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)

3
votes
1answer
64 views

$A$ and $B$ are positive matrices and $\|A+B\| = \|A\|+\|B\|$, then A and B have a common eigenvector

Suppose $A$ and $B$ are positive matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$. I'm wondering if there is a ...
0
votes
1answer
26 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
0
votes
1answer
19 views

Lower bound for the distance between matrices of different rank.

This is a follow up question to this: Norm of diference of matrices of different rank Suppose $A$ is a norm one $n\times n$ matrix of rank $k$. What is $\inf\|A-B\|$, where the infimum is taken over ...
0
votes
1answer
21 views

Solve a generalized eigenvalue problem in LDA

http://www.facweb.iitkgp.ernet.in/~sudeshna/courses/ML06/lda.pdf Page 6. I don't quite understand how that can be solved... I have tried following general one $$det(S^{-1}_{w}S_B-JI)=0$$ But I am ...
-4
votes
1answer
35 views

Is it unitary matrix or not? [on hold]

$A = \begin{bmatrix} \frac{i}{3^{1/2}} & \frac{1+i}{3^{1/2}} & 0\\ \frac{-1}{2^{1/2}} & 0 & \frac{i}{2^{1/2}}\\ \frac{1-i}{3^{1/2}} & \frac{1}{3^{1/2}} & 0 \end{bmatrix}$ Is ...
0
votes
1answer
20 views

Columns of a matrix linearly independent and spans

Let $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n \in \Bbb R^n$ and let $P$ be the $n\times n$ matrix whose columns are $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n$ I'm wondering why the followings are ...
-3
votes
2answers
31 views

An example of unitary matrix which is $3\times 3$ and complex

Please give me an example of unitary matrix which is $3\times 3$ and complex. If I get this example, i will finish my thesis.
0
votes
1answer
13 views

Hexagonal - number of cells

For $n = 2$; We have something like this: https://zapodaj.net/0cc6e3c190f32.png.html and number of calls is equal 7. But how designate for $n$ ? For $n = 3$; we have 19
0
votes
0answers
4 views

partial order and equivalence relation question [on hold]

Let A = ℤ+ x ℤ+ and R be a relation on A (that is, R ⊆ A xA) defined as follows. (a,b) ~ (x,y) if and only if a + y = b + x. Is R a partial order? Is R an equivalence relation?
2
votes
1answer
29 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A ...
-1
votes
0answers
21 views

Dynamical Systems problem

I have a problem that have been trying to solve but it's not going so good. I would like some guidelines on how to work myself around this problem: Two neighboring countries spy on each other and ...
0
votes
2answers
19 views

Proof the inequalitiy for the two Matrices $A, B$

Let $ A,B \in C^{nxn}$ then $$ 1 \le || (\lambda I-B)^{-1})(A-B)||\le ||(\lambda I-B)^{-1}||*||(A-B)||$$ for any eigenvalue $\lambda $ of $ A $ which is not an eigenvalue of $ B$ and any operator ...
2
votes
1answer
49 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
1
vote
1answer
32 views

A projection matrix which projects to a space $V$ with $\mathrm{dim}V=2$ has $3$ eigenvalues which span a space of dimension $=3$

I have found an exercise involving a $3\times 3$ projection matrix which projects to a space $V$ with $\mathrm{dim}(V)=2$. The matrix(or operator) is defined as $P(x)=v*(x, v)+u*(x, u)$. So, in my ...
0
votes
0answers
41 views

Is there a variant of Euler's theorem for matrices?

Euler's theorem says that whenever $a$ and $p$ are coprime integers, then $a^{\phi(p)} \equiv 1 \mod p$ This is particularly useful for evaluating large powers of $a$: $a^n \ \%\ p = a^{n\ \%\ ...
0
votes
0answers
19 views

Matrix Norm Confusion

I am looking at my textbook which considers an example but I am not sure how it derived the matrix norm with $||A|| = \sqrt{9/2 + (1/2)\sqrt{65}}$ and was hoping someone could provide the calculations ...
1
vote
2answers
26 views

Norm of diference of matrices of different rank

Suppose $A$ is a $n\times n$ matrix of rank $k$ that has Euclidean norm equal to $1$. Given $p<k$, and $\epsilon>0$, can we always find a norm one matrix $B$ of rank $p$ such that ...
0
votes
1answer
17 views

Frobenius norm of matrix $A^{T}A$ is $trace(A^{T}A)$? Where all of the values in the matrix $A$ are real

We know that frobenius norm of a matrix $A$ is given by $\|A\|_{F}=\sqrt{trace(A^{T}A)}$. Can we write frobenius norm of matrix $A^{T}A$ to be $\|A^{T}A\|_{F}=trace(A^{T}A)$, that is I am effectively ...
0
votes
0answers
24 views

Given the 3x3 matrix D2 is a second order differentiation matrix, for a function u defined and twice differentiable on the interval [a,b].. [closed]

Given that the 3x3 matrix D2 is a second order differentiation matrix, for a function u defined and twice differentiable on the interval [a,b].. u"(a) ≈ D2 * u(a) u"((a+b)/2) ≈ D2 * ...
1
vote
0answers
53 views

Cramer's rule doesn't work here?

I tried to solve the following system: $$A_2\cdot 2\mathrm{i}\sin( \beta a) = B_3\exp(- \alpha a)$$ $$\mathrm{i} \beta A_2 2\cos( \beta a) = - \alpha B_3\exp(- \alpha a)$$ Then I got $A_2=0 ...
0
votes
2answers
25 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
0
votes
0answers
14 views

Find 2 point getting far away each other from their intersection point

I want to know how to find 2 aircraft getting far away from their intersection point, from Dataset such as aircraft 6-7,11-2, 10-6,37-36,etc. Dataset: my algorithm is: calculate direction ...
0
votes
1answer
42 views

Linear Regression without X? :

(Have been working in matrix algebra) Given model: $ y_i = a + e_i$ ( $y_i= α+ϵ_i$ ) That is $y$ subset $i$ and error term subset $i$ Where the expected value of each error term for each entry ...
1
vote
1answer
30 views

Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
0
votes
1answer
13 views

Get vertex points of transformed rectangle knowing bounding box and transform matrices

(I'm not a mathematician so talk down to me). I have a rectangle that has been transformed by a series of matrix transforms. I can recover the transform matrices and get the x,y coordinates of each ...
0
votes
0answers
31 views

Change of Basis between linear Transformations

I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ...
0
votes
1answer
42 views

Rational canonical form of the matrix $A$

Let the matrix \begin{equation} A=\begin{bmatrix} 2 & 1 & 2 \\ -2 & -1 & -4 \\\ 1 & 1 & 3 \end{bmatrix}. \end{equation} So far I found the characteristic polynomial ...
0
votes
0answers
26 views

How to formalise a procedure involving Cartesian products of sets of vectors and transformation in matrices?

I am asking for an help to formalise with the correct notation the following procedure. Let $n\in \mathbb{N}$. Let $\{0,1\}^{n-1}$ be the set of vectors of dimension $(n-1)\times 1$ with each ...
5
votes
2answers
75 views

If $BA = I$, prove that $AB = I$ (using determinants)

I've seen this problem around here, but I wanted to check if this particular solution is right. So, if $BA = I$, then $det(B)det(A) = 1$, meaning neither $det(B)$ or $det(A)$ are equal to $0$. ...
-1
votes
0answers
18 views

Let $M=A^{T}A$ be a positive semidefinite matrix. Is it true that for $i \neq j$, $|m_{ij}| \leq \frac{m_{ii}+m_{jj}}{2}$ [duplicate]

Let $M=A^{T}A$ be a symmetric positive semidefinite matrix. Is it true that for $i \neq j$, $|m_{ij}| \leq \frac{m_{ii}+m_{jj}}{2}$? Where $m_{ij}$ is an element of matrix $M$, and $i$ represents the ...
0
votes
1answer
21 views

Square root of a complex symmetric matrix?

Is it possible to express a complex symmetric matrix $A$ as square of a matrix $B$ (i.e. $A = B^2$)? If $A$ were Hermitian, we could use Spectral Theorem to get $A = UDU^{-1}$ where $D$ has diagonal ...
0
votes
1answer
21 views

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$.

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$. $A$, $B$, and $C$ are square matrices of the same size. $\hat{c}_j$ is the $j$th column of $C$, $\hat{a}_k$ are the columns of $A$, ...
0
votes
0answers
32 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
1
vote
2answers
33 views

Find a matrix $M$ such that $M^TAM = I$

I have that my matrix $A = \begin{bmatrix}4&0&3\\0&1&0\\3&0&4\end{bmatrix}$, I've done diagonalization but now finding a matrix $M$ and its transpose acting as a conjugate for ...
0
votes
0answers
10 views

In a transformation matrix, why is $Y$-axis ($-\sin$) in left most column as opposed to right like X and Z [duplicate]

$4 \times 4$ Transform Matrix with axis columns $XYZ$ left to right $X$-axis rotation: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & \cos & -\sin & 0 \\ 0 & \sin & \cos ...
2
votes
1answer
16 views

determinant inequality for Hermitian matrix

$A \in \mathbb{C}^{M \times M}$ is a positive semidefinite matrix with all diagonal entries being $1$. and the vector $\mathbf{y} \in \mathbb{C}^{M}$ has entries $|y_{i}| < 1$. Prove that $$2 ...
1
vote
0answers
19 views

$k$-th diagonal element of an inverse matrix $(\textbf{H}^{\dagger}\textbf{H})^{-1}$

Let $\textbf{W}=\textbf{H}^{\dagger}\textbf{H}$ with $\{\cdot\}^{\dagger}$ being conjugate transpose operator. In some materials I have read so far, there is a common statement that the $k$-th ...
3
votes
2answers
42 views

Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
0
votes
0answers
25 views

Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
0
votes
0answers
44 views

Why isn't the identity/unit matrix upright?

I realize this is more of a typesetting problem then a mathematical one. I've already tried the TeX stack exchange and the question got canned. In ISO 80000-2:2009, variables and running numbers are ...
1
vote
1answer
38 views

Vector space endomorphisms in $\mathbb{R}[x]$ commuting with $E:f\mapsto f+f'$

I am wondering if every vector space endomorphism in $\mathbb{R}[x]$ that commutes with $E:f\rightarrow f+f'$ is invertible. (denoting $f'$ the derivative of $f$) To begin with, $E$ is invertible ...
-3
votes
0answers
19 views

Properties of Determinants - True or False [closed]

Can you help me answer these true or false questions for an n x n matrix A? I think that 3 and 10 are actually false Picture of the problem The determinant of a lower-triangular matrix A is the sum ...
0
votes
0answers
32 views

Divisibility of dimension by matrix equations.

Let ${\bf M}$ and ${\bf N}$ be two real $k\times k$ matrices such that ${\bf M}^2+{\bf N}^2={\bf M}{\bf N}$. Show that if $\det\left({\bf N}{\bf M}-{\bf M}{\bf N}\right)\neq 0$, then $3\mid k$.
0
votes
1answer
18 views

Diagonal block matrices of a positive definite block matrix

Let $R=\begin{bmatrix} R^{11} & R^{12} & R^{13} \\ R^{21} & R^{22} & R^{23}\\ R^{31} & R^{32} & R^{33} \end{bmatrix}$ be a symmetric positive definite matrix where $R^{ii}$, ...
2
votes
0answers
31 views

If $v^T A^{-1} u = -1$, then the matrix $A + uv^T$ isn't invertible

Let $A \in M_n(\mathbb{R})$ be an invertible matrix and $u,v \in \mathbb{R^n}$. By the Sherman Morrison formula, we know that if $v^TA^{-1}u \neq -1$ then $(A + uv^T)^{-1}$ exists. I want to prove ...
1
vote
1answer
15 views

Finding the inverse of a map given in vector form.

The question asks me to find the inverse map $ \mathbf\Phi^{-1} $, of: $$ \mathbf{\Phi}(\mathbf{x})= \mathbf{n\lor(x \lor n)} + \alpha\mathbf{(n \cdotp x)n} $$ for $\alpha$ such that the inverse ...
2
votes
6answers
40 views

Determine whether $w$ is in the $Span\{v_1, v_2, v_3\}$

my question is how to determine whether a vector $w$ is in the $span\{v_1, v_2, v_3\}$. In this case: $w = \begin{bmatrix} 9 \\ 6 \\ 1 \\ 9 \\ ...
2
votes
2answers
67 views

When this matrix is diagonalizable?

When this matrix is diagonalizable? ($a_i \in \mathbb{R}$) $$ \begin{pmatrix} &&&a_1\\ &&a_2&\\ &\ddots&&\\ a_n&&&\\ \end{pmatrix} $$ I think I should ...
0
votes
0answers
37 views

Let A=$\tiny\begin{pmatrix}1&1&1\\1&2&2\\ 1 & 2 &3 \end{pmatrix}$ and B=$\tiny\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 &0 \\ 1 & 1 &1 \end{pmatrix}$

Then (A) there exists a matrix C such that A = BC = CB (B) there is no matrix C such that A = BC (C) there exists a matrix C such that A = BC, but A $\neq$ CB (D) there is no matrix C such that A ...
0
votes
1answer
11 views

Inverse of a correlation matrix when all the correlations are equal

Let's have variables that multivariate normally distributed and have the same correlation among each other (different variance). Can we analytically derive the inverse of the correlation/covariance ...