Questions tagged [matrices]
For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.
53,649
questions
-2
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I have no idea how to work on this problem.
I need to find the change of basis matrix for bases $$B = \{\cos x, \sin x\},\hspace{0.5cm} B’ = \{e^{ix}, e^{-ix}\}$$ for the sub space $V = ker(D^2 + I)$ with $I$ being the identity map $$I = D(f)(x)...
- 7
0
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0
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19
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showing the vector form of the solution
Example of a matrix form of a system of linear equations
I am studying for linear algebra exam and there is one part I do not understand in this example. Why is the vector form of the solution: $$sx_1 ...
- 1
0
votes
1
answer
16
views
Moments of a complex matrix
How can I have higher order moments, for example, the skewness of a complex matrix, in terms of the trace and eigenvalues? As in this paper https://doi.org/10.1016/0024-3795(80)90258-X, they ...
0
votes
0
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19
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Change of basis in homogeneous coordinates
I am currently trying to understand the following Python code which computes a 3D rotation-translation matrix in homogeneous coordinates :
...
- 1,637
0
votes
0
answers
33
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How to mathematically model in MATLAB?
I am trying to create a code for the following statement in MATLAB:
Suppose there are 60 small base stations (small mobile towers) and 1 Macro base station (big mobile tower). Let us index these nodes ...
- 21
0
votes
0
answers
27
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Positive Definite Operators
Consider the symmetric linear operator A, for which its eigenvectors form a complete orthogonal set and all its eigenvalues are positive. Show that A is a positive definite operator.
Show that the ...
- 31
0
votes
0
answers
47
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Finding the rank of a 4 by 5 matrix
Problem:
Find the rank of the following matrix:
$$
\begin{bmatrix}
1 & 3 & -2 & 5 & 4 \\
1 & 4 & 1 & 3 & 5 \\
1 & 4 & 2 & 4 & 3 \\
2 & 7 & -3 &...
- 3,559
1
vote
2
answers
42
views
Efficient way to compute average of outer products.
I have a ser of vectors $v_i \in \mathbb{R}^d$, with $i \in \{1, ..., n\}$. I have to compute a weighted average of the outer products of these vectors:
\begin{equation}
\sum_{i=1}^{n} v_i v_i^T w_i
\...
- 21
0
votes
2
answers
31
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What are the conditions for a non-Hermitian matric's Rayleigh quotient to be less than its maximum eigenvalue?
What I am trying to prove is that: for a row-stochastic matrix P (not necessarily symmetric), whose every row sums $\sum_{j=1}^nP_{ij}=1$, the Rayleigh quotient of P, $R(\mathbf{x})=\frac{\mathbf{x}^{\...
0
votes
0
answers
17
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$2$-norm of a matrix obtained through discretization of the eigenvalue problem
Define $$A = \frac{1}{2h} \begin{pmatrix}
0 & 1 & & & & -1\\
-1 & 0 & 1 & & \\
&\ddots & \ddots & \ddots \\
& & & -1 & 0 & 1 \\
1 &...
- 1,693
0
votes
1
answer
25
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How can I decompose this matrix into the linear or bilinear product of two matrices?
Given the $(m_1 + m_2) \times (n_1 + n_2)$ block matrix $$A = \begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix}$$ where $A_{11} \in \mathbb R^{m_1 \times n_1},A_{12} \in \mathbb R^...
- 990
1
vote
0
answers
35
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Quick exponentiation of bit matrices
Is there a method for quickly rising to a power a matrix with only 0s and 1s? I am aware of the diagonalization method. However, it is general and requires a lot of work. Due to the constraint I ...
- 996
0
votes
0
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13
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Given that the columns of $A$ are linearly independent and that $P=A(A^TA)^{-1}A^T$, show that $C(P) = C(A)$
I need to show that the column space of a projection matrix of $A$ and the column space of $A$ are same. How could I prove it?
0
votes
0
answers
24
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Proof of a matrix inequality with sorted column vectors and permutation matrices
Consider two coloum vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$. The elements in vectors $x$ and $y$ are sorted (in ascending or descending order, but the sorting order of the two vectors ...
- 1
1
vote
0
answers
12
views
Find the maximum value of a linear system c = A*b (matrix multiplication)
Assuming there is a square matrix A and vector b about to be multiplied.
[A]{b}={c}
However I am interested only in the maximum value within the resulted vector c.
Is there a way to find that without ...
0
votes
0
answers
8
views
Find a "smaller" positive semidefinite rank 1 matrix
Given a positive semidefinite matrix $A\in \mathbf{C}^{n\times n}$ and a vector $x\in \mathbf{C}^{n\times 1}$, I would like to find a positive semidefinite matrix $B\in \mathbf{C}^{n\times n}$ such ...
- 27
1
vote
1
answer
39
views
Software for symbolic matrix operations
Is there a software for symbolic manipulation that can treat matrices as whole variables?
For instance:
solve A = B + C * A for A
where ...
- 11
0
votes
0
answers
18
views
Find out if eigenvalues and eigenvectors belong to a matrix, and if yes, find the matrix
So in my Linear Algebra course I am given the following eigenvalues with their eigenvectors and I have to find out if they belong to a matrix, and if yes, to find the matrix. So this is given:
$\...
0
votes
0
answers
9
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Need help in understanding the proof for Non-invariance of DLT Algorithm in the book by Richard hartely and Andrew Zisserman.
Background:
In section 4.4.2 of the book titled, "Multiple View Geometry for computer vision" on page 106, the author writes the proof, $$\tilde{A_i}\tilde{h}=(\tilde{\epsilon_{i1}},\tilde{\...
1
vote
1
answer
20
views
matrix representation for $φ: \mathbb{R}[t]→ \mathbb{R}^2 f(t) → \begin{pmatrix} f(1) − f(0)\\ f(2) − f(1) \end{pmatrix})$ to the basis$(1,t,t^2,t^3)$
I am having problems with this exercise. Can someone help me?
a) Determine the matrix representation for $φ: \mathbb{R}[t]degree≤3 → \mathbb{R}^2 f(t) → \begin{pmatrix} f(1) − f(0)\\ f(2) − f(1) \end{...
- 466
-1
votes
0
answers
26
views
Help with solving linear system matrices
I am currently studying linear algebra and I am having trouble with some exercises related to linear system matrices. I would appreciate any help or guidance on how to approach these questions.
The ...
- 117
0
votes
0
answers
23
views
Existence of at most one matrix $A \in M_{2,3}(\mathbb{R})$ such that $BA =\operatorname{diag}(1,1,0).$
Hey I want to check my solutions to this exercise:
a) Let $B\in M_{3,2}(\mathbb{R})$ be a matrix. Show that there is at most one matrix $A \in M_{2,3}(\mathbb{R})$ such that $BA=\begin{pmatrix} 1 &...
- 466
2
votes
1
answer
33
views
Determine the mapping matrix $A=M_B^B$ with the base $B=(1,t,t^2,t^3)$
Hey I have some problems with this problem
Consider all polynomials with $\leq \deg 3$ and the following map:
$f: \mathbb{R}[t] \rightarrow \mathbb{R}[t]$
$f(t) \rightarrow f(t+1)-f(t)$
a) Determine ...
- 466
0
votes
1
answer
19
views
Does this non-symmetric have only positive principal minors?
In a recent post (Is this non-symmetric matrix positive definite?), I asked whether the following non-symmetric $n \times n$ matrix:
$$ A = \begin{bmatrix}
1 & \delta_1 & \dots & \delta_1\\...
- 51
3
votes
3
answers
103
views
Derivative of $\varphi({\bf X}) = \sum_{i=1}^n \lambda_i({\bf X}) \log \lambda_i({\bf X})$
Let $\mathbb{S}_+^n$ denote the set of $n \times n$ symmetric positive definite matrices. Let scalar field $\varphi : \mathbb{S}_+^n \to \Bbb R$ be defined by
$$\varphi({\bf X}) := \sum_{i=1}^n \...
-4
votes
0
answers
21
views
Two functions f(x) and g(x) in Euclidean space are orthogonal, so how are |f(x)| and g(x) related? [closed]
Two functions $f(x)$ and $g(x)$ in Euclidean space are orthogonal, so how are $|f(x)|$ and $g(x)$ related?
0
votes
1
answer
49
views
If $M\in M_{m \times n}(\mathbb{R})$ is a rank $r \geq 1$ matrix then it can be written as a sum of exactly $r$ rank 1 matrices
I have problem with the proof of the following statement.
If matrix $M\in M_{m \times n}(\mathbb{R})$ has rank $r \geq 1$, then it can be written as a sum of exactly $r$ rank-$1$ matrices
I try in ...
- 11
2
votes
0
answers
36
views
Under what conditions is the matrix exponential $ e^A $ symplectic?
Let $ A \in \Bbb R^{2n \times 2n}$ and let the matrix exponential $ e^A $ be defined by
$$ e^A := \sum_{k=0}^{\infty}\frac{A^k}{k!}.$$
I want to ask under what conditions the matrix $ e^A $ is ...
1
vote
1
answer
30
views
Trace of off-diagonal blocks of a positive semidefinite matrix
Consider the matrix
$$A=\begin{pmatrix} A_1 & A_2 \\ A_3 & A_4 \end{pmatrix}$$
Let's suppose that $A$ is a real $n\times n$ positive semidefinite and satisfies $\|A\|\leq 1$, i.e., the largest ...
- 538
0
votes
1
answer
66
views
Determinant of a matrix is equal to the determinant of its transpose
For a square matrix A, I want to show that $$ | \mathbf{A}^T| = |\mathbf{A}| $$
Proof:
Let the same permutation that changes $ \varphi(j_1,...,j_n) $ into $ \varphi(1,...,n) $, change $ \varphi(1,...,...
1
vote
0
answers
40
views
Bounding spectral norm of the matrix $f(\Sigma)$
Suppose that we have a sequence of matrices $\{\Sigma_n\in\mathbb{R}^{n\times n}\}_{n=1}^\infty$ with uniformly bounded spectral norm, that is to say, the sequence $\{\|\Sigma_n\|\}_{n=1}^\infty$ is ...
- 41
0
votes
1
answer
51
views
Linear way to determine a matrix is invertible
I am working with a linear program and I am trying to come up with a symmetric matrix $S$ whose determinant is non-zero (alternatively that the matrix is invertible).
Calculating the determinant is ...
- 121
0
votes
0
answers
21
views
Elementary Column and Row operations for matrices with polynomial entries.
I'm not entirely sure how to perform elementary column and row matrices with polynomial
entries. I can do it for matrices with integer entries.
For instance, I was looking at a problem regarding ...
- 39
0
votes
0
answers
17
views
Non-symmetric positive semi-definiteness in literature
The article Density matrix on Wikipedia includes the following sentence
Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ...
- 779
0
votes
0
answers
49
views
Trace of matrix logarithm for two invertible matrices
Consider an unitary matrix $U$ and an positive definite, invertible and diagonalizable matrix $\rho$ .
Then, if the following identity holds (i.e., if there are additional $2\pi I$ factor),
\begin{...
- 1
1
vote
1
answer
47
views
Correctly accounting for Hadamard product
I have an equation of the following form:
$$ \frac{\partial a}{\partial t} = Da + Fe^{i\mu \theta}$$
where $a = a(\theta,t)$ is a function and $D$ is a linear operator.
When discretized for solving ...
- 383
0
votes
0
answers
39
views
Understanding proof of convexity of trace function
I am trying to understand Theorem 2 ($F(A,K) := \text{tr}(A^{-r}K^{\ast} A^{-p} K)$ is convex for $p,r \geq 0$ and $p+r \leq 1$) of the paper Convex trace functions and the Wigner-Yanase-Dyson ...
- 487
0
votes
2
answers
58
views
Given $X, Y \in \Bbb C^{n \times n}$ and $XY+2YX=3I$, show $[X,Y]$ is nilpotent
Given $X, Y \in \Bbb C^{n \times n}$ and $XY+2YX=3I$, show that $[X,Y]$ is nilpotent.
I tried with Jacobson Theorem but it does not work. Trying to calculate traces of powers by induction to see if ...
- 1,011
-1
votes
0
answers
10
views
Unitary similarity of a 2x2 matrix with an upper triangular matrix
Let $A \in \mathcal{M}_2(\mathbb{C})$ be a non normal matrix with eigenvalues $1$ and $-1$.
I've read in a proof that such a matrix is unitary similar to $\begin{pmatrix}1 & 2c \newline
0 & -...
1
vote
1
answer
24
views
Projection matrix product under operator norm
I've come across a question that is similar to this one, but somewhat "reversed" in the role of the unitary matrix.
Suppose I have matrix product $A\, P\, B$ where $A$, $B$ are generic real ...
- 93
0
votes
0
answers
56
views
Stuck while differentiating $x\mapsto\|x\|^2 + \lambda{\|Ax - b\|}^2$
I want to differentiate the following equation
$$ x \mapsto \|x\|^2 + \lambda{\|Ax - b\|}^2 $$
where the real symmetric matrix $A$ is not invertible. I have decomposed $A$ using the spectral theorem, ...
- 1
-2
votes
0
answers
33
views
Finding an orthonormal basis that minimizes a quadratic cost function
Given $\mathbf{A} \in \mathbb{R}^{m \times n}$, how to find a orthonormal basis $\mathbf{s}_1, \dots, \mathbf{s}_n \in \mathbb{R}^n$ such that
$$ \min_{\mathbf{s}_1,\dots,\mathbf{s}_n} \sum_{1=1}^n \...
0
votes
0
answers
25
views
Proving a set of matrices has strictly negative eigenvalues
I have a sequence $(A_1, A_2, \dots)$ of tridiagonal matrices of the form
$$A_n = \begin{pmatrix} a_1 & b_1 & 0 & \cdots & \cdots & 0
\\ c_1 & a_2 & b_2 & \ddots &...
- 35
0
votes
0
answers
11
views
Proof for <<Having pivot positions in all rows is equivalent to spanning all the space R^m>>
I know that for an equation like $Ax=b$ to have at least one solution for all $b \in \Bbb R^m$, there must be pivot positions in each row of the matrix $A$, But I'm struggling with proving this ...
0
votes
0
answers
25
views
Infinity norm and operator norm question
Suppose a matrix $A$ and a vector $v$ are given for appropriate size. In the Euclidean norm $\| \cdot \|_2$, we know that $$\|A v\|_2 \le \|A\|_{\text{op}} \cdot \|v\|_2$$ where $\|A\|_{\text{op}}$ is ...
- 87
0
votes
1
answer
51
views
Eigenvalues of an $n \times n$ symmetric tridiagonal matrix
How to show that an $n \times n$ tridiagonal matrix only has positive eigenvalues?
$$ \begin{pmatrix}
70 & -35 & 0 & \dots & 0 \\\
-35 & 120 & -35 & \ddots & \...
- 3
1
vote
1
answer
182
views
How to solve this least-squares-like problem?
Given the finite set of matrices $\Bbb X := \{ {\bf X}_1, {\bf X}_2, \dots, {\bf X}_N \} \subset \mathbb{C}^{N_t \times L}$ and the matrix ${\bf Y} \in \mathbb{C}^{N_r \times L}$,
$$ \left( \hat{H}, \...
- 21
2
votes
1
answer
54
views
Comparative statics in linear programming
Suppose $\Phi$ is a $3 \times 3$ non-negative matrix and let $\mu(\Phi)$ be a solution to the following maximization problem over a nonnegative matrix,
$$\begin{aligned}
\max_{(\mu_{ij}) \geq 0}& &...
- 31
0
votes
0
answers
29
views
Why are we looking for the lowest rank matrix in matrix completion tasks?
As the Wikipedia page on matrix completion states,
One of the variants of the matrix completion problem is to find the lowest rank matrix $X$ which matches the matrix $M$, which we wish to recover, ...
- 839
1
vote
1
answer
40
views
Eigenvalue Localization
Let $\mathbf{A} \in \mathbb{C}^{n \times n}$ be a normal matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Show there exists a permutation $\pi$ of $1, \dots n$ so that $$ \sum_{i=1}^n\left|a_{i ...
- 79