Questions tagged [symmetric-matrices]
A symmetric matrix is a square matrix that is equal to its transpose.
1,925 questions
2
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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
68
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Condition for symmetry of $A'(x)x$
I have a differentiable matrix function $x\in\mathbb{R}^n \mapsto A(x)\in\mathbb{R}^{n\times n}$ where $A(x)$ is symmetric for every $x$. Since $A$ is differentiable the Jacobian $A'$ exists:
$$(A'(x))...
- 2,386
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votes
0
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16
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Computing inverse of a matrix without needing too much memory
I have matrix that is $36\times36$ and it has a structure as below
$$
M=
\left[ {\begin{array}{cc}
B1 & B2 & B3 & B4 & B5 & B6\\
B2^H & B1 & B2 & B3 & B4 &...
- 369
1
vote
1
answer
47
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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
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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
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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
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0
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82
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A conjecture about the convexity of matrix functions
$f: S_{++}^n \to \mathbf{R}$ is given, where $S_{++}^n$ denotes the set of all $n$-dimensional positive definite matrices.
Suppose that $f$ is convex when $n=1$.
The conjecture is whether $f$ is ...
- 21
1
vote
1
answer
69
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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
$$
\mathbf{A}_{ii} = 1, \quad \mathbf{A}_{ij} \in [0,1], \quad i,j \...
- 514
2
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0
answers
53
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Decomposition of symmetric positive semi-definite matrices
Fix $A\in\mathrm{M}_n(\mathbb{R})$ a symmetric positive semi-definite matrix. Then there exists a matrix $M\in\mathrm{M}_n(\mathbb{R})$ such that
\begin{equation}
A=MM^\top+M^\top M.
\end{equation}
...
- 531
0
votes
1
answer
87
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Can any matrix be decomposed into the product of a symmetric matrix and another matrix? [closed]
For any matrix $X\in\mathbb R^{m\times n}(m\leq n)$ that has full rank, i.e. $rank(X)=m$,
can we find 2 matrices $Y\in\mathbb R^{m\times m}$ and $Z\in\mathbb R^{m\times n}$,
s.t. $X=YZ$, $\enspace ...
- 65
2
votes
0
answers
27
views
Can the Jordan algebra of equivariant maps on symmetric matrices generated by constants and endormorphisms map any two compatible elements
Let's $G$ be a subgroup of the orthogonal matrix group $O(n)$ acting by conjugation on $V$, the space of real symmetric matrices of size $n \times n$.
Let's endow $V$ with an (non-associative) algebra ...
- 360
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27
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Is there an explicit formula for the eigenvalues and eigenvectors of this particular symmetric matrix? [duplicate]
Consider a symmetric $n\times n$ matrix with the elements $A_{ij}=\min(i,j)$, i.e.:
$$A=\begin{pmatrix}
1 & 1 & \cdots & 1 \\
1 & 2 & \cdots & 2 \\
\vdots & \vdots & \...
- 338
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1
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38
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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
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1
answer
40
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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?
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0
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answers
36
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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
$$
A=
\...
- 15
2
votes
0
answers
51
views
Lower Bound on the Minimum Eigenvalue of a Hollow Symmetric Matrix with Entries in $[0,1]$
Let $\mathbf{A}$ be a hollow symmetric real $n \times n$ matrix, where each entry $A_{i,j} \in [0,1]$ and $A_{i,i} = 0$ for all $i$. I would like to establish a lower bound for the minimum eigenvalue $...
- 51
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Showing the bounds of two specific symmetric matrices
Let $M$ be an $n\times n$ symmetric matrix, partitioned as:
$$M = \begin{bmatrix} A& B\\ B^T&C\end{bmatrix}$$
where $A$ is $k\times k$. Let $\alpha = \sum\limits_{i,j=1}^k a_{ij}$, $\beta = \...
0
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28
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A question about tridiagonal complex symmetric matrix
I discovered a pattern when doing numerical experiments, a matrix with following form (tridiagonal, symmetric, real main diagonal,imaginary secondary diagonal):
$$
M=
\begin{bmatrix}
k_1 &ig_1 &...
- 1
0
votes
1
answer
43
views
Trace of product of symmetric matrix, Gram matrix and outer product of vectors
Suppose $u^1$, $u^2$, $u^3$ and $w$ are vectors in $\mathbb{R}^3$. Let $B$ be the matrix with entries $w_i w_j$ and $C$ be the matrix with entries the inner product $u^i \cdot u^j$. For the identity ...
- 417
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0
answers
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Approximating the eigenvalues of the multiplication of two matrix
I want to give the upper bound of the eigenvalue decay of a matrix $C=A^{-1} B$, to be more specific the upper bound of $\frac{\lambda_j}{\lambda_1}$ for each j. $\lambda_1\geq\lambda_2\geq\cdots\geq\...
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
$$Q_n :=\begin{bmatrix}
1-\frac{ε}{2} & \frac{ε}{2} & & &...
- 3,176
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votes
0
answers
50
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Eigenvalues of the product of diagonalizable matrices
I am trying to understand how I can compute the eigenvalues of $A^{-1}B$ where $A$ and $B$ are diagonalizable matrices (in patricular, they are symmetric tridiagonal matrices).
I know that $A$ can be ...
0
votes
1
answer
50
views
Efficient inversion of $A^T A + B^T B$
Consider matrices $A \in \mathbb R^{k \times n}, B(\eta) \in \mathbb R ^{l \times n}$, where $B$ depend on some parameters $\eta$. The matrix $A$ will be fixed.
My problem is to compute
$$
(A^TA + B(\...
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0
answers
51
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Rank of a random matrix $A=B^T B$ where $ B$ is a random matrix
I am working on the rank of a random matrix (by random, I mean each element of the matrix could come from a certain distribution (could be normal)). Suppose $A= B^T B$, where $B$ is a $n \times m$ ...
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votes
0
answers
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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 $\mathbf{X}\in\mathbb{R}^{n\times m}$.
$$
\mathbf{X}^\top\mathbf{A}\...
- 71
0
votes
0
answers
32
views
Norm inequality about positive definite matrix, along with its symmetric form and skew symmetric form
I have difficulty in solving the following problem (P4.2.3 in 《Matrix Computations》)
Suppose $A\in\mathbb{R}^{n\times n}$ be positive definite, and set T = (A+$A^T$)/2, show that $||A^{-1}||_2\leq ||T^...
- 1
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votes
0
answers
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Limit of a matrix function
I am going through a proof from this book. More specifically, I am looking at the last paragraph on page 62 (78 in the PDF). I will reproduce it here for convenience:
It claims that for a full rank ...
- 53
1
vote
0
answers
43
views
A relation about norms of a block matrix.
Suppose $M =
\begin{bmatrix}
A & B \\
B^{T} & C \\
\end{bmatrix}$,
where $A$ and $C$ are $n\times n$ real symmetric matrices and $B$ is an arbitrary $n\times n$ real matrix. By calculating ...
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1
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0
answers
81
views
Generalized Inverse of Submatrices
Consider a symmetric and positive semi-definite $n\times n$-matrix $\Sigma$. For $k = 1,\dots, n$, let $\Sigma_k$ denote $\Sigma$'s Leading Principal Minor of of order $k$, i.e., the submatrix of $\...
- 208
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votes
2
answers
135
views
Finding the determinant of a matrix under given conditions
Let $x_1, x_2, x_3, x_4, x_5 > 0$ and $A = [a_{ij}]_{5 \times 5}$ be a matrix such that
$$
a_{ij} =
\begin{cases}
|x_i - x_j|, & \text{if } i \neq j \\
x_i, & \text{if } i = j
\end{cases}
$...
- 509
0
votes
2
answers
59
views
Is the matrix $(Id - JQ)$ regular for any skew-symmetric matrix $J$ and symmetric positive semi-definite matrix $Q$? [duplicate]
Consider the matrix $(Id - JQ)$, where $Id \in \mathbb{R}^{n \times n}$ is the identity matrix, $J^T = - J \in \mathbb{R}^{n \times n}$ is a skew-symmetric matrix and $\mathbb{R}^{n \times n} \ni Q^T =...
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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
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1
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Number of Independent Components of a totally symmetric rank-four tensor
I saw a question that asked to derive the number of independent components of a rank-four tensor in d dimensions. I know the formula $\frac{(d+r-1)!}{(d-1)!(r!)}$, where d is the number of dimensions ...
- 101
1
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1
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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:
$$A = \left[ \begin{matrix}
a & -b& -c& d \\
b & e& -d& -f \\
c & -d& a& -b \\
d & f& b & ...
- 167
3
votes
1
answer
83
views
For what $A$ the linear map $f(B):=AB^T+BA^T$ surjective, defined as a linear map between suitable vector spaces defined below?
Let $V$ denote the space of $m\times m$ real square matrices. Let $W$ denote the space of $m\times m$ real symmetric square matrices. For an $m\times m$ square (not necessarily symmetric) matrix $A,$ ...
- 2,556
0
votes
0
answers
18
views
Kronecker product from vector half
I need to compute $D_p^\top(\Sigma\otimes \Sigma)D_p$, where $D_p$ is the duplication matrix and $\Sigma$ is a $p\times p$ symmetric matrix. When $p$ is large, the kronecker product becomes very large ...
0
votes
0
answers
36
views
Matrix with equal row sums and column sums
Background: While studying (finite) Markov chains I came across a characterization, different than the one using a transition matrix $p_i(j)$. Namely, we can specify the "diagram" with ...
- 839
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votes
2
answers
74
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Maximal eigenvalue of a real symmetric Toeplitz matrix, especially in the limit
Hi: In a research project I stumbled upon real symmetric Toeplitz matrices of the following type:
$$
A_n =
\begin{bmatrix}
a_1 & a_2 & a_3 & \cdots & a_n\\
a_2 & a_1 & a_2 &...
- 53
2
votes
0
answers
24
views
What is the criterion for a symmetric complex matrix to be orthogonal-diagonalisable?
For a symmetric matrix $M\in\mathbb{C}^{n\times n}$ to be orthogonal-diagonolizable, I mean that there exists $P\in\mathbb{C}^{n\times n}$ such that $PP^T=I$ and $PMP^T$ is diagonal.
Note that there ...
- 1,479
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votes
0
answers
66
views
The Limit of Power Law Toeplitz Matrices
Let $\mathbf A$ be an $N \times N$ symmetric Toeplitz with entries $A_{ij} = (1+|i-j|)^{-\chi}$ for an exponent $\chi$ above zero. Numerically, I can see that the minimum eigenvalue of $\mathbf A$ ...
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
0
answers
29
views
How to obtain the row sum of the inverse of a symmetrix Toeplitz matrix?
Given a (real) symmetric Toeplitz matrix $A$:
$$
A = \begin{bmatrix}
a & b & c & \ldots & y & z \\\
b & a & b & \ddots & & y \\\
c & b & \ddots & \...
- 1,963
1
vote
1
answer
51
views
Suppose that the eigenvalue of largest magnetude of $P$ is simple. Is it true that the the same happens to $U P$ for any unitary matrix $U$?
Suppose that $P = [p_{i,j}]_{1\leq i,j\leq n}$ is an $n\times n$ matrix such that:
$p_{i,j}\geq 0$ for all $i,j\in\{0,1,\ldots,n\};$
$\sum_{j=1}^n p_{i,j}=1$;
$P$ is a symmetric matrix, $p_{i,j}=p_{j,...
- 3,176
0
votes
1
answer
39
views
Eigenspace of sum of outer product.
I am investigating the eigenspace of the matrix:
$$ \frac{1}{k}\sum_{i=1}^k \mu_i \mu_i^T + \sigma^2I \in \mathbb{R}^{d\times d}$$
where $k < d$.
I want to ask if the eigenspace $V_k$ spanned by ...
- 83
0
votes
0
answers
27
views
Indefinite matrix and congruence transformation
Let $a,b \in \mathbb R$ and consider the matrix
$$M =\begin{bmatrix} a-1 & a \\ a & b+a-1 \end{bmatrix}. $$
I want to derive the sharpest conditions possible for $a,b$ for which the matrix $M$ ...
- 414
3
votes
2
answers
113
views
What is the connection between bilinear and quadratic forms.
I know that a bilinear form $B$ on the $\mathbb R$-vector space $\mathbb R^n$ is defined to be a map $B:\mathbb R^n\times \mathbb R^n\to \mathbb R$ which is linear in each coordinate.We know that a ...
- 3,801
0
votes
2
answers
100
views
Quadratic form of a real symmetric matrix is bounded
If $\lambda_1>\lambda_2>...>\lambda_r$ are the different eiegenvues of a real symmetric matrix $A\in M_{n\times n}(\mathbb{R})$.
$1.$
Show that the quadratic form associated to $A$ satisfies ...
- 1,392
3
votes
0
answers
52
views
Eigenvalues of product of diagonal matrices and Sylvester-Hadamard matrices
Set $n=2^k$ (for some integer $k$) and let $D={\rm diag}(d_1,d_2,\cdots,d_n)$ and $D' = {\rm diag}(d_1', d_2
,\cdots, d_n')$ be two diagonal matrices in $\mathbb C^{n \times n}$. Let us also presume ...
- 1,446
0
votes
0
answers
32
views
Reasoning for reduced SVD factorization
I am aware that for any $m \times n$ matrix $A$, we can write:
Known 1: $A = U\Sigma V^T$ where $U$ is orthogonal and $m \times m$, $V$ is orthogonal and $n\times n$, and $\Sigma$ is diagonal and $m \...
- 611
2
votes
1
answer
58
views
Rank of a matrix $A$
Let $$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_{...
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