For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
votes
1answer
50 views

Decide the range of eigenvalues for $A+B$

Let $A,B$ be $n\times n$ Hermitian matrices on $\mathbb{C}$ such that all eigenvalues of $A$ lie in $[a,a']$ and all eigenvalues of $B$ lie in $[b,b']$. Show that all eigenvalues of $A+B$ lie in ...
1
vote
3answers
47 views

How come least square can have many solutions?

I know there always exists a least-square solution $\hat{x}$, regardless of the properties of the matrix $A$. However, I keep finding online that least-square can have infinitely many solutions, if ...
-1
votes
2answers
72 views

Proof that for all symmetric matrices $A$ and $B$, $AB=(BA)^T$.

Recall that a matrix, $M$, is said to be symmetric if and only if $M=M^T$. I've been trying to use the homomorphic nature of the transpose operator to prove this proposition but this approach hasn't ...
2
votes
1answer
37 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
2
votes
2answers
51 views

derivative of a function including a vector

given a column vector including function of a parameters $x=\bigg(f(\beta_1),\ldots,f(\beta_m)\bigg)^T$ where $T$ denotes transpose of the vector. Can somebody tells me what is the derivative with ...
2
votes
1answer
29 views

Diagonal elements of subset of Hadamard matrices

I'm looking at Sylvester's construction of Hadamard matrices, where $H_{2^n} = \left[\begin{array}{c c} H_{2^{n-1}} & H_{2^{n-1}} \\ H_{2^{n-1}} & -H_{2^{n-1}} \end{array}\right]$, where ...
0
votes
4answers
130 views

Show that $A^{n}\to0$ if and only if $\|A\|^{n}\to0$

Let $\boldsymbol{A}$ be a square matrix. Show that $\lim_{n\to\infty}\boldsymbol{A}^{n}=0$ if and only if $\lim_{n\to\infty}\|\boldsymbol{A}\|^{n}=0$ for the spectral radius or for some operator ...
1
vote
3answers
35 views

How to determine a basis and the dimension for this vectorspace?

Determine a basis and the dimension for the following vectorspace: \begin{align*} W = \left\{A \in \mathbb{R}^{3 \times 3} \mid A \ \text{is a diagonal matrix and} \ \sum_{i=1}^3 A_{ii} = 0\right\} ...
2
votes
1answer
22 views

different ways to see why this matrix limit is correct

given that $0 < a < 1$ it follows that: $$\lim_{n\to\infty}\begin{pmatrix} a & (1-a) \\ (1-a) & a \end{pmatrix}^n = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 ...
2
votes
0answers
18 views

Formula Index Confusion

I am working on a computer vision project and need to implement the formula on the bottom of page 13 of http://www.dgp.toronto.edu/~donovan/stabilization/opticalflow.pdf My question pertains to the ...
1
vote
1answer
40 views

Checking if a matrix is in the span of other matrices

Problem: Expand the following set matrices \begin{align*} \left\{ \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}, \begin{pmatrix} 2 & 1 \\ -1 & 4 \end{pmatrix}, \begin{pmatrix} 0 & ...
0
votes
3answers
26 views

Number of real values of $x$ satisfying the following determinant?

The number of real values of $x$ which make the following determinant equal to $0$ are ? $$ \text{det}\left(\begin{matrix} x & 3x + 2 & 2x-1 \\ 2x-1 & 4x & ...
1
vote
1answer
28 views

Definition of Reducible matrix and relation with not strongly connected digraph

I connot quite understand the definition of reducible matrix here. We know $A_{n\times n}$ is reducible, when there exists a permutation matrix $\textbf{P}$ such that: $$P^TAP=\begin{bmatrix}X ...
-6
votes
2answers
74 views

If two matrices commute, do these two matrices have an inverse [on hold]

If two matrices, $A$ and $B$ commute: $AB = BA$, is this sufficient for $A$ to have an inverse, or $B$ to have an inverse. Or put another way, does $AB=BA$ imply that $A^{-1}$ exists, in general. ...
4
votes
1answer
108 views

Why is this not a valid proof?

A thread I saw recently has led me to believe that this is not a valid proof of the fact that for matrices $A$ and $B$, $AB=I\implies BA=I$. Suppose $AB=I$. Then $$A^{-1}AB=A^{-1}I$$ $$B=A^{-1}$$ ...
1
vote
2answers
62 views

Proving that $\hat{x} = x$ in least square

I am trying to show that if $Ax=b$ has a unique solution $x$, then the least square solution $\hat{x}$ is the exact one (i.e., $\hat{x} = x$). My attempt: We know that for $A x = b$ to have an exact ...
-1
votes
0answers
19 views

reduction to canonical form of the quadratic form that corresponding to the matrix [closed]

Do reduction to canonical form of the quadratic form that corresponding to the matrix: $$\begin{bmatrix} 1 & -1 & 0 \\ -1 & -2 & -1 \\ 0 & -1 & 2 \\ \end{bmatrix}$$ $$\in ...
0
votes
0answers
15 views

Spectral radius and matrix norm inequality as its consequence

I am trying to undestand a proof and there is one part that's holding me back. By assumption we have that spectral radius $\rho(A) < 1$. Hence, following inequality should hold $$\|A^k\| < C ...
0
votes
0answers
28 views

what is the probability that the rows of a matrix sum to a given vector?

Given a random matrix of 1s and 0s with the upper half set to zeros, as shown: $$ \left( \begin{eqnarray} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 1 & ...
-1
votes
0answers
11 views

What is the generator matrix for the ternary perfect covering code K3(13,1) ? [closed]

What is the generator matrix for the ternary perfect covering code K3(13,1) = 59049 ? How can I build the matrix? Thank you all for your valuable time.
1
vote
0answers
13 views

closed form solution to best invertible matrix which minimizes product

Let $U, X \in \mathbb{R}^{n_1 \times r}$ and let $V, Y \in \mathbb{R}^{n_2 \times r}$. Consider the optimization problem $$ \begin{align*} \min_{A, B, \Sigma \in \mathbb{R}^{r \times r}} \left\{ \| ...
0
votes
0answers
29 views

What is a double folded matrix?

I am reading this paper and on page 2 they mention: In the model, 144 TCR and 36 interneurons were simulated in a doublefolded matrix, i.e. in a matrix without boundaries as shown in Fig. 1 ...
-5
votes
0answers
33 views

inversrse of a matrix [closed]

Find the inverse of matrix $A$ using elementary transformations, where $$A= \begin{pmatrix}2 &-1& 4\\4 & 0 &2\\3 &-2 &7\end{pmatrix}$$
1
vote
1answer
29 views

If $S:(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$ Why does $0 \in \sigma (S)$? and what is $\dim(R(S))$ in $\ell^2$?

Consider the unilateral shift $S$ on $\ell^2$ defined by $$(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$$ Why dose $0$ is eigenvalue of $S$? and what is $\dim(R(S))$ in $\ell^2$?( Range of $S$)
3
votes
2answers
48 views

Prove $\det(A - nI_n) = 0$.

Problem: Prove that $\det(A - n I_n) = 0$ when $A$ is the $(n \times n)$-matrix with all components equal to $1$. Attempt at solution: I tried to use Laplace expansion but that didn't work. I see the ...
0
votes
1answer
36 views

Calculation of determinant using its properties [duplicate]

The task is to calculate the following determinant by using the properties of a determinant: $$\begin{vmatrix} n^2 & (n+1)^2 & (n+2)^2 \\ (n+1)^2 & (n+2)^2 & (n+3)^2 \\ ...
-1
votes
2answers
30 views

$LU$ Decomposition of antidiagonal matrix

I cannot find the $LU$ decomposition of anti-diagonal matrix $$\begin{bmatrix} 0 &0 &0 &1 \\ 0 &0 &2 &0 \\ 0 &3 &0 &0 \\ 4 &0 &0 &0 \end{bmatrix}.$$ ...
2
votes
3answers
75 views

Prove that $|GL_n(\mathbb{F})|< q^{n^2}$.

Let $\Bbb F$ be a finite field, say $|\Bbb F|=q$; then we know that $|GL_n(\Bbb F)| < \infty$. But how can we prove that $|GL_n(\mathbb{F})|< q^{n^2}$? I'm guessing because there $n^2$ ...
3
votes
1answer
41 views

Prove that $\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$

So I need to prove that: $$\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$$ where $A$, $B$ are two orthogonal matrices, but it seems I'm missing something.
10
votes
1answer
120 views

Determinant of a special $4\times 4$ matrix

Let $f(x)=\sum_{k=1}^{4}a_{k}x^{k},\varepsilon =\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}.$ $\qquad\qquad 4\times 4$ matrix $$T=\begin{bmatrix} 1& a_{2}& a_{3}& a_{4}\\ 1& ...
2
votes
3answers
43 views

What is Homogeneous Coordinates? Why is it necessary in 2D transformation?

What is Homogeneous Coordinates? Why is it necessary in 2D transformation of objects in computer graphics? The concept of homogeneous coordinates in effect converts the 2D system a 3D one. So, why ...
0
votes
0answers
24 views

Why doesn't line fitting seem to work in polar coordinates

I have 2 points, $(r_1, \theta_1)$ and $(r_2, \theta_2)$. They are plotted and I'm trying to find a curve in the form of $r=\theta\beta_1+\beta_2$ to connect both of them. This is basically performing ...
0
votes
3answers
41 views

Matrix proof question

Suppose that $A,B,C$ are $n\times n$ matrices and the matrix $ABC=I_n$. Show that $A,B, $ and $C$ are invertable matrices. Not sure how to show $AA^{-1}=I_n$ $BB^{-1}=I_n$ $CC^{-1}=I_n$
0
votes
0answers
20 views

Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
0
votes
1answer
38 views

Invertibility Proof for matrix

Suppose that A is a square matrix that satisfies $A^n=0$ for some positive integer n. Show that $I-A$ is invertible and $(I-A)^{-1}=I+A+A^2+...+A^{n-1}$. Not sure how to start the problem.
2
votes
1answer
42 views

Desk Reference Question: Linear Algebra and its Applications by Lay or Strang? Or Handbook of Linear Algebra?

I would like a good working desk reference for linear algebra for someone in applied mathematics (no proving abstract theorems). I saw the Handbook of Linear Algebra as well, but I am concerned it may ...
0
votes
0answers
14 views

Incidence matrix of invertible substitution

A statement I am reading says: An invertible substitution $\sigma$ over $\{1,2\}$ is non-primitive iff $M_{\sigma}$ (it's incidence matrix) has one of the following forms:$$\left( \begin{array}{cc} 1 ...
2
votes
1answer
18 views

Singular Value Decomp inequality

Let $\newcommand{\<}{\langle}$ $\newcommand{\>}{\rangle}$ $T \in L(V)$ be a linear operator on an n-dimensional real inner-product space of $(V,<.,.>)$ whose singular value decomposition ...
2
votes
1answer
36 views

$\log (A + \delta A) = ?$ (as an expansion in $\delta A$), where $A$ and $\delta A $ are matrices

$A$ and $\delta A$ are two non-commuting matrices and I am seeking a power series expansion to 2nd order in $\delta A$. After writing it as $\log (A (1 + A^{-1}\delta A) )$, I am unable to figure out ...
0
votes
1answer
29 views

Find the derivative of$Y^{-1}XY$

Let X be an $n\times n$-matrix of the form $X(x)=\left (f _{ij}(x) \right )$, where the functions $f _{ij}:V_{K}^{n}\rightarrow \mathbf{K}$ are analytic, and let $Y$ be in $GL_{n}(\mathbf{K})$. Show ...
2
votes
2answers
56 views

Questions about solutions to $Ax=b$

I would like to answer the following questions: Let $A$ be an arbitrary $n\times n$ matrix and let $Ax=b$ have more than two solutions. Does it follow that $Ax=b$ is solvable for every $b$? ...
3
votes
1answer
97 views

Describe the space of solutions to a simple matrix equation

I would like to describe the space of solutions to the following matrix equation. Here $U$ and $V$ are two unknown real matrices with $n$ lines and $p$ columns (i.e. they belong to ...
0
votes
3answers
50 views

Proving that $AB\neq 0$ if $\operatorname{rank} A =2, \operatorname{rank} B=3$

Let $A_{2\times4}, B_{4\times4}$ such that $\operatorname{rank}A =2$, $\operatorname{rank} B=3$. Prove that $AB\neq 0$. We know that rank $(AB) \le \min\{\operatorname{rank}A, ...
0
votes
0answers
10 views

Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of: $$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ...
0
votes
0answers
21 views

Looking for some condition to have non-negative definite matrix

Let $\cal I$ be a finite set of indices and for every $i \in {\cal I}$, there exist a matrix $A_i \in \mathbb{R}^{k \times m}$, such that $A_i$ has only one 1 in each column. Let $p \in \mathbb{R}^m$ ...
0
votes
1answer
14 views

Relationship between eigenvectors and singular vectors of a Hermitian matrix?

What is the relationship between the eigenvectors and singular vectors of a Hermitian matrix? Intuitively, I would expect them to be the same (modulo scaling). However, this doesn't seem to be the ...
0
votes
0answers
60 views

Let $B_{m \times n} = P_{m \times m} A_{m \times n}$ where $P$ is a product of elementary matrices. When is $P$ unique?

Let $A$ and $B$ be two $ m \times n$ matrices , such that $B = PA$ where $P$ is a product of elementary matrices. Now, when when $rank(A) < m$ does it necessarily mean that $P$ is not unique and ...
0
votes
1answer
38 views

Algorithm to find non-zero matrix $N$ such that $N \times M = 0$

Given a $p \times q$ matrix $M$, is there an algorithm to find the $q \times p$ matrix $N$ such that: $$N \times M = 0$$ Assuming $M$ is non-zero, the trivial solution is $N = 0$. I am looking for ...
0
votes
0answers
19 views

Convex functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
3
votes
0answers
71 views

What does it actually mean by a “Characteristic Polynomial”?

Please can you describe in layman's term, what does it actually mean by a "Characteristic Polynomial"? Is it a property only of Matrices? What does it describe about a Matrix, that is, what can we ...