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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6
votes
0answers
28 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 ...
4
votes
0answers
52 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
23 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
22 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
24 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
70 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
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 ...
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 ...
-1
votes
2answers
99 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
87 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
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
0answers
23 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
32 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
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 ...
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 ...
4
votes
1answer
630 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
36 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
509 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
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 & ...
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 ...
0
votes
1answer
41 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.
5
votes
2answers
586 views

Largest eigenvalue of a symmetric positive definite matrix with rank-one updates

I have a $n \times n$ symmetric positive definite matrix $A$ which I will repeatedly update using two consecutive rank-one updates of the form $A' = A + e_j u^T +u e_j^T$ where $\{e_i: 1 \leq i \leq ...
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 & ...
-5
votes
0answers
33 views

inversrse of a matrix [closed]

Find the inverse of matrix $A$ using elementary transformations, where $$A= \begin{pmatrix}2 &-1& 4\\4 & 0 &2\\3 &-2 &7\end{pmatrix}$$
-6
votes
3answers
60 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. ...
3
votes
1answer
144 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
39 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
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
18 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 ...
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
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 ...
-1
votes
0answers
28 views

What's the continuation technique? [closed]

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
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.
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 ...
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\{ \| ...
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
1answer
1k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
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 $S$)
1
vote
3answers
62 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 ...
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 $$\begin{bmatrix} 0 &0 &0 &1 \\ 0 &0 &2 &0 \\ 0 &3 &0 &0 \\ 4 &0 &0 &0 \end{bmatrix}.$$ ...
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.
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
40 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 $AA^{-1}=I_n$ $BB^{-1}=I_n$ $CC^{-1}=I_n$