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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.

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How to compute $RA$ in $O(n^2)$ operations instead of $O(n^3)$ using Householder reflection

I am currently writing a program that performs QR decomposition on a matrix $A$. The guidelines to my assignment tell me that once I calculate $R$ using Householder reflections, there is a way to ...
Amrutha Paval's user avatar
1 vote
3 answers
40 views

calculating the determinant of a matrix with $1+a_i$ on the main diagonal [duplicate]

How should we verify the following determinant equality? $$ \left|\begin{array}{cccc} 1+a_{1} & a_{2} & \cdots & a_{n} \\ a_{1} & 1+a_{2} & \cdots & a_{n} \\ \vdots & \...
flower7182's user avatar
-1 votes
0 answers
29 views

Permutations in determinants

While solving the system of three linear equations in three unknowns through gaussian elimination. I was able to compute the determinant of a 3×3 matrix. But this algorithm doesn't shed any light on ...
Alv's user avatar
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1 vote
1 answer
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Projection of vector

The projection of a vector $x$ onto a vector $u$ is $proj_u(x) =\frac{\langle x, u \rangle}{\langle u, u \rangle}u.$ Projection onto $u$ is given by matrix multiplication $proj_u(x)=Px$ where $P=\frac{...
Brett's user avatar
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1 answer
25 views

On the sum of definite matrices

Let $A,B$ be two $n\times n$ symmetric positive semi-definite matrices. I have the intuition that: $$A-B$$ has, at most, $\operatorname{rank}(A)$ positive eigenvalues, and at most $\operatorname{rank}(...
a06e's user avatar
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1 vote
2 answers
32 views

Eigenvalues and eigenvectors of certain rank two Hermitian matrix

Let $n\geq 2$ and $A\in\mathbb{C}^{n\times n}$ such that $$ A = \begin{bmatrix} 0 & x^*\\ x & 0 \end{bmatrix} $$ for some $x\in\mathbb{C}^{n-1}\backslash\{0\}$. Is there an explicit form for ...
PIII's user avatar
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0 answers
11 views

matrices with entries in a division ring is simple [duplicate]

I want to proof that matrices with entries in a division ring is simple. From the website, I find that left ideal of $M_n(D)$ is bijection to the D submodule of $D^n$, where D is division ring, so my ...
lee's user avatar
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1 answer
30 views

The inner product of column vectors is the same as matrix multiplication

I am very much new on the topic of inner product: Definition. The inner product of vectors $x, y \in \mathbb{R}^n$ is $\langle x, y\rangle =\sum_{i=1}^{n} x_ky_k=x_1y_1+x_2y_2+\dots+x_ny_n$ I can't ...
Brett's user avatar
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2 votes
1 answer
110 views

Does there exist such matrix

Does there exist $a,b\in\mathbb{R}$ s.t $$\begin{pmatrix} a& -b\\ b & a\end{pmatrix}^{20}+23\begin{pmatrix} a & -b \\ b & a\end{pmatrix}=\begin{pmatrix} 20 & -23 \\ 23 & 20\...
lee max's user avatar
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5 votes
3 answers
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What problems will arise if we define matrix multiplication this way?

$a\in \mathbb{R}^n$ $a=(a_1,a_2,\dots,a_n)$ lets define this to be equivalent to $(a_1,a_2,\dots,a_n,0,0,0 \dots,0)$ (finite many zeros) by this I think we can make an $n \times m$ matrix $A$ ...
pie's user avatar
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0 votes
0 answers
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Convergence of power method for dominant complex eigenvalue

Suppose $A\in\mathbb{R}^{n\times n}$ is a diagonalisable matrix. Let the eigenvalues of $A$ are given by, $$|\lambda_1 | = |\lambda_2| > |\lambda_3| \geqslant |\lambda_4|\geqslant \cdots\geqslant |\...
Newrion's user avatar
  • 361
1 vote
2 answers
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How to find the basis of $U$?

Let $U=\mathrm{span}\{u_1,u_2,u_3,u_4\} \subseteq\mathbb{R}^4$ where $u_1=\begin{pmatrix} 2 \\ 4 \\ 4 \\ 2 \end{pmatrix};$ $u_2=\begin{pmatrix} 3 \\ 5 \\ 3 \\ 1 \end{pmatrix};$ $u_3=\begin{pmatrix} 3 \...
Brett's user avatar
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33 views

Smallest $k$ s.t. $\operatorname{rank} A^m=\operatorname{rank} A^k$ for all $m>k$.

Problem: Let $A\in \mathbb C^{n\times n}$ be a matrix. Determine the smallest number $k\in \mathbb N$ (dependent on $A$), such that $\operatorname{rank} A^m=\operatorname{rank} A^k$ for all $m>k$. ...
categoricallystupid's user avatar
1 vote
0 answers
12 views

Find appropriate rotation matrix

Consider the following elements in linear algebra: $\pmb{G}: L \times K$ matrix $\pmb{F}: T \times K$ matrix $\pmb{H} = \pmb{G} \pmb{F}^{\intercal}: T \times L$ matrix. The apex $\intercal$ denotes ...
user9875321__'s user avatar
0 votes
1 answer
27 views

Proving A*Α=E for chosen coefficients.

I have this problem : Let $ζ = e^{2πi} $ and $A \in Mat(n;\mathbb{C})$ so that $[A]_{jk} = \frac{ζ^{jk}}{\sqrt{n}}$ for every $j, k$. I have to prove $A^{*}A=E$. (Where $A^{*}=\bar{A}^t$.) This is ...
Ally's user avatar
  • 3
0 votes
1 answer
32 views

Matrix Transformation of a Polynomial

I'm having trouble understanding matrix representations of polynomials. The question in particular is: $$ f → (x^{2} − 1)f'' + 2xf'. $$ I know for this question we have to use the standard bases of n ...
KiS's user avatar
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1 answer
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An elementary inequality problem with 6 variables

I am trying to prove/disprove the following statement. Given $x,y,z \in \mathbb{R}$ and $0\leq a,b,c <1$ such that $x+y+z =0$ and $a^2x+ b^2y+c^2z<0$. Then $ax+by+cz<0$. In my context, $[x,...
abcxyzf's user avatar
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1 vote
1 answer
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A case problem about rank-1-perturbation of diagonal matrices

I have the following prediction for rank-1 perturbations of diagonal matrices, but I don't know how to prove (or disprove it). Given $v:= [v_1,...,v_K] \in (0,1]^K$, we define $a:= \sum_{k=1}^K v_k = \...
abcxyzf's user avatar
  • 171
1 vote
1 answer
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On invertibility of a real matrix

Given that $A$ is a real 'symmetric' matrix. I need to prove or find a counter example for the statement that $I+iA$ is always invertible, where $I$ is the identity matrix of the order same as that of ...
Yathi's user avatar
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-1 votes
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Solve for variable matrice in matrice equation (linear algebra)

Say I have three matrices $A (3\times3)$, $B (2 \times 2)$ and $C (3 \times2)$. The question is to find $X$ in the matrice equation $A^2XB = C$. But I don't quite understand how to solve this? My ...
janedoe121212's user avatar
1 vote
0 answers
39 views

If A,B,C are three matrices of order $2$ , such that $|A+B|−|C|=−10 , |B+C|−|A|=25 , |C+A|−|B|=15$ then value of $|A+B+C|$ is equal to

If $A, B, C$ are three matrices of order $2$, such that $|A+B|-|C|=-10$, $|B+C|-|A|=25$, $|C+A|-|B|=15$ then value of $|A+B+C|$ is equal to ($|A|$ denotes the determinant of matrix $A$) My Approach: ...
mathophile's user avatar
  • 3,555
2 votes
2 answers
127 views

Theorem on non-diagonalisable matrix

My professor gives me the theorem on non- diagonalisable matrices: Let a matrix the $A \in M_{n\times n}(\mathbb{R}).$ $A$ has $k$ independent eigen vectors $\Leftrightarrow$ A is similar to $$ \begin{...
Brett's user avatar
  • 121
0 votes
0 answers
58 views

Find th determinant of a matrix with x under diag, 1 on diag and numbers to n above diag

$$ \begin{pmatrix} 1&2&3&\cdots&n \\ x&1& 2 &\cdots&n-1\\ x&x&1&\cdots&n-2\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x&x&x&\...
kate's user avatar
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1 vote
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Proof that if $I, J \trianglelefteq R$ then $M_n (I) \cdot M_n (J) = M_n (IJ)$

I want to proof the following proposition. If $I, J \trianglelefteq R$ then the product of matrix ideals $M_n (I) \cdot M_n (J) = M_n (IJ)$. Let's consider the matrices $A = (a_{ij}) \in M_n (I)$ and $...
Nickeil's user avatar
  • 61
0 votes
1 answer
28 views

Property of the matrix operation to get reduced row echelon form?

If you have a matrix $A$, you can perform row operations to turn it into the reduced row echelon form $\hat{A}$. I believe you can encode these row operations into a single matrix $R$ such that $RA = \...
user3667125's user avatar
5 votes
0 answers
91 views

If $B^3=B$, is $B$ diagonalizable?

Let $B\in M_n(\mathbb{F})$ such that $B^3=B$. Is $B$ diagonalizable? If $B^3=B$, then $B^3-B=0$. Consider the polynomial $p(x)=x^3-x$. If $p(B)= B^3-B=0$. Since we know that the minimal polynomial of ...
user926356's user avatar
  • 1,346
-3 votes
0 answers
55 views

$lim_{ n \to \infty} [(1/2,1/2,0); (0,1/2,1.2);(1/2,0,1/2)]^n*[1,2,3]$ [closed]

this matrix is not diagonalizable over Real number, how should I approach now?
Ketan Choudhary's user avatar
0 votes
0 answers
29 views

How to find the general form of a vector acted upon by linear and non linear operations

I stumbled upon this problem, originally I considered it easy to solve, I tried, and failed, and now I am beginning to believe it is quite a formidable adversary; so I am posting here, in hope of ...
Noumeno's user avatar
  • 261
0 votes
0 answers
22 views

Doubling the numbers within an image matrix

I have an image Matrix tag that looks like this ...
MarkyMark's user avatar
1 vote
0 answers
23 views

Perron Eigenvalue versus Determinant for a class of circulant-like matrices.

Let $\mu_0(dx), \ldots, \mu_{k-1}(dx)$ be $k$ non-null (strict) subprobability measures on $[0,1]$, with max of their support $\eta_0, \ldots, \eta_{k-1}$. Let also $[n] \in \{0, \ldots, k-1\}$ be $n$...
Olivier's user avatar
  • 1,201
1 vote
1 answer
39 views

Forming a matrix (i.e., linear operator) based on implicit description of coefficients

I'm implementing an algebraic multigrid solver based on "A Multigrid Tutorial, 2ed", and in chapter 3, the formation of the prolongation operator (a matrix denoted $P = I_{2h}^{h}$) is ...
Jared Frazier's user avatar
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0 answers
53 views

Under what kind of conditions is the following matrix non-singular?

Given $m$ complex numbers $z_1,\ldots,z_m \in \mathbb{C}^m$, I am interested in the singularity of the following $2m \times 2m$ real matrix \begin{equation} M := \begin{bmatrix} 1 & \cdots & 1 ...
fs l's user avatar
  • 55
1 vote
2 answers
71 views

Jacobian matrix from mass action kinetic ODE network

I have an ODE system consisting of >1000 equations for which I would like to obtain the Jacobian matrix. For simplicity, let's consider the following example: ...
hr8tpa's user avatar
  • 11
1 vote
1 answer
48 views

Diagonalizability of a special matrix involving a parameter k

I'm taking a linear algebra course this semester and we just got our second assignment. I've been working on one particular problem and I think I've made some progress, but I want to make sure my ...
Arihant Tripathy's user avatar
4 votes
2 answers
389 views

What is canonical spectral theorem?

My teacher has given me the following definition of the canonical spectrum theorem: Let a matrix the $A \in M_{n\times n}(\mathbb{R})$, and set of eigenvalues, $\sigma(A)$={$\lambda_1$,$\lambda_2$........
Brett's user avatar
  • 121
1 vote
0 answers
31 views

Finite strictly decreasing sequence and positive semidefiniteness

Suppose we have finite strictly decreasing sequence $a_1 > a_2 > \dots > a_n$ of positive numbers. When will a matrix \begin{bmatrix} a_{1} & a_{2} & a_{3} & \dots & a_{n}...
user9264047's user avatar
3 votes
1 answer
40 views

Proving Similarity of 2 Matrices

I am given two matrices $U$ and $V$. I'm also given that $U=JK$ and $V=KJ$, where $J$ is an invertible matrix. I am supposed to prove similarity but I'm unsure if my proof is sufficient. $$V=KJ$$ $$VJ^...
Grey's user avatar
  • 394
-4 votes
0 answers
49 views

How to show that the inverse of a positive matrix $A$ can never be non-negative? [closed]

Consider a matrix $A \in \mathbb R^{n \times n}$ where $a_{ij}$>0 for $1\leq i,\,j\leq n$, i.e., the matrix $A$ is a positive matrix. How to show that its inverse can never be non-negative? I ...
Fathi's user avatar
  • 25
0 votes
1 answer
72 views

What does it mean to multiply rectangular matrices?

I understand matrix multiplication as linear transformation of one or more vectors based on the transformation matrix. This can be visualized when both the matrices are square and are of same ...
Pranay Varanasi's user avatar
0 votes
0 answers
26 views

Can I diagonalise a block matrix of this form analytically?

I have a block matrix $$ \left(\begin{matrix} A & B \\ B^{\dagger} & 0 \end{matrix}\right) $$ where $A$ and $B$ are Hermitian square block matrices. Is there a way to diagonalise this ...
Al141224's user avatar
0 votes
0 answers
37 views

How to Derive the Characteristic Polynomial of a Companion Matrix?

I am working on a problem involving the characteristic polynomial of a companion matrix and need some help understanding the derivation. Here is the matrix in question: $ C(p) = \begin{pmatrix} 0 &...
Herrpeter's user avatar
  • 1,326
0 votes
0 answers
23 views

How to maximize the sparsity of orthogonal matrix?

Given a unit vector $v\in\mathbb{R}^n$, it's needed to find an orthogonal matrix $Q\in \mathbb{R}^{n\times m}$ ($m \leq n$) of maximum sparsity. What are the bounds of sparsity for such matrix? What ...
Rusurano's user avatar
  • 792
0 votes
0 answers
44 views

Properties of onto transformation matrix

I totally disagree with the following statement given by David Lay in the book Linear algebra and its applications: If $T: \mathbb{R}^n \to \mathbb{R}^m$ maps $\mathbb{R}^n$ onto $\mathbb{R}^m$, then ...
Evgen's user avatar
  • 101
1 vote
1 answer
53 views

How many homomorphism are from Z to the ring of matrices?

Let be $(\mathbb{Z},+,\cdot)$ and $(\mathcal{M}_{2x2}(\mathbb{Z}) , + ,\cdot)$ rings, and $\phi: \mathbb{Z} \longrightarrow \mathcal{M}_{2x2}(\mathbb{Z})$ a function. How many ring homomorphisms $\phi$...
pucky's user avatar
  • 21
2 votes
1 answer
32 views

Is the product of the modulus of the largest entry in an invertible matrix and the modulus of the largest entry in its inverse at least 1?

Let $A$ be an invertible matrix with complex entries. Is it true that $$\left(\max_{i,j} |A_{ij}|\right)\left( \max_{i,j} |(A^{-1})_{ij}|\right) \geq 1,$$ where $(A^{-1})_{ij}$ is the $ij$th entry in $...
Eli Seamans's user avatar
-2 votes
1 answer
44 views

How do I prove that distinct eigenvalues produce distinct eigenvectors? [closed]

Given an n x n invertible matrix A and nonzero constants λ₁ and λ₂ with λ₁ ≠ λ₂, if there exist nonzero vectors v₁ and v₂ with ...
Pierre Carvan's user avatar
1 vote
1 answer
36 views

Upper bound of operator norm of Hadamard (Schur) product

As in Extension of the Schur product theorem to operators, we can define two compact self-adjoint integral operators $A$ and $B$ on $L^2(\Omega)$, by $$ A \phi(x)=\int_{\Omega} a(x, y) \phi(y) d y ; \...
user1247096's user avatar
0 votes
0 answers
42 views

Exponential decay of |A|, where $\rho(A) <1$. [closed]

Suppose that the map T: $\xi_1=\eta\left(\tau\left(\xi_0\right), \xi_0\right) $ is of the form $$ T: \xi_1=A \xi_0+\Xi\left(\xi_0\right), \quad \xi_0 \in \pi, $$ where $\Xi\left(\xi_0\right)$ is of ...
San Lam's user avatar
0 votes
0 answers
18 views

n-step of a symmetric doubly stochastic transition matrix [closed]

I am trying to explain a special 'transition matrix' that is symmetric and doubly stochastic. One important feature of this kind of matrix is that its n-step brings each value closer to a constant. ...
Wei Yan's user avatar
0 votes
0 answers
45 views

Symplectic approximation to a given matrix

I am interested in understanding methods to rigorously compute the closest symplectic matrix approximation to an arbitrary matrix $A$. A symplectic matrix $S$ satisfies the condition $ S^T J S = J $, ...
Dante Perès 's user avatar

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