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
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0answers
4 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 ...
1
vote
0answers
5 views

For certain positive semidefinite matrices, subtracting the outer product of their row-sums does not change the positive semidefiniteness

Let $e$ denote the vector of all ones, $J=ee^T$ and $\langle A,B\rangle = trace(AB^T)$. Consider a symmetric positive semidefinite (psd) matrix $A\geq 0$ (that is, $a_{ij} \geq 0$ for all entries) ...
-5
votes
4answers
40 views

If $A$ is a $3\times3$ Matrics Then $\left |(2A)^{-1} \right |=?$ [on hold]

If $A$ is a $3\times3$ matrics.And $\left | A \right | = -7$.Then what's the value of $\left |(2A)^{-1} \right |$ Please help to do this math easily.I tried a lot but still no idea come into my ...
-1
votes
0answers
24 views

What matrix property has been used here?

In above equations, let us suppose first equality is true. Can someone show me how second equality came? What property of matrices has been used here. Matrices' order is: $A$ is $r*|A|$ matrix. ...
0
votes
0answers
20 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 ...
0
votes
1answer
33 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 ...
3
votes
0answers
32 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
25 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 $B$ ...
1
vote
0answers
18 views

matrix derivative with tensor product

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
0
votes
0answers
18 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 $\theta_1$, ...
0
votes
1answer
58 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? ...
5
votes
0answers
20 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
3
votes
2answers
29 views

Rotate a unit sphere such as to align it two orthogonal unit vectors

I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors). I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed ...
1
vote
1answer
18 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 ...
0
votes
1answer
22 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
vote
1answer
19 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
0answers
24 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
0
votes
2answers
95 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 ...
1
vote
0answers
16 views

Discrete version of Sylvester's Law of Inertia

Given a matrix $A\in\mathbb{C}^{n\times n}$, we denote by symbol $\mathrm{In}_d (n_<,n_>,n_1)$, the discrete inertia (or inertia w.r.t. the unit circle) of $A$, where $n_<,n_>$ and $n_1$ ...
1
vote
0answers
19 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
31 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 ...
2
votes
3answers
82 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 $1$. ...
0
votes
0answers
20 views

Getting VAR parameters from a research paper.

Many econometrics papers provide the parameters used in their VAR model. If I notate my VAR model as $$z_{t+1} = c + B z_{t} + \Sigma \epsilon_{t+1}$$ where $\epsilon \sim N(0, I)$, then I need to ...
0
votes
1answer
45 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
43 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
62 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
34 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
25 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
125 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
34 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
17 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
37 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 & ...
0
votes
1answer
17 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
3answers
57 views

If two matrices commute, do these two matrices have an inverse

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
102 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
61 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
18 views

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

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
12 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 ...
-1
votes
0answers
26 views

What's the continuation technique? [on hold]

when I read a paper about matrix completion I saw this words"This motivates us to use the continuation technique ".So there exist a method named continuation technique? TKS
0
votes
0answers
25 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) ?

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
10 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
32 views

inversrse of a matrix [on hold]

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
28 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
46 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 \\ ...