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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2
votes
1answer
135 views

Eigenvectors of matrices which commute with a projection

Just a quick question. Cant seem to prove it or find any relevant references! Maybe it's really simple :\ Is the following statement true (for square matrices of the same finite dimension)? If there ...
0
votes
1answer
380 views

Computing the Frobenius normal form

I was wondering whether someone could give me an example how one actually determines the Frobenius normal form of a given matrix. Further, it seems hard to find an example where the new basis is ...
0
votes
3answers
61 views

matrix of all possible differences

I need to compute the matrix of differences of a vector, just like here in section "Matrix of differences". Is that actually correct? I thought that, in order to add up (or subtract) two ...
2
votes
1answer
203 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
0
votes
1answer
29 views

Sum of every element's product in a matrix whose entries are either $\pm1$

There is a matrix $A$ which has $19$ rows and $19$ columns. Every element has a value either $1$ or $-1$. Defined that $R_k$ is the product of every elements in the $k^{th}$ row and $C_k$ is the ...
10
votes
2answers
167 views

Determinant of $4\times4$ Matrix

I tried to solve for a $4 \times 4$ matrix, but I'm unsure if I did this properly, can anyone tell me if I did this correct? Or if there were any mistakes where at? Also, I know this is an inefficient ...
4
votes
2answers
698 views

roots of minimal and characteristic polynomial

Why is it, that for the matrix $A \in \text{Mat}(n\times n, \mathbb{C})$ the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have same roots? Since $\chi_A(t) = \mu_A(t) ...
2
votes
1answer
117 views

Determine invariant subspaces

imagine that a matrix of an endomorphism has the characteristic polynomial $(\lambda-2)^2(\lambda-3)$ now i was wondering whether all invariant subspaces can be determined by $0,V$ and $\ker(A-2)^2, ...
0
votes
1answer
168 views

Solve equation involving matrices and Kronecker product

I have what appears to be a simple equation: $$ C_u = H(C_y \otimes R)H^T $$ where the matrix dimensions are compatible, i.e. $\dim(C_u)=\dim(C_y)=n \times n$, $\dim(H)=n \times np$, $\dim(R)=p ...
1
vote
1answer
500 views

Cyclic vector space

In class we defined what it means that there is a creating element $v$ of a vector space, such that for an endomophism $A$ on $V$ we have: ${\rm span}(v,Av,...,A^{n-1}v)=V$. Also we said that if the ...
3
votes
3answers
84 views

Determinant of unspecified matrices

Suppose A and B are $5\times 5$ matrices with $\det(A) = -1/3$ and $\det(B) = 6$, find the determinant of $ 2AB$. Solution: $$= \det(2AB) $$ $$= 2^5 \det(A)\det(B) $$ $$= (32)(-1/3)(6)$$ $$= -64$$ ...
1
vote
1answer
48 views

Basis given the rank of A is equal to n

Let $A$ be an $m \times n$ matrix with columns $C_1,C_2,\dots,C_n$. If $\operatorname{rank} (A) = n$, show that $\{ A^T C_1,\dots,A^T C_n\}$ is a basis for $\mathbb{R}^n$. I'm really confused ...
2
votes
1answer
90 views

I cannot understand how this matrix works or how it is defined

I'm currently reading Ranking a Stream of News and have trouble on page 100 (don't be afraid, the math starts at 99). I cannot understand a matrix they define. In this article, the authors are ...
7
votes
3answers
2k views

Eigenvalues and Eigenvectors of $2 \times 2$ Matrix

Let's say I have a $2 \times 2$ matrix (actually the structure tensor of a discrete image - I): $$ \begin{bmatrix} \frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial ...
4
votes
1answer
181 views

Invertible antisymmetric matrix and identities

A link to the page is available here. The relevant bit is on P. 15 of the book. I would really appreciate it if somebody could help! It is probably something quite obvious, hence left out by the ...
7
votes
5answers
323 views

About positive semidefiniteness of one matrix

It is not clear how to prove that the matrix $(\min(i,j))_{i,j=1,\dots,n}$ is (or is not) positive semidefinite. There are some facts from Horn and Johnson's book Matrix Analysis: if $A \in M_n$ is ...
0
votes
4answers
152 views

Evaluation of a specific determinant.

Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$. $$A_2 \begin{pmatrix} 1& 1\\ 1& 2 \end{pmatrix}; A_3 = ...
9
votes
4answers
2k views

What is the fastest way to find the characteristic polynomial of a matrix?

Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$. I know that: the coefficient of $\lambda^n$ is $(-1)^n$, the coefficient of ...
1
vote
0answers
53 views

Why process noise model is$ \dfrac{T^4}{4}$ in Kalman filter.

I am using Kalman filter for filtering noise on 2D object movement. I read a lot of examples, but no one has been explained, why noise model is distance powered by 2: $$ s \times s = \dfrac{T^2}{2} ...
1
vote
1answer
198 views

REVISITED$^1$: How is the determinant of a matrix $A\in M_{2\times 2}(\mathbb{R})$ considered a bilinear form?

I'm trying to prove that $B(X,Y)=\det (X+Y) - \det (X) - \det (Y)$ is a blinear form on the vector space $A$ is from, and also trying to determine if it is an inner product space. I think if I know ...
0
votes
1answer
117 views

Matrix operations - filter out diagonal elements of a matrix using multiplication operation

Is there a way to filter out only the diagonal elements of a matrix $A$, by doing a matrix multiplication with some matrix $B$ like this: $A*B$ = $D$ where $D$ matrix contains only diagonal elements ...
0
votes
2answers
133 views

Rotation matrix - rotate a ball around a rotating box

I've a 3D box: center point = (a,b,c), width = w, height = h, ...
7
votes
4answers
410 views

REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?

Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution in ...
2
votes
0answers
198 views

Example of Google page ranking algorithm?

I read about the Google page ranking algorithm from here http://en.wikipedia.org/wiki/PageRank . My question is why only outbound links are used in page rank calculation? Do inbound links not ...
20
votes
4answers
490 views

Can every nonsingular $n\times n$ matrix with real entries be made singular by changing exactly one entry?

I was just thinking about this problem: Can every nonsingular $n\times n$ matrix with real entries be made singular by changing exactly one entry? Thanks for helping me.
2
votes
2answers
81 views

Let $A,B$ be elements of $M_2(\mathbb{R})$. Give an example to show that $A+B$ can be invertible if $A,B$ are both non-invertible

The goal for this problem is to show that even if two matrices $A$ and $B$ are non-invertible, $A+B$ can be invertible. I tried to show this using a proof, but I ended up actually proving that this ...
2
votes
2answers
88 views

Let $A_{\alpha}$ be the $\alpha$-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$

Let $A_{\alpha}$ be the alpha-rotation matrix. Prove $A_{\alpha}^T = (A_{\alpha})^{-1}$ In other words, prove $A_{\alpha}$ transpose = $A_{\alpha}$ inverse. First of all, what is a ...
3
votes
1answer
51 views

If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$

So this question is basically a proof. If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ ...
10
votes
2answers
383 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
1
vote
2answers
191 views

Space spanned by matrices

I have a set of 5 by 5 matrices, M1,M2,...,M19 ,M20. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should approach the ...
1
vote
2answers
147 views

Matrix equation involving a Pauli matrix

I should solve the following problem: find the matrix $A$ that satisfies the following equation: $$\sin(\pi A)+\cos(\pi A)^2= \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right)$$ How ...
2
votes
2answers
97 views

Regarding a Paper by Paul A. Clement on Tridiagonal Matrices

I've asked this question at MathOverflow and was told it'd be better suited for here. In Paul A. Clement's (1959) paper, A Class of Triple-Diagonal Matrices for Test Purposes, SIAM Review, Vol. ...
0
votes
1answer
84 views

Quaternion Matrix Multiply

My question is about applying rules of quaternions to quaternion matrices. I know that for some rotation quaternion q = [w, x, y, z], I can find the rotation of ...
9
votes
1answer
209 views

Calculating the eigenvalues of a matrix

How to find the eigenvalues of $$\begin{bmatrix} 0 & 1 & & &\\ k & 0 & 2 & &\\ & k-1 & 0 & 3 &\\ & ...
2
votes
0answers
55 views

Fast algorithm to determine invariant subspaces

Imagine that there is an endomorphism $A \in \text{End(V)}$ and I want to find out the biggest non-trivial decomposition of the vector space into invariant subspaces. I am looking for the best way to ...
11
votes
3answers
412 views

Eigenvalues of some peculiar matrices

While I was toying around with matrices, I chanced upon a family of tridiagonal matrices $M_n$ that take the following form: the superdiagonal entries are all $1$'s, the diagonal entries take the form ...
0
votes
1answer
33 views

Matrix operations - equivalent operation for a given operation

This is the given problem, I need to write a code for this: $(M*Q) \circ (N*Q) $ where $M,Q,N$ are known matrices, "$\ast$" denotes matrix multiplication and "$\circ$" denotes elementwise division. ...
2
votes
1answer
808 views

Is it possible to determine if this matrix is ill-conditioned?

I want to better understand ill-conditioning for matrices. Say we're given any matrix $A$, where some elements are $10^6$ in magnitude and some are $10^{-7}$ in magnitude. Does this guarantee that ...
2
votes
1answer
111 views

which of the following options is correct?

Let $y(t)=\begin{pmatrix} y_1(t)\\ y_2(t) \end{pmatrix}$ satisfy $\dfrac {dy}{dt}=Ay; t>0, y(0)=\begin{pmatrix} 0\\ 1 \end{pmatrix}$ where $A$ is a $2 \times 2$ constant matrix with real ...
0
votes
1answer
83 views

Second Derivative of a matrix

Pardon me for not knowing LateX representation, I have following function, where $\mu$ and $\Sigma$ are both Matrices. $$ h = \mu^T \Sigma \mu $$ which is a function of $\alpha$, such that its ...
0
votes
2answers
71 views

Is there a special term for an array consisting only of ones?

Is there a special term for an array consisting only of ones? Sorry for the rather elementary question. I am getting into MapReduce programming and am trying to frame my code to be nice and neat.
0
votes
1answer
44 views

Symmetric Matrices $B_{n \times n}$ of the form $A^{T}A$ for some matrix $A_{m \times n}$?

Is every symmetric matrix $B_{n \times n}$ of the form $A^{T}A$ for some matrix $A_{m \times n}$? What I've done so far is: For any matrix $A_{m \times n}$, the matrix $A^{T}A$ is symmetric because ...
0
votes
0answers
97 views

List of n objects & their similarities. How to group them in sets based on their similarities?

I have a list of $n$ objects, and I know the similarity (as an index from $0$ to $1$) between any two objects. Question: how to create $i$ groups ($i< n$) of objects based on their ...
3
votes
1answer
117 views

Question about Noncommutative Rings by I. N. Herstein

Here's an example that I do not quite understand it fully, it's on page 6 of the book. And here's what it says: Let $F$ be a field, and $F_n$ be a ring of all $n\times n$ matrices over $F$. We ...
0
votes
1answer
85 views

Definition of $\exp(A)$ in terms of spectral decomposition.

I am read this question Plugging a matrix multiplied by an imaginary number in the exponential function. Here the question'author defined $$\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$$ What ...
3
votes
1answer
207 views

Classification of Matrices

I need a little help on solving matrices. I actually just want to confirm my answer Given matrices : $$ A= \begin{bmatrix} 2 & 0 & 8 & 9 & 7 \\ 0 & 0 & 1 & 2 ...
1
vote
1answer
2k views

characteristic polynomial of companion matrix [duplicate]

I have a matrix in companion form, $A=\begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots & ...
3
votes
1answer
91 views

Finding the value of $c$ that makes this system of equations have a solution

Find the value of $c$ that makes the system of equations below have a solution. $$\begin{align*} u + v + 2w &= 2 \\ 2u + 3v - w &= 5 \\ 3u + 4v + w &= c \end{align*}$$ I have ...
0
votes
0answers
216 views

Second Derivative of determinant of a matrix

I have to find out Second order Derivative of $(XAX')^{-1/2}$, wrt $X$. After first derivative itself, it becomes too complicated. Please Help.
1
vote
0answers
74 views

Simplified matrix notation

There is a mathematical notation to define an array that you can write using standard keyboard characters on one line? for example:   1 3 8 7 10 17 22 6   5 10 23 8 11 98 7 12