All Questions
Tagged with symmetric-matrices linear-algebra
1,228 questions
2
votes
0
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74
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$|Ax|\leq |Bx|$ iff $A\leq B$
Let $A,B$ be positive semidefinite self-adjoint operator on some finite inner product space. Is it true that $|Ax|\leq |Bx|$ for every vector $x$ iff $A\leq B$ [this notation means $B-A$ is positive ...
- 1,194
1
vote
1
answer
47
views
Given a positive definite matrix S, and a symmetric matrix A with small errors as S. How to prove that A is positive definite?
This is problem 4.14 in Multivariable Calculus with Applications by Peter D. Lax & Maria Shea Terrell.
A symmetric 2 by 2 matrix $A$ has been computed numerically with small errors as a symmetric ...
- 13
2
votes
1
answer
40
views
Eigenvectors of the product of a diagonal matrix and a symmetric matrix
We say that a set of vectors $\{\phi_k\}_{1 \leq k\leq n}$ is $A$-orthogonal if $\phi_i^T A \phi_j = 0$ whenever $i \neq j$. I am trying to prove the following result:
Let $M$ be an invertible ...
- 1,018
1
vote
1
answer
60
views
Conjugate to transpose via symmetric matrix
I am wondering for which square matrices $A \in M_n(\mathbb{F}_p)$ there exists an invertible symmetric matrix $S \in GL_n(\mathbb{F}_p)$ such that
$$S^{-1}AS = A^t?$$
It is a classical result, over ...
1
vote
1
answer
69
views
Prove a lower bound on eigenvalues of a class of matrices
I am not sure if this is trivial or not. Let $\mathbf{A} \in \mathbb{R}^{n \times n}$ be real and symmetric, with the property that
$$
\mathbf{A}_{ii} = 1, \quad \mathbf{A}_{ij} \in [0,1], \quad i,j \...
- 514
2
votes
0
answers
53
views
Decomposition of symmetric positive semi-definite matrices
Fix $A\in\mathrm{M}_n(\mathbb{R})$ a symmetric positive semi-definite matrix. Then there exists a matrix $M\in\mathrm{M}_n(\mathbb{R})$ such that
\begin{equation}
A=MM^\top+M^\top M.
\end{equation}
...
- 531
0
votes
0
answers
27
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Is there an explicit formula for the eigenvalues and eigenvectors of this particular symmetric matrix? [duplicate]
Consider a symmetric $n\times n$ matrix with the elements $A_{ij}=\min(i,j)$, i.e.:
$$A=\begin{pmatrix}
1 & 1 & \cdots & 1 \\
1 & 2 & \cdots & 2 \\
\vdots & \vdots & \...
- 338
0
votes
1
answer
38
views
Computing the operator norm of a rank one matrix
Suppose I have $x$ is i.i.d. from some distribution, and we know that $||x|| \leq l$, and $\lambda_{max}(x x^\top) \leq c$.
I am interested in computing $||x x^\top||_{op}$. My attempt is the ...
- 336
0
votes
1
answer
40
views
Semi-definiteness of symmetric matrix with some zero diagonal elements
Let we have a symmetric matrix
$$
A=A^T\in\mathbb R^{n\times n}
$$such that some of its diagonal elements are zero.
Is it true that $A$ is semidefinite only if it is diagonal?
- 51
0
votes
0
answers
36
views
Prove that the Jacobi method converges for every $2 \times 2$ symetric definite positive matrix
Prove that the Jacobi method converges for every $2 \times 2$ system given by a symetrical definite positive matrix.
I tried building the iteration matrix $B_J = -D^{-1} (L+U)$ given a matrix
$$
A=
\...
- 15
2
votes
0
answers
51
views
Lower Bound on the Minimum Eigenvalue of a Hollow Symmetric Matrix with Entries in $[0,1]$
Let $\mathbf{A}$ be a hollow symmetric real $n \times n$ matrix, where each entry $A_{i,j} \in [0,1]$ and $A_{i,i} = 0$ for all $i$. I would like to establish a lower bound for the minimum eigenvalue $...
- 51
0
votes
0
answers
34
views
Showing the bounds of two specific symmetric matrices
Let $M$ be an $n\times n$ symmetric matrix, partitioned as:
$$M = \begin{bmatrix} A& B\\ B^T&C\end{bmatrix}$$
where $A$ is $k\times k$. Let $\alpha = \sum\limits_{i,j=1}^k a_{ij}$, $\beta = \...
0
votes
0
answers
28
views
A question about tridiagonal complex symmetric matrix
I discovered a pattern when doing numerical experiments, a matrix with following form (tridiagonal, symmetric, real main diagonal,imaginary secondary diagonal):
$$
M=
\begin{bmatrix}
k_1 &ig_1 &...
- 1
8
votes
0
answers
287
views
Bounds on the Eigenvalues of Perturbations of a Symmetric Matrix
Let us fix $\varepsilon\in (0,1)$ and $\beta\in\mathbb R$. Consider the $2 n\times 2n$ symmetric tridiagonal probability matrix
$$Q_n :=\begin{bmatrix}
1-\frac{ε}{2} & \frac{ε}{2} & & &...
- 3,176
0
votes
0
answers
50
views
Eigenvalues of the product of diagonalizable matrices
I am trying to understand how I can compute the eigenvalues of $A^{-1}B$ where $A$ and $B$ are diagonalizable matrices (in patricular, they are symmetric tridiagonal matrices).
I know that $A$ can be ...
0
votes
1
answer
50
views
Efficient inversion of $A^T A + B^T B$
Consider matrices $A \in \mathbb R^{k \times n}, B(\eta) \in \mathbb R ^{l \times n}$, where $B$ depend on some parameters $\eta$. The matrix $A$ will be fixed.
My problem is to compute
$$
(A^TA + B(\...
0
votes
0
answers
30
views
Given S.P.D. matrix $A\in\mathbb{R}^{n\times n}$, solve $X^\top A X=\mathbf{I}_{m\times m}$ for matrix $X\in\mathbb{R}^{n\times m}$.
Context:
Given a symmitric positive definite (SPD) matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$, solve the following for matrix $\mathbf{X}\in\mathbb{R}^{n\times m}$.
$$
\mathbf{X}^\top\mathbf{A}\...
- 71
0
votes
0
answers
37
views
Limit of a matrix function
I am going through a proof from this book. More specifically, I am looking at the last paragraph on page 62 (78 in the PDF). I will reproduce it here for convenience:
It claims that for a full rank ...
- 53
1
vote
0
answers
43
views
A relation about norms of a block matrix.
Suppose $M =
\begin{bmatrix}
A & B \\
B^{T} & C \\
\end{bmatrix}$,
where $A$ and $C$ are $n\times n$ real symmetric matrices and $B$ is an arbitrary $n\times n$ real matrix. By calculating ...
- 115
1
vote
0
answers
81
views
Generalized Inverse of Submatrices
Consider a symmetric and positive semi-definite $n\times n$-matrix $\Sigma$. For $k = 1,\dots, n$, let $\Sigma_k$ denote $\Sigma$'s Leading Principal Minor of of order $k$, i.e., the submatrix of $\...
- 208
0
votes
0
answers
32
views
prove $x'X^{ +}=0$ where $X^{+}$ is the Moore-Penrose generalized inverse and $x'X=0$.
$X$ is an $n*n$ symmetric matrix and it is given that $x'X=0$ where $x$ is an $n*1$ vector. Let $X^{+}$ be the Moore-Penrose generalized inverse of $X$.
How to prove $x'X^{
+}=0$??
I am struggling ...
- 1
1
vote
1
answer
62
views
How to recover the vector $x$ when multiplying it with a matrix $A$ that has a specific structure
I have the matrix with specific structure as below:
$$A = \left[ \begin{matrix}
a & -b& -c& d \\
b & e& -d& -f \\
c & -d& a& -b \\
d & f& b & ...
- 167
3
votes
1
answer
83
views
For what $A$ the linear map $f(B):=AB^T+BA^T$ surjective, defined as a linear map between suitable vector spaces defined below?
Let $V$ denote the space of $m\times m$ real square matrices. Let $W$ denote the space of $m\times m$ real symmetric square matrices. For an $m\times m$ square (not necessarily symmetric) matrix $A,$ ...
- 2,556
0
votes
1
answer
78
views
Rank of matrix $a_i \cdot b_j + a_j \cdot b_i$ [duplicate]
Find the rank of the matrix $d_{ij} = a_i \cdot b_j + a_j \cdot b_i$, where $a$ and $b$ are arbitrary vectors.
I noticed that
$C = a \cdot b^T = \begin{bmatrix}
a_{1} \\
... \\
...
- 371
1
vote
1
answer
51
views
Suppose that the eigenvalue of largest magnetude of $P$ is simple. Is it true that the the same happens to $U P$ for any unitary matrix $U$?
Suppose that $P = [p_{i,j}]_{1\leq i,j\leq n}$ is an $n\times n$ matrix such that:
$p_{i,j}\geq 0$ for all $i,j\in\{0,1,\ldots,n\};$
$\sum_{j=1}^n p_{i,j}=1$;
$P$ is a symmetric matrix, $p_{i,j}=p_{j,...
- 3,176
0
votes
1
answer
39
views
Eigenspace of sum of outer product.
I am investigating the eigenspace of the matrix:
$$ \frac{1}{k}\sum_{i=1}^k \mu_i \mu_i^T + \sigma^2I \in \mathbb{R}^{d\times d}$$
where $k < d$.
I want to ask if the eigenspace $V_k$ spanned by ...
- 83
3
votes
2
answers
113
views
What is the connection between bilinear and quadratic forms.
I know that a bilinear form $B$ on the $\mathbb R$-vector space $\mathbb R^n$ is defined to be a map $B:\mathbb R^n\times \mathbb R^n\to \mathbb R$ which is linear in each coordinate.We know that a ...
- 3,801
0
votes
2
answers
100
views
Quadratic form of a real symmetric matrix is bounded
If $\lambda_1>\lambda_2>...>\lambda_r$ are the different eiegenvues of a real symmetric matrix $A\in M_{n\times n}(\mathbb{R})$.
$1.$
Show that the quadratic form associated to $A$ satisfies ...
- 1,392
3
votes
0
answers
52
views
Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices
Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2
,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
- 1,446
0
votes
0
answers
32
views
Reasoning for reduced SVD factorization
I am aware that for any $m \times n$ matrix $A$, we can write:
Known 1: $A = U\Sigma V^T$ where $U$ is orthogonal and $m \times m$, $V$ is orthogonal and $n\times n$, and $\Sigma$ is diagonal and $m \...
- 611
2
votes
1
answer
58
views
Rank of a matrix $A$
Let $$A=\begin{pmatrix}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{13} & a_{14} & a_{24} \\
a_{13} & a_{14} & a_{24} & a_{34} \\
a_{14} & a_{24} & a_{...
- 646
2
votes
2
answers
122
views
$A^3=A$, ${\alpha }^{T}{A}^{T}{A\alpha } \leq {\alpha }^{T}\alpha$ implies $A$ is symmetric?
Let $n\times n$ real matrix $A$ satisfies ${A}^{3} = A$; and for any $\alpha\in\Bbb R^n$, we have ${\alpha }^{T}{A}^{T}{A\alpha } \leq {\alpha }^{T}\alpha$, prove that $A$ is symmetric.
My attempt: $...
- 3,713
1
vote
2
answers
68
views
Congruent diagonalization using row and column operations
Let $$A=\begin{pmatrix}
1 & 2 & 3\\
2 & 4 & 6\\
3 & 6 & 9 \end{pmatrix}.$$
Find an invertible matrix $P$ such that $P^tAP$ is diagonal.
Let me start by saying that I already ...
- 1,392
0
votes
1
answer
39
views
Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?
I know in general, matrix multiplication is not commutative, but would it be true in this special case?
$D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
- 199
1
vote
0
answers
50
views
How to decompose the Hessian matrix of a 3rd degree polynomial into 2 or more vectors/matrices?
I am trying to figure out what can be said about the spectral radius of the Hessian of a 3rd degree random polynomial defined over a unit hypercube, drawing on known results in random matrix theory. ...
- 113
0
votes
1
answer
47
views
Determine if inner product over a real vector space has a certain form
Verify if the following statement is true: Every inner product on $\mathbb{R}^n$ has the form $\langle v,u\rangle = v(Au),$ where $A$ is a symmetric matrix with positive entries on the diagonal.
I ...
- 1,392
0
votes
0
answers
40
views
Find values which make two matrices congruent
For which values of $a\in\mathbb{R}$ are the following matrices congruent?
$$A=\begin{pmatrix} 1&4-a-a^2\\
2& -1 \end{pmatrix}$$
$$B=\begin{pmatrix} -a-1 & 3\\
3 & -5
\end{pmatrix}$$
...
- 1,392
1
vote
0
answers
21
views
Fastest way to divide by a symmetric positive matrix
Say $P_{yx}$ is a general $(n_y,n_x)$ matrix.
Say $P_y$ is symmetric, positive definite, of size $(n_y,n_y)$.
I want to compute $GT=P_y\backslash P_{yx}$ (matrix left division, or perhaps more ...
- 280
1
vote
1
answer
40
views
A matrix inequality $X - \frac{1}{u^\top Xu} Xuu^\top X \succeq Y - \frac{1}{u^\top Yu} Yuu^\top Y$
Problem. Let $n\ge 2$. Let $X, Y$ be $n\times n$ real symmetric positive definite (PD) matrices with $X \succeq Y$ (i.e. $X - Y$ is positive semi-definite (PSD)). Let $u\ne 0$ be a $n\times 1$ real ...
- 42.7k
1
vote
1
answer
65
views
If $B$ is PD in null space of $A\in \mathbb{R}^{m\times n}$ then exists $r \in \mathbb{R}$ such that $B+rA^TA$ is PD
If a symmetric matrix $B$ is PD in null space of $A\in \mathbb{R}^{m\times n}$, with
rank$A=m<n$, then exists $r \in \mathbb{R}$ such that $B+rA^TA$ is PD.
What I've tried is the following: If $x\...
- 460
1
vote
0
answers
75
views
A question about permutation similarity of symmetric matrices
Let $A , B \in M_{4 \times 4}(\mathbb{R}_{\geq 0})$. Matrices $A, B$ are $\textbf{permutation similar}$ if there exists a permutation matrix such that $A= PBP^T$. Define $\mathrm{diag}(A)$ to be the ...
2
votes
2
answers
73
views
$E$ is finite-dimensional Euclidean space over $\mathbb{R}, x \in E, x \neq 0$. Then $\{Ax: A = A^* \succeq 0, \|A\| \leq 1\}$ is a closed ball
Let $E$ be a finite-dimensional Euclidean space over $\mathbb{R}$ and let $x \neq 0$ be a vector in $E$. Show that the set $K=\{Ax: A = A^* \succeq 0, \|A\| \leq 1\}$ is the closed ball of radius $\...
- 3,023
0
votes
0
answers
39
views
Equivalence for a matrix to be symmetric
Let $A\in\mathbb R^{n\times n}$ with the QR-decomposition $A=QR$ ($Q$ orthogonal, $R$ upper triangular matrix). I asked myself if the following statement holds:
$$A\in\mathbb R^{n\times n}\text{ ...
- 21
0
votes
1
answer
55
views
$A-B$ semi positive definite can tell us the information of the positive inertia index
Suppose $f,g$ are symmetric real quadratic forms on $\mathbb{R}^n$ such that $f(x)\geq g(x)$ for all $x\in\mathbb{R}^n$. Prove that the positive inertia index of $f\geq$the positive inertia index of $...
0
votes
0
answers
48
views
Inverse and Determinant of Matrix $Axx^TA+cA$
Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
- 2,953
0
votes
0
answers
33
views
How to compute Sylvester form of a matrix representing a symmetric bilinear form?
Can somebody state a step-by-step algorithm to, given a symmetric n x n-matrix A, (congruently) diagonalize A such that the entries of the diagonal are 1, -1 and 0 corresponding to the signature of A ...
- 11
1
vote
0
answers
36
views
Singular values as min max of absolute rayleigh quotient
Consider a real symmetric matrix M, satisfying $\mathbf{1}^\intercal M = \mathbf{1}^\intercal$ having eigenvalues $1=\lambda_1 \gt\|\lambda_2\| \geq \|\lambda_3\| .... \geq \| \lambda_n\|$, then can I ...
0
votes
1
answer
31
views
Lyapunov Special Symmetric Case [closed]
Consider the Lyapunov equation: $$AX+XA = B$$ and assume that $A$ is symmetric positive definite and $B$ is symmetric.
I am not able to proof that $X=X^{T}$ holds.
Would be grateful if somebody could ...
- 19
1
vote
1
answer
84
views
Proving rank of $A-\lambda vv^t $ is $r-1$.
Let $A$ be an $m\times m$ symmetric real matrix of rank $r$ s.t. $r\ne m$.If $\lambda$ nonzero is an eigenvalue of $A$ with corresponding unit column vector $v$ s.t. $Av=\lambda v$.Then prove the ...
user1313860
0
votes
2
answers
54
views
Writing convention of Courant–Fischer theorem
Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
user1290851