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)

1
vote
0answers
15 views

Similarity classes of matrices

Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? ...
4
votes
1answer
25 views

Given block matrix $M$, show determinant relationship between $M$ and the block elements of $M.$

Given that $M = \begin{pmatrix} A & B \\ C &D \end{pmatrix}$ and $M^{-1} = \begin{pmatrix} P & Q \\ R & S \end{pmatrix},$ where $A, B,\dots$ are $k \times k$ matrices, show that ...
1
vote
2answers
29 views

Symbol for excluding one element of a vector

Let $x_i \in x := (x_1,\ldots,x_n)$. I want to define $x_{-i}$ such that $x_{-i}$ contains every element of $x$ except $x_i$, i.e. $x_i \not \in x_{-i}:=(\ldots,x_{i-1},x_{i+1},\ldots)$. Is there ...
0
votes
1answer
32 views

prove $||A_n^{-1}||$ is bounded

Let $(A_n)_{n=1}^\infty{} \in GL_{n\times n}(\mathbb{R}) $. $\lim_{n\rightarrow\infty} A_n = A$; $A\neq0$ is invertible. I have a notion that for any norm $$||\cdot||:GL_{n\times n}(\mathbb{R}) ...
0
votes
0answers
24 views

Gaussian process for machine learnig

Here is my question in the equation 2.11 A is N by N matrix, so there is not feasible if N is large the textbook say in the euqation 2.12, we only need to invert size n by n. But I think K is 1 by ...
1
vote
1answer
43 views

Expected number of times to get arbitrary arrangement of coins

I'm thinking about a question: We consider tossing coins repeatedly. Using $+1$ to denote front and $-1$ back, given a positive interger $m$ and $\sigma=(\sigma_1,\dots,\sigma_m)$ where ...
7
votes
5answers
115 views

Compute $\det{T}$ where $T(X)=AX+XA$

Consider the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$ Compute the determinant $\det ...
5
votes
0answers
31 views

Probability that two random matrices span the full matrix algebra

Given two matrices $A$ and $B$ drawn at random in $\mathbb{R}^{n\times n}$, what is the probability that the matrix algebra generated by $A$ and $B$ is the full matrix algebra $\mathbb{R}^{n\times ...
0
votes
1answer
28 views

Bound for symmetric part of matrix

Let $A $ be an $n \times n $ matrix such that $AA^T \geq x^2I, x\geq 0 $, which means that the matrix $AA^T-x^2I$ is positive semidefinite. Can we show that $(A+A^T)/2 \geq xI$? Thanks
0
votes
1answer
20 views

A fact about symmetric matrices and square roots

Is it true that if $A$ is symmetric then any square root is symmetric? I can't prove this using basic symbolic computation, so what if we insist that $A$ is diagonalizable, or even positive definite?
2
votes
1answer
17 views

Continuous dependence of matrix elements

I've stumbled upon several solution of linear algebra problems which use notion of "continuous dependence" of matrix polynomials on matrix elements. For instance (translated, so any inaccuracies are ...
0
votes
0answers
17 views

Linear recurrent sequences and matrices.

Let $k$ be a field, let $d$ be an integer greater than $1$, let $(v,x)\in k^d\times k^d$ and let $A\in k^{d\times d}$ be invertible. For all $n\in\mathbb{N}$, let define the following element of ...
0
votes
1answer
28 views

Show that $MX = 0$ where $M =\mathbb I_n − X(X'X)^{-1}X'$ [on hold]

Show that $MX = 0$ where $$ M = \Bbb I_n - X(X'X)^{-1}X' $$ Can someone help me?
0
votes
1answer
39 views

Are matrices 2D by definition?

On the one hand, I read on Wikipedia that [A] matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. However, googling "3D matrix" ...
1
vote
0answers
29 views

Orientability of differantiable manifold of orthogonal matrices

I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal ...
0
votes
1answer
26 views

Finding a polynomial to satisfy a matrix equation

Is there a canonical way of finding a polynomial $p$ such that $$ p\left(\begin{bmatrix} 1& 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 ...
1
vote
1answer
22 views

Determinant comparison about skew-symmetric matrices

Suppose $S$ is a real skew-symmetric matrix, show that $\det(I+S) \geq 1$, where equality holds iff $S=0$. My idea is to define a function $f(t)=\det(I+tS)$, for a fixed $S \neq 0$, and then show ...
0
votes
1answer
12 views

Matrix for Mixed Boundary Value Problem

My friend and I have been working on numerical solving the following equation $$-u'' = f$$ with $x \in [0,1] $ , $ u'(0) = 0$, $u(1) = 0$. Analytically, we found the eigenvalues and eigenfunctions ...
1
vote
0answers
23 views

Understanding Matrix and Vector Notation

I am trying to understand the Matrix and Vector Notations on page 2 here: (the page is also pasted below, to make it easier to explain the problem). Problem: For equation (2), I think it should be ...
0
votes
2answers
32 views

Prove/Disprove question on matrix vector multiplication and linear independence [on hold]

If $\left\{Bv_1, \ldots , Bv_k\right\}$ is a linearly independent set in $\mathbb{R}^k$ where $B$ is a $k \times n$ matrix, then $\left\{v_1, \ldots ,v_k\right\}$ is a linearly independent set in ...
1
vote
1answer
25 views

General structure of a 3 by 3 persymmetric matrix with zero eigenvalues

Here is an interesting problem that might be extended to higher dimensions, I am looking for different simple ways of describing a 3 by 3 matrix with one or two zero eigenvalues. This persymmetric ...
2
votes
1answer
23 views

Sum of powers of eigenvalues

The sum of the eigenvalues $\lambda_k$ of an $n\times n$ matrix is equal to the trace of the matrix, i.e. $$\sum_{k=0}^{n-1}\lambda_k=\text{tr}(A).$$ Is there a "closed form" sum of positive integer ...
2
votes
0answers
18 views

Defective eigenvalues problem - determining defect given a relation among elements, deducing third linearly independent eigenvector, etc.

Suppose I have the following matrix: $$\begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 1 \\ -2 & -4 & -1 \\ \end{bmatrix}$$ The only eigenvalue of which ...
1
vote
1answer
21 views

Calculate the matrix from minimal polynomial and eigenspaces

I need to find a matrix $A \in M(6,\mathbb C)$ that satisfies following: $e_1+e_2+e_3\in \ker(L_A-3\cdot \operatorname{id})^3 \setminus \ker(L_A-3*id)^2$ $\operatorname{span}(e_1+e_4,e_5+e_6)= ...
0
votes
0answers
6 views

Name for a “pseudo-diagonally dominant matrix”

I'm doing a literature search, and I'm just wondering if there is a name for (Hermitian positive definite) matrices which have the sum of the off-diagonals in any given row dominated by the diagonal ...
0
votes
1answer
19 views

Finding a unit length vector that maximises a sums to zero for a linear equation

I have a weight vector $w^T = [1, 4, 9]$ and the linear unit: $u = w^Tv$, where $u$ is the output. q1) How do I find the unit length vector $v$ that maximises the output from the unit? q2) Also how ...
0
votes
0answers
14 views

GMM with full and diagonal covariances

I have Gaussian Mixture Model-- distribution with probability density function, that is a weighted sum of Gaussian probability density functions: \begin{equation} p(X)=\sum_{i=1}^k ...
0
votes
1answer
23 views

Spectral theorem for matrices…

From the spectral theorem we know that if $A$ is a symmetric matrix then there exists an orthogonal matrix $M$ such that $A=M^{-1}DM=M^TAM$ My question is: if I have the matrix $A$ how do I find ...
4
votes
3answers
84 views

Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular

Problem: Let $A$ and $B$ be real orthogonal matrices, $n$x$n$, where $n$ is an odd number. Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular. What have I done so far: ...
3
votes
2answers
28 views

Eigenvalue of a matrix and a polynomial of that matrix

Let $A$ be a $n \times n$ matrix over $F$, and let $c_1, ... c_n$ be its eigenvalues. Show that for every polynomial $g(x) \in F[x]$, the eigenvalues of $g(A)$ are $g(c_1), ... , g(c_n)$. I think by ...
1
vote
0answers
27 views

When is $adj(A)$ nilpotent?

Are there any conditions regarding the $adj(A)$ being nilpotent for some square matrix A?
0
votes
0answers
19 views

Equations involving orthogonal matrices

My variables are: $O_1,\dots,O_M\in \mathbb{O}(d)$ I have equations like: $O_{1_1} x_{1,1}+ \dots + O_{1_{m_1}} x_{1,m_1} = x_{1,(m_1+1)}$ $\vdots$ $O_{n_1} x_{n,1}+ \dots + O_{n_{m_n}} x_{n,m_n} ...
3
votes
2answers
36 views

Given a matrix $A$ with $\operatorname{tr} (A) = 0$, prove that there is a B such that $\forall 1\leq i\leq n :(B^{-1}AB)_{i,i}=0$

I've tried using some matrices $B^{-1}$ that switch the rows, but the $B$ at the end placed the elements back in the diagonal (in different order) so I couldn't find a rule.
1
vote
1answer
31 views

eigenvalues of $g(A)$

Let $A\in M_n(F)$ and let $c_1,\dots,c_n$ be eigenvalues of $A$. Prove that for each polynomial $g(x) \in F[x]$, eigenvalues of $g(A)$ are $g(c_1), \dots g(c_n)$. (Hint: triangulate $A$) I don't ...
5
votes
3answers
56 views

If $\lambda$ is the eigen-value of a $n\times n$ non-singular orthogonal matrix $A$, then prove that $\frac{1}{\lambda}$ is also an eigen-value.

QUESTION: If $\lambda$ is the eigen-value of a $n\times n$ non-singular matrix $A$ and $A$ is a real orthogonal matrix, then prove that $\frac{1}{\lambda}$ is an eigen-value of the matrix ...
1
vote
1answer
88 views

Calculating SVD By Hand

When calculating the SVD of the matrix $$A = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix}$$ I followed these steps $$A A^{T} = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix} ...
1
vote
1answer
28 views

How to solve a form of tridiagonal system of equations (with fringes?)

I have read about the Thomas algorithm. Right now, I am trying to use it to solve the following linear system $$\begin{bmatrix} b_1 & c_1 & 0 & \dots & 0 & a_1\\ a_2 & b_ 2 ...
1
vote
2answers
42 views

If $A$ is a matrix with negative eigenvalues, then $\exists M$ : $A = -MM^T$

Let $A$ be a symmetric matrix with all its eigenvalues negative. Prove that there exists a matrix $M$ such that : $A = -MM^T$. Now, regarding my question, I have found another older question, that ...
0
votes
1answer
48 views

Properties of determinants

Prove using properties of determinants : \begin{equation*} \left|\begin{matrix} b^2 + c^2 & a^2 & a^2\\ b^2 & c^2 + a^2 & b^2\\ c^2 & c^2 & a^2 + b^2 \end{matrix}\right| = ...
1
vote
2answers
44 views

Prove that $U$ is a vector-subspace

If $U$ is the set of all matrices that are commutative with the matrix $A$, show that $U$ is a vector subspace of the space $M^\mathbb{R}_{3\times 3}$ $$A=\begin{pmatrix}2&0&1\\ ...
0
votes
1answer
18 views

Problem with change of basis of an polynomial.

Good morning, i have a problem solving this: Express $a_{0}+a_{1}x+a_{2}x^{2}$ in terms of basis: $1,x-1,x^{2}-1$ I make this: ...
2
votes
0answers
12 views

Positive Definite Matrix Induced by Lorentz Matrix

Assume $G$ is a Lorentzian matrix, which means it has signature $(+,-,\cdots,-)$, and $v$ is a unit timelike vector, i.e. $v^TGv=1$. So do we have that matrix $2Gvv^TG-G$ is positive definite? Any ...
1
vote
1answer
29 views

Kernel and geometric multiplicity relation

Say I have a square matrix $A$ with one eigenvalue $\lambda_1$. The minimal polynomial is $(\lambda-\lambda_1)^k$ and $\dim(\ker(A))=\alpha$. What can I know about the geometric multiplicity of ...
-5
votes
1answer
73 views

Consider the Matrix $A$, Find $A^{100}$ [on hold]

Consider the Matrix $A$ = $ \left( \begin{array}{ccc} 1 & -2 & 8 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right)$ or $A$ = $ \left( \begin{array}{ccc} 1 & 0 & 0 \\ ...
3
votes
2answers
60 views

Prove two complex matrices have null trace

Let $A,B \in \mathbb{C}^{2 \times 2} \setminus \{O_2\}$, where $AB=-BA$ and $\det(A+B)=0$. Prove that $\operatorname{tr}(A) = \operatorname{tr}(B) = 0$ (where $\operatorname{tr}$ is the trace). ...
1
vote
0answers
23 views

Set of all positive definite matrices with off diagonal elements negative

Please, help me to find the set of all positive definite matrices (PDM) of which off diagonal elements are negative. Considering the case with n=2, the symmetric mat $A=[a_1, a_2;a_2, a_3]$ needs to ...
0
votes
0answers
8 views

matrix function diagonalization

Like diagonalization of a constant matrix is it possible to diagonalize a matrix function $\phi(t)$ if $t\in(0,T)$ i.e., if there exist $ P(t)$ suchthat $P^{-1}(t)\phi(t)P(t)=D(t)$ in all cases? or is ...
5
votes
3answers
66 views

What are all the uses of the determinant?

I've learned how to calculate the determinant but what is the determinant used for? So far, I only know that there is no inverse if the determinant is 0.
0
votes
1answer
23 views

Matrix for rounding to the nearest whole number

There are two barbers in a town. Of the people who go to the 'good' barber, $92\%$ will go to the good barber again the next time. Of the people who go to the 'bad' barber, $18\%$ will go to the ...
1
vote
1answer
24 views

Constructing a determinantal inequality

The following is from page 3410 of the paper Quadratically constrained attitude control via semidefinite programming. Consider a polynomial: $$\mu_1(p_1^Tx)^2+ \cdots + \mu_n(p_n^Tx)^2\leq a$$ ...