\top X a)X - Xaa^\top X$

Let $a\in\mathbb{R}^n$ be a nonzero vector, $X\in\mathbb{R}^{n\times n}$ be positive definite. What are the eigenvalues and eigenvectors of $(a^\top X a) X - Xaa^\top X$?
5
votes
1answer
136 views

How do the eigenvalues change if we change the diagonal entries of the matrix?

Suppose $A \in M_n(\mathbb R)$ is stable. By stable, we mean the eigenvalues are all on the left open half plane of $\mathbb C$. Now if we decrease the value of $A_{11}$, does the matrix remain stable?...
2
votes
1answer
62 views

Infinite Matrix Multiplication

$\{a_{ij}\}_{i,j}\in \mathbf{C}$. Suppose $x=(x_1,x_2, . . . )$ be a sequence. Define a new sequence $Ax$ by $(Ax)_i = \sum_{j=1}^{\infty} a_{ij}x_j$ (if it makes sense). Consider the linear map $A:x\...
0
votes
1answer
21 views

Suppose a square matrix $A$ has spectral radius $\rho(A) < 1$. Fixing the last row and scaling other entries by $r \in (0,1)$, will $\rho(A)<1$?

Suppose $A \in M_n(\mathbb R)$ has spectral radius small than $1$, i.e., $\rho(A) < 1$. Denote $A = \pmatrix{a_1^T \\ \vdots \\ a_n^T}$, where $a_j^T$ denotes the $j^{th}$ row of $A$.Putting $B=\...
1
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2answers
26 views

Show: $\sum_{i,j=1}^n |a_{ij}|^2=trA^*A=\sum^n_{i=1}\sigma_i^2$

I'm trying to prove the above fact for an arbitrary matrix $A$, with eigenvalue $\lambda_i$, and singular values $\sigma_i$. My approach so far: the trace of a matrix is the sum of its eigenvalues, ...
0
votes
0answers
29 views

There exists $H$ such $S=XH^t+HX^t$ for every simetric $S$

Given the function $f(X)=X*X^t$ then If $X^t=X^{-1}$ for every simetric matrix $S$ there exists $H$ such that $f'(X)H=S$ So i know that $f'(X)H$ is simetric regardless of $X$ being orthogonal, in ...
2
votes
3answers
68 views

Show some eigenvalue properties for $A=xy^*$

Let $x,y$ be given vectors of dimension $n \times 1$, $A=xy^*$, and $\lambda=y^*x$. I’m trying to demonstrate the following: $\lambda$ is an eigenvalue of $A$. If $\lambda \ne 0$, it will be the ...
11
votes
1answer
255 views

Is the function $A \mapsto \sum\limits_{j=0}^{\infty} \langle A^j v, A^j v \rangle$ differentiable everywhere?

Suppose $v \in \mathbb R^n$ is a fixed vector. We define a scalar-valued function on $n \times n$ matrices $f: M_n(\mathbb R) \to \mathbb R$ by \begin{align*} A \mapsto \sum\limits_{j=0}^{\infty} \...
1
vote
1answer
28 views

$3 \times 3$ Diagonal matrix as the square of an $3 \times 3$ non diagonal matrix

It is easy to see if the matrix is $2 \times 2$, by just considering a system of equation with $4$ equations. However, for a $3 \times 3$ matrices, a system of equation with $9$ equations might not ...
0
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0answers
42 views

To express distance matrix of a graph in terms of known matrices.

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. Let $A$ denote the adjacency matrix of $G$ and $D$ denotes the distance matrix of $G$. If the graph $G$ has the diameter at most $2$ ...
0
votes
0answers
26 views

Square root of a specific matrix over $\Bbb Z$

Let $ B^2 = \begin{bmatrix} -2&0&0 \\ -1&-4&-1\\ 2&4&0\\ \end{bmatrix}^2 = \begin{bmatrix} 4&0&0\\ 4&12&4\\ -8&-16&-4\\ \end{bmatrix} = A similar to \...
0
votes
0answers
13 views

Transition matrix of a specific matrix

Let $ A = \begin{bmatrix} 3&3&3 \\ 3&3&3\\ 3&3&3\\ \end{bmatrix} $ I know that its rational canonical form is $ \begin{bmatrix} 0&0&0 \\ 0&0&0\\ 0&1&...
2
votes
2answers
40 views

Square root of a specific matrix in M(3,Z) over Z

Let $ \begin{bmatrix} a&b&c \\ d&e&f\\ g&h&i\\ \end{bmatrix}^2 = \begin{bmatrix} x&0&0\\ 0&0&-y\\ 0&1&-z\\ \end{bmatrix} $ Where all of the ...
0
votes
1answer
39 views

Does there always exist matrix $X \in M_{m \times r}(\mathbb R)$ such that $AXB$ is in upper/lower triangular form?

Suppose we have two fixed real matrices $A \in M_{n \times m}(\mathbb R)$ and $B \in M_{r \times n}(\mathbb R)$ with $n > m$ and $r < n$. Moreover, $A$ has full column rank $m$ and $B$ has full ...
0
votes
0answers
23 views

Square root and similarity between integer matrices

Prove or disprove: Let $A, B \in \operatorname{M}(3,\mathbb{Z})$ and $A \sim B$. $A$ has a square root in $\operatorname{M}(3,\mathbb{Z})$ iff $B$ has a square root in $\operatorname{M}(3,\...
0
votes
1answer
30 views

Show that $B \in \Bbb M_3 (\Bbb Z).$ [closed]

Let $A \in \Bbb M_3 (\Bbb Z)$ be such that $A=B^2,$ for some $B \in \Bbb M_3 (\Bbb R).$ Show that $B \in \Bbb M_3 (\Bbb Z).$
0
votes
1answer
30 views

Similarity of matrices and its square root over $\mathbb Z$

I already ask this but now its "for all" Prove or disprove: $A \in M(3,\mathbb{Z})$ has a square root with integer entries if and only if $XAX^{-1} \in M(3,\mathbb{Z})$ has a square root with ...
0
votes
1answer
16 views

Rational canonical form conjugation

I know that if A is an nxn intger matrix, with A = XRX^-1 where R is the rcf of A. Then R is also an integer matrix. My question is Does X and X^- are integer matrices as well ?
0
votes
1answer
43 views

Square root of a matrix $A$ and matrices similar to $A$

Prove or disprove: $A \in \Bbb M(3,\mathbb{Z})$ has a square root with integer entries if and only if $XAX^{-1} \in \Bbb M(3,\mathbb{Z})$ has a square root with integer entries, for some invertible $...
0
votes
0answers
10 views

Similarity of block diagonal matrices

Let M and N be block diagonal matrices with the same size and partition If M ~ N , then there exist an invertible matrix S such that MS = SN. My question is, does S is also block diagonal matrix ? ...
0
votes
1answer
11 views

Rational canonical form of integer matrix

In the paper Computing Rational Forms of Integer Matrices by MARK GIESBRECHT† AND ARNE STORJOHANN It says that, When A ∈ Qn×n has all integer entries, the Frobenius form F of A has all integer ...
0
votes
2answers
38 views

Diagonalizable Matrix $A$ with Eigenvalues scalar=c, is $A=cI$

Let $A$ be an $n \times n$ diagonalizable matrix such that all of it's eigenvalues are equal to a scalar, $c.$ Then $A=cI$. Is this true? Why or why not? I'm thinking not, because I can come up with ...
0
votes
1answer
16 views

Rational canonical form of some square of a matrix

Let $$ B = \begin{bmatrix} -2 & 0 & 0 \\ -1 & -4 & -1 \\ 2 & 4 & 0 \\ \end{bmatrix} $$ Then $$ B^2 = A = \begin{bmatrix} 4 & 0 & 0 \\ 4 & 12 & 4 \\ -8 & -...
0
votes
0answers
14 views

Counterparts to Schwarz matrices

Schwarz matrix (see Definition $3.1$) is a class of tridiagonal matrices which has special properties: the inertia $\{n_0, n_+, n_{-}\}$ of a Schwarz matrix is completely determined by the sign of the ...
0
votes
0answers
17 views

The spectrum of a compression of nonsymmetric matrix

If $A\in\mathbb{R}^{n\times n}$ with a spectrum $eig(A)\subset\mathbb{C}$. Define a matrix $P\in\mathbb{R}^{n\times m}$, where $m<n$ and $P$ is a nonnegative matrix, i.e. its entries are either ...
7
votes
4answers
433 views

To find the inverse of a special kind of matrix.

In a matrix analysis problem, I encountered the following special kind of matrix $$ \begin{bmatrix} 0 & 1 & a & a & a & a \\ 1 & 0 & a& a& a& a \\...
0
votes
0answers
60 views

Eigenvalue Comparison

Let $A$ be a real matrix, if some of its diagonal entries perturbed by subtracting postive real numbers, then what can we say about the eigenvalues of perturbed matrix in comparison to eigenvalues of ...
2
votes
1answer
113 views

Are the sets $\{X: \max_i \text{Re}\lambda_i (B+AX) < 0\}$ and $\{X: \rho(B+AX) < 1\}$ homeomorphic?

Suppose $A \in M_{n \times m}(\mathbb R)$ and $B \in M_n(\mathbb R)$ are fixed with $m < n$. Denote two sets $\mathcal E, \mathcal F$ by \begin{align*} \mathcal E &= \{X \in M_{m \times n}(\...
1
vote
0answers
60 views

Showing convexity of a set in $\mathbb{C}^k$

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert Space and $T\in\mathscr{B(\mathcal{H})}$. Suppose $$\mu_k(T):=\left\{ \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ ...
4
votes
1answer
66 views

Is there any inequality between 2-norm condition number and Frobenius norm condition number for rectangular matrix?

What I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$n - ...
0
votes
1answer
15 views

Eigenvetor Property of a Matrix

If a matrix $A$ is complex orthogonally similar to an upper triangular matrix, that is, $A=QUQ^T, Q^TQ=I$ and $U$ is upper triangular matrix, then there exist at least one eigenvector $x$ of $A$ such ...
1
vote
0answers
42 views

Proving a matrix inequality involving a extended matrix and the pseudoinverse under the background of signal processing

Consider two positive definite Hermitian matrices $\pmb W_1$ and $\pmb W_2$ such that $\pmb W_1 \succ \pmb W_2$, which means $\pmb W_1-\pmb W_2$ is positive definite. There exist two full cloumn rank ...
1
vote
1answer
45 views

A matrix inequality involving pseudoinverse

I am trying to solve a problem in the context of signal processing, which leads to the following question. Consider two positive definite Hermitian matrices $A$ and $B$, and a full cloumn rank matrix $...
0
votes
2answers
49 views

linear combination of some matrices is identity matrix

Assume $T$ is a $n\times n$ matrix over number field $\mathbb{F}$. If $\lambda$ is not an eigenvalue of $T$, we know $T-\lambda E$ is invertible matrix where $E$ is the identity matrix. Now if we have ...
0
votes
1answer
41 views

Eigenvalues of an adjacency matrix

I am trying to prove or disprove the following: Let $G$ be a non-bipartite AND connected $k$-regular graph where the size of the second-largest eigenvalue of the adjacency matrix $A_G$ of $G$ is $\...
0
votes
0answers
36 views

Eigenvector relations between matrices whose Gramian Matrices are the same

I have two symmetric real matrices I am interested in, $A_{n\times n}$ and $B_{m\times m}$. If I do the following operations: $$EAE^{T}=FBF^{T}$$ where $E$ and $F$ are of dimensions $p\times n$ and $...
0
votes
0answers
16 views

Can we prove that every neighborhood of the $0$ companion matrix that contains a Hurwitz matrix?

Let $\mathcal C \subset M_n(\mathbb R)$ be the set of companion matrices which we can identify by $\mathbb R^n$. Let us denote the set of companion matrices with eigenvalues lying on the open left ...
3
votes
1answer
120 views

Classification of $n\times n$ real matrices up to congruence

As we known, for matrix similarity $A= P^{-1}B P$, we can classify equivalence class in $M_n (\mathbb{C})$ by Jordan Canonical Form. Matrix congruence, $A = P^T B P$ with $P$ invertible, is also an ...
2
votes
0answers
25 views

$A$ is irreducible doubly non-negative, then $rank(A) \geq n-1$.

Let $A = D + \begin{bmatrix} 0 & F \\ F^T & 0 \\ \end{bmatrix} $ be in the block form, where $D$ is diagonal and $F$ is $k \times (n-k)$. Let $A$ be doubly non-negative, i.e. ...
2
votes
0answers
32 views

Injectivity and inverse of the map $A \mapsto A_c$, the map sending a matrix to its companion form

Let $\mathcal E \subset M_n$ denote the set of matrices such that $e_1$ (standard basis vector) is a cyclic vector for all $A \in \mathcal E$. That is if $A \in \mathcal E$, $\{e_1, Ae_1, \dots, A^{n-...
-1
votes
1answer
38 views

$ \begin{bmatrix} D_1 & C \\ C^T & D_2 \\ \end{bmatrix} $ always positive semidefinite?

Is block matrix of the form $$ \begin{bmatrix} D_1 & C \\ C^T & D_2 \\ \end{bmatrix} $$, where $C$ is a nonnegative matrix (entrywise nonnegative) and $D_1$ and $D_2$ are ...
1
vote
1answer
26 views

A guess about invertible sign matrix

For any $n\times n$ invertible matrix $A=(a_{ij})$, which satisfies $a_{ij}\in\{0,1,-1\}$. Assume $A^{-1}=(b_{ij})$. I guess $\vert b_{ij}\vert \le 1$, because I have found it is right when $n=1,2$. ...
1
vote
1answer
34 views

Are the real and imaginary parts of an invertible matrix has to be invertible too?

Let $\mathbf{A} \in \mathbb{C}^{N}$ be a complex matrix. Assuming that its inverse exists. Does it imply that both $\mathbf{A}_{R}^{-1}$ and $\mathbf{A}_{I}^{-1}$ exist? where $\mathbf{A} = \mathbf{A}...
0
votes
2answers
32 views

Let A,B be nxn matrix If $(A^2)(B^2) = (B^2)(A^2)$ , is $AB=BA$?

Let $A,B$ be $n\times n$ matrix If $(A^2)(B^2) = (B^2)(A^2)$ , is $AB=BA$?
2
votes
2answers
139 views

Differentiability of largest eigenvalue for a $C^1$ function

I encountered following interesting statement: If $f: [a, b] \to M_n(\mathbb C)$ is a $C^1$ ( $C^1$ in the interior and left/right differentiable over the end points) function over an interval $[a, b]...