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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
3 views

Gram schmidt swapping two vectors

The question has background here but it's really just a linear algebra question. Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process (with no normalization of vector) ...
jacopoburelli's user avatar
0 votes
1 answer
16 views

Derivative of trace involving product and function of matrix

I have the following matrix expression: $$\mathrm{Trace}\left( \mathrm{A}^\intercal f(\mathrm{X})\right)$$ In the above, $f(\mathrm{X})$ is given by: $$f(\mathrm{X}) = \left(\mathrm{B}^\intercal \...
HeyCool08's user avatar
0 votes
0 answers
23 views

prove that the matrix multiplication yields sufficiently small entries

Let's define the following matrices: \begin{align} \Sigma_t= \sum_{k=0}^{t-1} x_k x_k^T \in \mathbb R^{n \times n}, \end{align} \begin{align} \tilde\Sigma_t= \sum_{k=0}^{t-1} w_k x_k^T \in \mathbb R^{...
Fathi's user avatar
  • 23
1 vote
1 answer
75 views

What actually is an augmented matrix, and why do they work?

I'm finishing a course in applied linear algebra and realized I don't understand how we can augment matrices and why it works. I know their uses and have no issue with Gaussian elimination, finding ...
greysonbowser's user avatar
5 votes
2 answers
420 views

Is it possible to prove that this matrix is invertible?

I am trying to determine whether the following, $ n $ by $ n $ matrix: $$ A = \begin{pmatrix} \Delta_{11} - 1 & \Delta_{12} & ... & \Delta_{1n} \\ \Delta_{21} & \Delta_{22} - 1 & .....
Balázs Markó's user avatar
0 votes
1 answer
78 views

Why is $A^{-1}$ existing a necessary condition for $x$ to be unique in $Ax = b$? [duplicate]

Consider the equation $Ax = b$ where $A \in \mathbb{R}^{m \times n}$, and $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. Say we know $A$ and $b$. I am wondering why, in order for us to uniquely ...
Princess Mia's user avatar
  • 2,431
2 votes
1 answer
47 views

I want to bound the entries of the matrix $\mid A^{-1} \mid \mid A \mid$

I want to show that the entries of the matrix multiplication $\lvert A^{-1}\rvert \lvert A \rvert$, where $A\in \mathbb R^{n \times n}$ are less than one. The absolute value in $\lvert A \rvert$ is ...
Fathi's user avatar
  • 23
0 votes
2 answers
25 views

Defining majorization of vectors using doubly stochastic matrix

On Wikipedia, it says that given two non-increasing vectors $a,b\in\mathbb{R}^d$, $a$ majorizes $b$ if and only if there exist a doubly stochastic matrix $D\in\mathbb{R}^{d\times d}$ such that $b=Da$. ...
Sagi Buchbinder Shadur's user avatar
0 votes
1 answer
64 views

Nature of polynomials that are satisfied by a $n\times n$ matrix with degree greater than $n$

We all are very familiar with the fact that minimal polynomial and characteristics polynomial of a matrix is satisfied by the matrix itself. Now, I am stuck while doing this. a) If $A \in M(3,\mathbb{...
Debmalya Roy's user avatar
0 votes
1 answer
32 views

Linear transformation of equation using inverse matrix method

I can't seem to understand why the following method works: Find the transformation equation of $y=4x-3$ after undergoing the transformation matrix $$ T = \begin{bmatrix} 4 & 3 \\ 1 & -1 \\ \...
dravenpog's user avatar
0 votes
1 answer
66 views

Trace condition implies matrix is unitary

Question: Let $U_a$ ($a=1,2,\cdots,n^2$) be unitary $n \times n$ matrices, and suppose that there exists an $n \times n$ matrix $M$ (EDIT: w.l.o.g. we can restrict $M$ to be diagonal[2]) such that $$\...
Ruben Verresen's user avatar
-3 votes
0 answers
26 views

How to make matrix diagonally dominant for gauss-seidel [closed]

I have to solve this with Gauss-Seidel but because it's not diagonally dominant it does not converge. How can I make it happen? CA + CB + CC + CD = 1 /// -0.3CA + 0.02CB + 0.05CD = 0/// 0.1CA - 0.82CB ...
Kev Polk's user avatar
-1 votes
0 answers
41 views

Determinant of exponential matrix equal to exponential of the trace

I need to prove the identity: $\det(e^{A}) = e^{Tr(A)}$, directly from Taylor expansion and without use the eigenvalues. My attempt was: $\det(e^{A}) = 1 + \det(A) + \det(A^2)/2! \ + \ ... \ = 1 + \...
VladStoker's user avatar
2 votes
2 answers
152 views

Finding a basis of a Subspace

I have a subspace $U = \langle x^2-x+4,x-1,x^2+x \rangle $ of $P_2$ over $\mathbb R$. I need to find a basis of $U$. We know already that these $3$ vectors span $U$ so we need to check for linear ...
adisnjo's user avatar
  • 29
0 votes
2 answers
29 views

Specific value for an invertible matrix

I have been given the following matrix, and have been tasked with finding the values of "a" that makes it invertible. I know that for a matrix to be invertible, then the determinant of said ...
Markus J's user avatar
0 votes
0 answers
16 views

Perspective and warp transformations

I'm looking for how to map a rectangular image to perspective and warp transformation. An example of how it works can be found here: https://mockover.com/editor/ So the input is the 4 corners of the ...
Son Nguyen's user avatar
0 votes
1 answer
55 views

What does it mean when a system of Linear Equations have more than one solution?

Consider 3 linear equations where one is a linear combination of other two(which are not parallel). Say $a$, $b$ and $a+b$. Now $a+b$ is also a line right? Then how $a$, $b$ and $a+b$ can have more ...
Pranay Varanasi's user avatar
0 votes
0 answers
39 views

How to prove that a matrix smallest eigenvalue is positively correlated with some of its elements?

A block matrix: $$\bf{J}= \begin{bmatrix} (\bf{A}-\bf{Q}) & (\bf{P}-\bf{K})\\ \bf{P} & (\bf{A}+\bf{Q}) \end{bmatrix},$$ where $\bf{A}$ is an $n$-dim positive definite real matrix, and its non-...
Kuonji's user avatar
  • 1
3 votes
0 answers
92 views

Eigenvalues of a 8x8 Matrix

This is the 6th problem of the TACA in August, 2023. One have 2 hours to solve 15 problems. I am wondering how to calculate the eigenvalues of the following 8 by 8 matrix by hand. Note that this is ...
Yue Yu's user avatar
  • 51
1 vote
1 answer
20 views

Integer Linear Programming - Dividing n people into m groups of specific sizes

I've recently asked this question about dividing n people into m groups for the specific model I used to solve the assignment problem of dividing the people into groups (boolean variables xij that ...
Zufra's user avatar
  • 195
1 vote
1 answer
19 views

Properties of matrices with a multiplicative order

I have been trying to find any article or sources talking about the structure and properties of matrices with a multiplicativw order, i.e. a matrix $A$ has a multiplicative order of $n$ if and only if ...
IV-301's user avatar
  • 49
1 vote
0 answers
12 views

Results of invertibility of a matrix involving the Szego kernel

In the context of reproducing kernel Hilbert spaces, the Szego kernel is the function $k(z_i,z_j)=\frac{1}{1-z_j\overline{z_j}}$. Given two sets of points $\{z_1,\ldots,z_n\},\,\{w_1,\ldots,w_n\}\...
Math101's user avatar
  • 4,578
6 votes
1 answer
103 views

Confusion regarding vector/matrix multiplication in index notation

I came across this question and answer (sorry I don't have an electronic source for it, only a paper copy). After reading the answer it had me questioning the notation one uses to denote row/column ...
digital's user avatar
  • 175
0 votes
1 answer
117 views

$\det(A^2-B^2) \leq \det(A^2)$ when $B$ is of rank 1

This is an exercise problem given at linear algebra class. As a problem before this I was able to show that $$\det(A + uv^T) = \det A + v^T (adj A )u$$ when $A$ is order $n$ square matrix and $u,v$ ...
mathhello's user avatar
  • 918
0 votes
1 answer
26 views

Compute inverse of a special 2 by 2 block matrix.

Let $$X\in\mathbb{R}^p,\quad \tilde{X} = (1, X^{\top})^{\top}\in\mathbb{R}^{p+1},\quad \tilde{\Sigma}=\mathbb{E} \left[\tilde{X} \tilde{X}^{\top}\right]\in\mathbb{R}^{(p+1)\times (p+1)} $$ $$ (\tilde{...
maskeran's user avatar
  • 523
3 votes
1 answer
64 views

Find values of $a,b$. such that a matrix is diagonalizable

Find all values of $a,b\in\mathbb{R}$ such that $A$ is diagonalizable. $$A=\begin{pmatrix} -1 & a & b\\ 0 & 1 & 2\\ 0 & 2 & 1\\ \end{pmatrix}.$$ So far, I have that: $$det(A-\...
user926356's user avatar
  • 1,296
1 vote
0 answers
29 views

Reading $B^{-1}$ from simplex table

In uni I'm following a course on optimalisation and I have come across a problem. I am given the following minimalisation problem: and the corresponding final Simplex table: I now need to determine ...
Jord van Eldik's user avatar
0 votes
0 answers
27 views

$3\times3$ real matrix decomposition to SVD using two unit quaternions and scale vector

I've been trying to search about doing $3\times3$ real matrix SVD, but instead of decomposing it into matrices, represent the two rotations as unit quaternions with the singular values as separate ...
Venom's user avatar
  • 1
-2 votes
0 answers
26 views

Further maths matrix question [closed]

The matrices p = [3/7, 3/7][1/7, 3/7] and Q = [11, 3][5,2] and R = [0,0][1,1] represent the transformations T, U and V respectively. a. A single transformation W is obtained by combining these ...
Peter Lux's user avatar
0 votes
1 answer
40 views

How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Known $(B,C_0,C_1)$, how to calculate ...
X.H. Yue's user avatar
1 vote
0 answers
58 views

How to go about finding the minimal conditions guaranteeing that such a matrix is invertible?

I am working on an economic input-output model, and I want to find the conditions under which a system of linear equations determining the equilibrium yields a unique solution. I have an equation of ...
Balázs Markó's user avatar
3 votes
1 answer
21 views

Further explanation wanted on 'double Gaussian elimination' to triangularize a matrix.

I am trying to learn a more efficient way to triangularize a matrix. I found the following answer here on StackExchange which I found interesting, talking about 'double Gaussian elimination': Short ...
Newbie1000's user avatar
0 votes
2 answers
85 views

$n$ linear equations for $n+1$ unknowns

Let us consider a linear system \begin{eqnarray*} a_{11}x_1 + \dots + a_{1n}x_n + b_1y &=& c_1 \\ &\dots&\\ a_{n1}x_1 + \dots + a_{nn}x_n + b_ny &=& c_n \end{eqnarray*} for ...
Quiriacus's user avatar
-2 votes
0 answers
11 views

Using adjacency matrices to calculate graph rotations [closed]

Given the adjacency matrix $A$ of a tree $T$, is there a way of transforming that adjacency matrix in an efficient way to perform a rotation on that graph?
lo9ud's user avatar
  • 1
1 vote
0 answers
41 views

If $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$?

I already asked this question for general $B$ and it was answered negatively here, so if the statement is true, one has to use the assumption on $B$ as well. In the original question, I already ...
mathemagician99's user avatar
0 votes
0 answers
55 views

Block Matrix Inverse and Imaginary Number

I came here to ask some help regarding the following question. Let $M \in \mathbb R^{n \times n}$ be a "symmetric" and positive semidefinite matrix, and $M^{(n)}$ is obtained from $M$ by ...
jason 1's user avatar
  • 757
-3 votes
0 answers
57 views

Proving, if possible, that $A^2 = A$ implies $(I_3 - 2A) = (I_3 -2A)^{-1}‎$, for $A\in M_R(3)$ [closed]

Prove, if possible, that for $A\in M_R(3)$: $$A^2 = A \quad\implies\quad (I_3 - 2A) \;=\; (I_3 -2A)^{-1}‎$$
Angel B.'s user avatar
2 votes
0 answers
48 views

The Invariance of the Distribution of a Matrix

I'm currently stuck to the following statement. Given $W\in\mathbb R^{p\times d}$ having i.i.d entries from $N(0,1/d)$, define $Q\in\mathbb R^{p\times p}$ as follows: $$Q_{i,j} = \begin{cases}w_i^...
jason 1's user avatar
  • 757
2 votes
1 answer
15 views

Equation for Counting Unique RREF "Cases" for any (m x n) Matrix

Given a real matrix $A$ of size $(m \times n)$, I am seeking to determine the number of unique reduced row echelon form (RREF) "Cases" that exist for a given matrix size. Two RREF matrices ...
FaffyWaffles's user avatar
-1 votes
0 answers
18 views

SVD for finite and infinite dimensional matrix [closed]

Suppose I have a matrix $A_n$ that is symmetric and full rank, then I could apply eigenvalue decomposition on this matrix $A_n = U_n \Sigma_n U_n'$. Now suppose, the size of $A_n$ grows to infinity(We ...
Jerry's user avatar
  • 1
-2 votes
1 answer
49 views

Basis trick for Subspaces [closed]

If I have a vector space $V$ of dimension $4$ over the real numbers , and there is a subspace $U$ of $V$ of dimension $3$, then if I find $3$ linearly independent vectors, will they automatically span ...
adisnjo's user avatar
  • 29
1 vote
1 answer
17 views

How to derive the relation about Jordan decomposition of a matrix?

Assume that $v$ is an eigenvector with an eigenvalue of $0$ in matrix $H$, and its Jordan decomposition $H^J=SHS^{-1}$ satisfies $$ H^J=\left( \begin{matrix} 0& 1& \cdots& 0\\ 0& ...
SHBooKP's user avatar
  • 194
1 vote
1 answer
20 views

Positive definiteness of a diagonal matrix and boundedness of specific set

Problem: Let $D\in \mathbb R^{n\times n}$ be a diagonal matrix. Show that the set $$E=\{x\in \mathbb R^{n\times 1} \mid x^tDx\leq 1 \}$$ is bounded if and only if the diagonal entries of $D$ are ...
categoricallystupid's user avatar
2 votes
1 answer
79 views

Matrix representation of conic sections

The quadratic equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, i. e. the equation satisfied by conic sections, can be represented in the form $a^T Q a$, where $a$ is the column vector $$ \begin{...
Davide Masi's user avatar
-1 votes
1 answer
44 views

We know if $||A||$ is operator norm of $A$, then $||Ax|| \leq ||A||.||x||$ Is there any general rule when does it hold with equality for all $x$? [closed]

We know if $||A||$ is operator norm of $A$, then $||Ax|| \leq ||A||.||x||$ Is there any general rule when does it hold with equality for all $x$ For example: if $A$ is k$I$ for any scaler $k$, it ...
Debojjal Bagchi's user avatar
1 vote
0 answers
41 views

When are all Eigenvalues of a Matrix negative?

Suppose I have the following matrix where: All diagonals are negative In each row, the absolute value of each diagonal element is greater than the sum of all other element in that row Suppose this ...
stats_noob's user avatar
  • 3,182
-1 votes
0 answers
32 views

Is there a way to find the eigenvalues of a matrix using character table? [closed]

I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
Marisha Singh's user avatar
0 votes
1 answer
55 views

Sequences of nested triangles [closed]

Let $ T_0 = \triangle A_0B_0C_0 $ and $ \lbrace T_n = \triangle A_nB_nC_n\rbrace_{n\geq 0} $ be the sequence of nested triangles where $ T_{n+1} $ is formed by taking the midpoints of the sides of $ ...
math.enthusiast9's user avatar
1 vote
0 answers
21 views

$\|B\|_{op} \le \|A\|_{op}$ when $B_{i,j} = v_i^TAv_j$

I'm currently stuck to the following statement. Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix whose diagonal elements are only 0 or 1, and let $B$ be $n\times n$ matrix such that $B_{i,j} = ...
jason 1's user avatar
  • 757
1 vote
1 answer
33 views

A question on positivity of eigenvalues for a matrix with some random chosen entries

Let $0<c<1$. Is it possible to construct a symmetric $n\times n$ matrix $A=(a_{ij})$ with $a_{ii}=1$ for all $1\leq i\leq n$ and $a_{ij}\in ${$1, c$} for all $i\neq j$ for some $n\geq 1$ such ...
ougao's user avatar
  • 3,683

1
2 3 4 5
1120