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
0answers
73 views

The minimal polynomial of A is dividing $x^{2013} -1$, prove A is diagonalizable over the complex field

$A $ is $nxn$ real matrix. The minimal polynomial of A is dividing $x^{2013} -1$. I need to prove that: (1). A is diagonalizable over the complex field. (2). If A is diagonalizable over the reals, ...
2
votes
0answers
65 views

Rank Of A Matrix Under Special Conditions

Let A be a $N*N$ matrix. Now A is defined in a special manner: Each row of A is defined by two integers L and R ($0\le L,R\le {N-1}$), such that all elements from the $L^{th}$ to the $R^{th}$ are all ...
2
votes
0answers
72 views

Proving commutation relation in Algebraic Bethe Ansatz

I have a problem with proving a certain commutation relation. For my Bachelor's thesis I give a more mathematically rigurous 'treatment' of a select set of chapters of a paper by L.D. Faddeev. Noting ...
2
votes
0answers
34 views

Property of Perron root of non-negative matrix

Let's have $A_{1}=\begin{bmatrix}3 & 1 & 0 & 0 & 0\\ 1 & 2 & 1 & 1 & 0\\ 0 & 1 & 3 & 0 & 1\\ 0 & 1 & 0 & 4 & 0\\ 0 & 0 & 1 ...
2
votes
1answer
71 views

How much linearly independent? or linearly dependent?

I want to improve a rank-deficient matrix by augmenting a row vector to it. However, unfortunately, I have only very 'similar' vectors.. For example, my matrix is somewhat like.. \begin{bmatrix} 1 ...
2
votes
1answer
17 views

Division By Single Item Matrices

Okay, I'm given $$P=\begin{bmatrix} 1 & 1 \end{bmatrix}^T$$ and given 4 different sections of problem. Before I continue, I want make sure my suspicion is correct: $$P=\begin{bmatrix} 1 \\ 1 ...
2
votes
0answers
329 views

How Would Arnold Explain the Jordan Normal Form to a 6 Year Old?

How would Vladimir Arnold explain the Jordan normal form, to a six year old, in full detail starting from nothing in a way that somehow explains everything in a deeper way, probably including topology ...
2
votes
0answers
48 views

linear algebra formulation help

I asked this on mathoverflow as well and apologies for cross-posting. I am trying to compute this so-called bending energy matrix. The bending energy of a thin plate in 3D is given by: $$ BE = ...
2
votes
0answers
167 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
2
votes
0answers
49 views

Continuity of the orthogonal matrix-valued function

Given $d<k$. Suppose that $H:\mathbb{R}^k\rightarrow {\cal M}(\mathbb{R})_{d,k}$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$. Now define a ...
2
votes
1answer
133 views

The density of diagonalizable matrices of $M_n(\mathbb{C})$ problem.

For any matrix $A = (a_{ij})_{1\leq i,j\leq n} \in M_n(\mathbb{C})$, we pose $||A|| = \max_{1\leq i,j\leq n} |a_{ij}|$. $1.$ Show that $||.||$ define a norm on $M_n(\mathbb{C})$ and that $\forall A, ...
2
votes
1answer
76 views

Matrix for linear map involving polynomials

I need to find the matrix corresponding to the linear map $f:V_3 \rightarrow V_3$, where $V_3$ is the vector space of all polynomials of degree less than or equal to 3, $$f(p(X))=p(X)-p'(X)$$, with ...
2
votes
0answers
196 views

Decompose $A=D+N$ with $DN=ND$, $N$ nilpotent, $D$ diagonalizable

Can anyone help me out with the following question: For the matrix $A$ give a diagonalizable matrix $D$ and a nilpotent matrix $N$ so that $A=D+N$ and $ND=DN$. $\begin{bmatrix} 1 & 4 \\ -1 & ...
2
votes
0answers
75 views

How to express B-spline basis function in matrix format

Can someone help me with the following Question? I am not sure if B-spline/NURBS can express as basic function in matrix, as, $$ x(t) = B(t)c $$ $$ B(t) = [b_1(t) ... b_M(t)] $$ in which x(t) is ...
2
votes
1answer
70 views

probability matrix with trace $1$ is square of probability matrix

Consider as probability matrix a matrix $M \in [0,1]^{n \times n}$ while every row sums up to $1$. Statement: Consider a $2\times 2$ probability matrix $M' \in [0,1]^{2 \times 2}$. Show, that the ...
2
votes
0answers
50 views

Conjugacy classes of unipotent $\mathbb{Z}\times\mathbb{Z}$ in $GL_3(\mathbb{Q})$

Let $G=\mathrm{GL}_3(\mathbb{Q})$. Now, consider all subgroups in $G$ of the form $\mathbb{Z}\times\mathbb{Z}$ consisting only of unipotent elements (elements whose eigenvalues are all $1$). How ...
2
votes
0answers
75 views

Solving a set of matrix equations

There are $k$ matrix equations with the same unknown $\mathbf{X}$: $\mathbf{A}_i(\mathbf{D}_i-\mathbf{X})^{-1}\mathbf{B}_i=\mathbf{C}_i$ where $i=1,2,...,k$. $\mathbf{A}_i$ is a $m\times n$ matrix. ...
2
votes
0answers
175 views

Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
2
votes
1answer
84 views

about symmetric and hermitian matrices

Using the frobenius norm $\langle A,B\rangle=\textrm{Tr}(B^{\star}A)$, where $B^{\star}=\overline{(B^T)}$, How can I prove that $\mathbb{S}(\mathbb{C})^{\perp}=A\mathbb{S}(\mathbb{C})$, that is that ...
2
votes
0answers
173 views

Diagonalization of Vandermonde matrix

Is there a method to diagonalize (at least some) $ n \times n $ Vandermonde matrices? For example invertible matrices which has method to invert them with Cramer method for example, but there is some ...
2
votes
0answers
28 views

$C$ Hermitian, show $\text{Tr}(C)=0 \iff \exists P,Q \text{ Hermitian s.t.} \,\, PQ-QP=iC$ [duplicate]

Let $C$ be an $n\times n$ Hermitian matrix. Show that $$\text{Tr}(C)=0 \iff \exists P,Q \text{ Hermitian s.t.} \,\, PQ-QP=iC.$$ Ideas: The right-to-left direction I have no problem with. For the ...
2
votes
1answer
233 views

Consequences when the commutator is a scalar multiple of the identity matrix

I just stumbled over the question below. As to the first, I could easily find out the answer (D) by invoking the commutation relation. But I don't figure out how to solve other two. Could anybody give ...
2
votes
1answer
87 views

Congruence of a matrix

Let $X=\begin{bmatrix}a&b&0&0\\ c&d&0&0\\0&0&-a&-b\\0&0&-c&-d\end{bmatrix}$ where $a,b,c,d\in \mathbb{Z}$. For a such given $X$, is there a $4\times ...
2
votes
0answers
58 views

rank of formal derivation

Let $K$ be a field, $n \in \mathbb{N}_{>0}$ and $K[x]_{ \leq n} $ he space of polynomials above $K$, that have a maximum degree of $n$. We define the formal derivation as follow: $\frac{d}{dx}= ...
2
votes
2answers
130 views

About linear transformations

Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$ $$T(A)(B)=AB-BA$$ I have to prove that this is a linear ...
2
votes
0answers
66 views

How many rotation matrices with simple rational entries

This is a follow-on from this earlier question, which asked for examples of simple rotation matrices. I'm interested in rotation matrices whose entries are simple rational numbers, because these are ...
2
votes
1answer
57 views

How to prove a set is convex

Let $E = \{x\mid (x - c)^{T} P^{-1} (x-c) \le 1 \}$, where $P$ is symmetric positive definite. Show that $E$ is convex. Here is what I did. It seems like $E$ is an ellipsoid. We want to show that ...
2
votes
2answers
250 views

Can I use determinants to show that two vector sets span the same subspace?

I have two sets of vectors, like these: $v_1 = (1, 6, 4)$ $v_2 = (2, 4, -1)$ $v_3 = (-1, 2, 5)$ in set $V$ $w_1 = (1, -2, 5)$ $w_2 = (0, 8, 9)$ in set $W$ I need to show that $V$ and $W$ span the ...
2
votes
1answer
432 views

Relation between Algebraic multiplicities and rank of a matrix

A is a 6x6 matrix, $rank(A-3I) = 4$, the minimal polynomial of A is $(x-1)^2(x-3)^2$ I need to write the Jordan matrix options for A. How can I use the given information about the rank, what does ...
2
votes
0answers
71 views

What would this set look like

Let $S\subseteq\mathbb{R}^{3}$ be the set of $\left(x,y,z\right)$, $x\ge y\ge z$ , which are the three eigenvalues of $diag\left(1,2,3\right)+Udiag\left(-1,-2,-4\right)U^{T}$, where $U$ is an ...
2
votes
0answers
70 views

Show $r(F)=r(F^2)$ implies $Im(F) \cap Ker(F)=\{0\}$

I wonder if I've made some mistakes in the proof of the following or if there is some simpler solution. Problem: Let $V$ be a finite dimensional vectorspace and $F:V \rightarrow V$ a linear operator. ...
2
votes
1answer
49 views

Proving that $M_p(M_q (K)) \cong M_{pq} (K)$.

My textbook finishes the proof of one of the theorems with the following fact: $$ M_p(M_q (K)) \cong M_{pq} (K), $$ where $K$ is a field, and it says that it is true by "block multiplication ...
2
votes
1answer
70 views

Using matrix theory to solve this problem

I'm sorry that I couldn't find a better title for this. I was wondering if my solution is valid for the following problem, or if I've made some mistake. Problem: Let $N=\{a_1, \dots, a_n\}$ be a ...
2
votes
0answers
167 views

Algorithm for reversion of power series?

Given a function $f(x)$ of the form: $$f(x) = x/(a_0x^0+a_1x^1+a_2x^2+a_3x^3+a_4x^4+a_5x^5+...a_nx^n)$$ Let $A$ be an arbitrary (any) infinite lower triangular matrix with ones in the diagonal: $$A ...
2
votes
1answer
61 views

the rank of matrix products including a commutation matrix

Given a full rank matrix $A \in \mathbb{R}^{M \times N^2}$ where the rank of ${A}$ is ${\rm rk}(A)= M \leq N^2$ and the commutation matrix $K_{NN}$. I need to find the rank of a matrix product ...
2
votes
2answers
83 views

Why doesn't the minimal polynomial of a matrix change if we extend the field?

Why doesn't the minimal polynomial of a matrix change if we extend the field? I appreciate any help or proof.
2
votes
0answers
41 views

Semidefinite Program formulation

I have the following problem and would like to formulate that as an SDP. I am not sure how to approach this : A set $S$ is given such that : $$ S = \{P \in R^{n \times m} : ||p_i - c_i|| \leq d_i ...
2
votes
0answers
37 views

Showing a Hermitian preserving map is not necessarily positivity preserving?

Say I have a linear map, which is not positivity preserving $$\phi: \mathscr H \to \mathscr H$$where $\mathscr H$ is the set of $n \times n$ Hermitian matrices. Then does there exist a positive ...
2
votes
0answers
91 views

How to compute the eigenvalues of a block matrix with a special structure and when the submatrices are square.

I have the next block matrix $$ M = \begin{bmatrix}A & B \\ C &D\end{bmatrix}, $$ where $A, D$ are Hurwitz (eigenvalues with negative real part) square matrices of different dimensions and ...
2
votes
3answers
130 views

Do these matrices have any name?

Assume $A$ is a square matrix defined as follow: $$A=\sum_{i} u_{i}u_{i}^T$$ where for each $i$, $u_i$ is a non-negative column vector. Do the matrices of these forms have any special name?
2
votes
0answers
46 views

How to calculate only the first row of $B$ square matrix in $AB=I$ without evaluating the whole one?

I am dealing with the following matrix equation. $$AB= I.$$ All are square matrices. $A$ is a known tridiagonal matrix, $I$ is identity matrix. Since $B$ is unusually large, I wonder if it is ...
2
votes
0answers
49 views

Parametric QR factorization: $\mathbf{D}(\alpha)\mathbf{V}^*$ a diagonal times a constant unitary matrix

Given a constant unitary matrix $\mathbf{V}^*$ and a parameter diagonal matrix $\mathbf{D}$, can the QR factorization of many different $\mathbf{D}\mathbf{V}^*$ be performed efficiently? Here ...
2
votes
0answers
73 views

Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
2
votes
0answers
68 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
2
votes
3answers
186 views

Chain rule for matrix exponentials

I need help in proving the following theorem: If $M(t)$ is an $n \times n$ matrix of differentiable functions, then $$ \frac{d}{dt}\left( \exp(M(t))\right) = \frac{d}{dt}M(t) \exp(M(t)) = ...
2
votes
0answers
134 views

Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$ \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$ Applying the Woodbury matrix identity ...
2
votes
0answers
200 views

Using Rref to find the inverse of a matrix.

Since, I can't divide vectors to deduce an inverse matrix I have dismissed that approach. I did find that if I multiply all of the matrix row operators It will yield the inverse. Since I did the logic ...
2
votes
0answers
362 views

“A dominant eigenvalue of a non-negative matrix has a non-negative eigenvector”

I have the non-negative 3x3 matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 4 & 1 \\ 3 & 2 & 1 \end{bmatrix}$. I've calculated the eigenvalues of this matrix, ...
2
votes
3answers
251 views

system of equation with 3 unknown

Solve $$\begin{matrix}i \\ ii \\ iii\end{matrix}\left\{\begin{matrix}x-y-az=1\\ -2x+2y-z=2\\ 2x+2y+bz=-2\end{matrix}\right.$$ For which $a$ does the equation have no solution one solution $\infty$ ...
2
votes
0answers
31 views

comparison between matrices

Let $M_1$ and $M_2$ be two symplectic matrices of dimension $2n_1\times 2n_1$ and $2n_2\times 2n_2$ respectively; Let $P_1>0$ and $P_2>0$ be the positive definite solutions of the following ...