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
37 views

Positive definiteness of a linear combination of semi positive definite matrices?

Assume I have the following condition: $M=\sum_{i=1}^N M_i$ is a positive definite matrix while $M_i$ is semi-positive definite matrix. If we introduce positive integer $\alpha_i>0$ such that $...
3
votes
1answer
121 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 $M_{n,p}(\mathbb{R})...
1
vote
1answer
70 views

Matrix with entries equal to $1$ and $-1$ (Sign Matrix)

What can we say about the determinant and (or) maximum eigenvalue of a matrix with entries equal to $1$ and $-1$. Further assume that the rows and columns are linearly independent. Are there special ...
0
votes
1answer
54 views

How do I solve for vector $P$ in the matrix equation $s=A'B^{-1}A$?

I would like to rearrange the matrix equation $s=A'B^{-1}A$ into the form $A=f(s,B)$ (i.e., some function of $s$ and $B$), where s is scalar, $A$ is $n\times 1$, $A'$ is the transpose of $A$, and $B$ ...
0
votes
0answers
47 views

Exponentiation of Pascal's Triangle(in matrix form)

I want to find a pattern in subsequent exponentiations of the pascal triangle shown in the form below Matrix P[K+1][K+1]: $$ \begin{matrix} \binom{0}{0} & 0 & 0 & 0\cdots &...
3
votes
0answers
61 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
0
votes
1answer
69 views

Characteristic polynomial of A, if $\det(\operatorname{adj}(\operatorname{adj}(A))) = 81$?

Let $A$ be a square real matrix whose eigenvalues are positive integers, with $$\det(\operatorname{adj}(\operatorname{adj}(A))) = 81 \, .$$ What is the characteristic polynomial of A? Any hints? ...
0
votes
0answers
39 views

Span of a projection matrix $P(\theta, \phi)$

I have a projection matrix which depends on two parameters, $\theta$ and $\phi$. I am interested in finding out if the relationship between space spanned by the projection matrix for say $\theta_1$, $\...
4
votes
0answers
59 views

Matrix which represents the product of ideal classes of 2 matrices.

Let $f(x)$ be an irreducible monic degree $n$ polynomial with $\mathbb{Z}$-coefficients and $\Theta$ be a root of $f(x)$. There is an old theorem of Latimer and MacDuffee that there is a 1-1 ...
1
vote
1answer
57 views

Solving for a Binary Matrix: A somewhat unusual method needs justification, and mabye interpretation.

Introduction: Define a "Bit Map" to be a matrix whose entries can only be $0$ or $1$. Then numbers above and beside each column and row indicates how many entries are "filled" with a one. For ...
2
votes
1answer
55 views

More efficient algorithm for matrix rearrangement (MatLab)

Say I have the following matrix: $$A = \begin{bmatrix}0.1 & 2 \\ 0.1 & 4 \\ 0.1 & 6 \\ 0.2 & 3 \\ 0.2 & 2 \\ 0.2 & 7 \\ 0.3 & 10 \\ 0.3 & 7 \\ 0.3 & 5 \end{bmatrix}...
0
votes
1answer
25 views

Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?
-1
votes
2answers
88 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
44 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 $H_{2^...
2
votes
2answers
52 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 ...
0
votes
2answers
113 views

What is that matrix?

Let an inner product on $\mathbb{R}^n$ be given by its Gramian matrix $G$. Let $A:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear operator with $\mathop{\rm rank} A=k$ (We denote its matrix also by ...
2
votes
3answers
122 views

Find the necessary and sufficient condition for $A^m\to0$

Let $A$ be $n\times n$ matrix on $\mathbb{C}$. Find a necessary and sufficient condition for $A^m\to0$ as $m\to\infty$. My thought: I think it should be that eigenvalues of $A$ are less than $1$. ...
1
vote
1answer
65 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 $[a+b,...
1
vote
0answers
45 views

Differentiating quadratic form containing vector raised to powers elementwise, can I avoid Hadamard notation?

Say $\mathbf{M}$ is a symmetric, p.d. 2x2 matrix, and $\mathbf{x}$ is a 2x1 vector. The familiar quadratic form is of course given by: $A=\mathbf{x'}\mathbf{M}\mathbf{x}$ (where $A$ is a scalar), and ...
0
votes
0answers
40 views

What does this matrix notation mean?

What does $|\textbf{M}|$ mean, where $\textbf{M}$ is a matrix? I am under the impression that you can element-wise divide $\textbf{M}$ by $|\textbf{M}|$ to normalize it in some way, kind of like how $|...
1
vote
3answers
116 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 $A$...
0
votes
4answers
138 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 norm....
2
votes
1answer
50 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 ...
1
vote
3answers
39 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\} \...
1
vote
1answer
123 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 & -...
2
votes
0answers
22 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 ...
0
votes
1answer
118 views

Disjoint matrix multiplication

I'm studying matrix product algorithms. I've seen that there is a concept of disjoint matrix multiplication. What does it consist in? Thank you.
0
votes
3answers
35 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 & 3x+...
3
votes
1answer
186 views

An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
1
vote
1answer
50 views

Optimizing a function of a matrix

Let \begin{equation} \begin{aligned} W= & \underset{X}{\mathrm{maximize}} & & \log \left|X + K_1\right|- \alpha \log \left|X + K_2\right|\\ & \mathrm{subject \; to} & & 0 \...
4
votes
1answer
121 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
82 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 ...
0
votes
0answers
46 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 & ...
0
votes
0answers
37 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 ...
1
vote
0answers
27 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\{ \| ...
11
votes
1answer
153 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& a_{1}&...
1
vote
1answer
32 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 $S$)
1
vote
3answers
74 views

what is the geometry behind the matrix multiplication?

What is the geometry behind the matrix multiplication? The questions that I am having is the follows. $\bullet$ I accept that we are viewing $\mathbb R^4$ as ...
0
votes
1answer
47 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
42 views

$LU$ Decomposition of antidiagonal matrix

I cannot find the $LU$ decomposition of anti-diagonal matrix $$\begin{bmatrix} 0 &0 &0 &1 \\ 0 &0 &2 &0 \\ 0 &3 &0 &0 \\ 4 &0 &0 &0 \end{bmatrix}.$$ ...
4
votes
1answer
67 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.
0
votes
0answers
56 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
59 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 $AA^{-1}=I_n$ $BB^{-1}=I_n$ $CC^{-1}=I_n$
7
votes
1answer
167 views

Prove $AB=I$ implies $BA=I$ using Fitting's Lemma

I know this question has already been asked, but I need a proof that for $A,B \in M_n(K)$, $$AB=I_n \Rightarrow BA=I_n$$ using Fitting's lemma. I thought of using the fact that $K^n$ is a ...
1
vote
2answers
2k views

Find the inverse a matrix with trigonometic entries

What is the inverse of \[ \begin{pmatrix} 1&0&0\\0&\cos x &\sin x\\ 0 &\sin x &-\cos x \end{pmatrix} \] Please help me to solve the above problem.
2
votes
1answer
29 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 ...
6
votes
1answer
217 views

Matrix with non-negative eigenvalues (and additional assumption)

Let $A \in \mathbb{R}^{n \times n}$ be positive semi-definite ($A = A^\top \succcurlyeq 0$) and with positive diagonal elements ($A_{i,i} > 0$ for all $i$). Assume that both the column and the row ...
0
votes
0answers
64 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 ...