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), ...

learn more… | top users | synonyms (2)

0
votes
0answers
10 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
0
votes
0answers
16 views

How to make a $4 \times 6$ matrix from a $6 \times 6$ diagonal matrix in MATLAB

I am having a diagonal matrix A = diag(a,b,c,d,e,f) which is a $6 \times 6$ matrix. From this I want to make $4 \times 6$ matrix specified by: ...
0
votes
2answers
18 views

Is there a way to turn this summation into a matrix multiplication?

I have two vectors $\mathbf{s}, \mathbf{p}$ of length $n$, and I need to compute a vector $\mathbf{\pi}$ defined by $$\pi_i=\sum_{j=1}^is_j(p_j-p_i)$$ for $i$ from $1$ to $n$. I suspect this ...
1
vote
0answers
17 views

First derivative of energy function

I need to calculate the first derivative of the following energy function (also found here on page 3): $$ E(d)=∑(d(x,y)-k(x,y)))^2+ ∑w1(x,y)((∂^2 d(x,y))/(∂x^2))^2 $$ I started to rewrite this ...
2
votes
2answers
14 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) ...
4
votes
2answers
48 views

Is this operator $A = \pmatrix{1&1\\0&1}$ self-adjoint?

Is this operator $$A = \pmatrix{1&1\\0&1}$$ self-adjoint? I think not, because $$\pmatrix{1&1\\0&1}^T\neq A$$ where $T$ is the transposition of the matrix. What do you all think?
-1
votes
0answers
31 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 $r*|\mathcal{A}|$ ...
2
votes
1answer
14 views

Number of components needed for 3D rotation

Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9. Is it possible to express a 3D rotation using less than 3 real ...
2
votes
0answers
23 views

Homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$

Let $\phi_1$ and $\phi_2$ be two injective ring homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$. Show that there exists a $g\in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\phi_1(x)g^{-1}$ for all ...
0
votes
1answer
16 views

How to prove infinite solution vs no solution for singular matrix problem.

In the problem Ax=B My coefficient matrix is \begin{bmatrix} α-1 & 1-α\\ α & -α \end{bmatrix} x is \begin{bmatrix} ln K\\ ln L \end{bmatrix} b ...
0
votes
0answers
5 views

generating orthogonal parity check matrix, from a random generator matrix

I have a matrix G of dimension 13x20. It is a full rank matrix. It is not in the standard form of a generator matrix. Now for the parity check matrix 'H', I need a standard representation H=[-P;I]. ...
4
votes
1answer
73 views
+50

Determinant of a Certain Block Structured Positive Definite Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
0
votes
0answers
17 views

Matrix representation of another matrix

Let $\mathbf{c}\in \mathbb{R}^n$ and $\mathbf{X}(s)= \begin{bmatrix} X_{11}(s) & X_{12}(s) & \cdots & X_{1n}(s) \\ X_{21}(s) & X_{22}(s) & \cdots & X_{2n}(s) \\ \vdots & ...
0
votes
1answer
13 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 ...
-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 ...
3
votes
1answer
91 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
1answer
35 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 ...
1
vote
1answer
60 views

Constructing a bases for $U_{1}+U_{2}$

I have the following: Let $L_{1}(x_{1},x_{2},x_{3},x_{4})=(3x_{1}+x_{2}+2x_{3}-x_{4}, 2x_{1}+4x_{2}+5x_{3}-x_{4})$ and $L_{2}(x_{1},x_{2},x_{3},x_{4})=(5x_{1}+7x_{2}+11x_{3}+3_{4}, ...
0
votes
1answer
27 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
24 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
34 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
59 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? ...
1
vote
0answers
21 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
22 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 $\theta_1$, ...
5
votes
0answers
21 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
32 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 ...
2
votes
0answers
41 views
+50

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
20 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 ...
1
vote
1answer
20 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
23 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
64 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
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 ...
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
0answers
27 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 ...
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 ...
2
votes
3answers
83 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
0answers
17 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 $n_1$ ...
0
votes
1answer
46 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
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 ...
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 ...
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
4answers
126 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 ...
4
votes
1answer
618 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
2
votes
1answer
35 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
2answers
505 views

Rotating the gradient

Suppose I have a triangle T in 3dimensional space and i want to rotate it in arbitrary ways. The coordinates for T are given by $f: T_R \in \mathbb{R}^2 \rightarrow T \in \mathbb{R}^3 $ where $T_R$ is ...
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 ...
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 & ...