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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0
votes
1answer
19 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 ...
5
votes
2answers
119 views
+100

Given a symmetric positive-definite matrix $M$, find all $A$ such that $A^\top M A=M$

Given $M$ a real symmetric positive-definite matrix, I would like to characterise all matrices $A$ such that $A^\top M A=M$. Note that the question of finding $A$ solutions to $A^\top M A=M$ for all ...
1
vote
1answer
8 views

Duality theorem between Cycle Space and Cut Space in terms of Matrices?

The book Graphs and Matrices by Bapat formulates linear algebra on graph theory, yet I cannot find important theorems such as Duality theorem between the cycle space and the cut space (Diestel p.26, ...
1
vote
2answers
487 views

Cardinality of a set of matrices

Consider the set $S$ of $3\times3$ matrices with binary coefficients, that is, the coefficients are integers modulo $2$. Compute $|S|$. I am not sure what is this question trying to ask. Am I ...
1
vote
2answers
28 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 ...
1
vote
0answers
30 views

Perturbation of the principal eigenvector of a PSD matrix

I have a $n \times n$ PSD matrix $A$ and $\tilde{A}=A+E$ be its symmetric perturbation. Let $\|E\|_2=\epsilon.$ Let $(\lambda,u)$ be the principal eigenvalue, eigenvector pair of $A$ and ...
1
vote
2answers
36 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\\ ...
5
votes
3answers
57 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
44 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
votes
0answers
20 views

How many solutions will the system have?

I performed 2 column operations: Col(1)2 and Col(3)-4 and in order for matrix A to be equal to the augmented matrix the variables x1=x2=x3=1.Part b) Since one row in matrix is a multiple of another ...
1
vote
2answers
139 views

Finding the reflection that reflects in an arbitrary line y=mx+b

How can I find the reflection that reflects in an arbitrary line, $y=mx+b$ I've examples where it's $y=mx$ without taking in the factor of $b$ But I want to know how you can take in the factor of ...
0
votes
1answer
16 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: ...
1
vote
1answer
23 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 ...
2
votes
0answers
8 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 ...
3
votes
2answers
47 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). ...
-5
votes
1answer
63 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 \\ ...
0
votes
1answer
22 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
0answers
14 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 ...
1
vote
0answers
31 views

Matrices and cofactors question

From Mathematical Methods for Engineers and Scientists 1 by K.T. Tang: Example 4.6.1. Evaluate ...
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 ...
2
votes
1answer
56 views

Finding centralizer of a matrix in general linear group.

I saw the following question from Gallian's book on abstract algebra. I am required to find the centralizer of the matrix $$A= \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ ...
4
votes
2answers
100 views

Prove or disprove: For $2\times 2$ matrices $A$ and $B$, if $(AB)^2=0$, then $(BA)^2=0$.

The question is in the title. my thoughts on the question: I know that $AB=O$ does not imply that $BA=O$, so my first impression was that it is false. I tried the counter-example I know but it ...
0
votes
1answer
46 views

If $(AB)^2=0$ then $(BA)^2=0$ (for $3\times2$ matrix $A$ and $2\times3$ matrix $B$)

Let $A$ be a $3 \times 2$ matrix, and $B$ be a $2\times 3$ matrix. If $(AB)^2 = 0$, then how can I show that $(BA)^2 = 0$? I have done some calculation but it's too much. I was wondering if there is ...
-4
votes
1answer
27 views

Cardinality of the set of $2\times 2$ real matrices with determinant $1$ [on hold]

Please, I need the cardinality of the set of $2\times 2$ real matrices with determinant $1$. I have no idea where to start.
1
vote
3answers
50 views

Is there any way to know the algebraic multiplicity of the $0$ eigenvalue in the minimal polynomial when the rank is $1$? [duplicate]

Say I have a matrix $A$ of $r=rank(A)=1$ I know that in the characteristic polynomial the algebraic multiplicity of $(\lambda-0)$ is $n-r$ which in my case is $n-1$ Is there a rule about the ...
1
vote
1answer
22 views

Constructing a determinantal inequality

The following is from page 3410 of the paper Quadratically constrained attitude control via semidefinite programming. Consider a polynomial: $$\mu_1(p_1^Tx)^2+ \cdots + \mu_n(p_n^Tx)^2\leq a$$ ...
0
votes
1answer
65 views

Upper Triangular Matrices of Monotone Vectors

I am looking for references to the following problem (I'm actually interested in general $n$, but will use $n=3$ as an example): consider a finite set, for example, $N = \{1,2,3\}$, and the associated ...
0
votes
2answers
137 views

How to transform between two layout forms of matrix calculus?

I'm trying to derive a very simple matrix derivative: Take the derivative of $\operatorname{Tr}(A' X)$ with respect to $X$. However, I got two different answers by following different methods. ...
1
vote
1answer
32 views

Finding the values of a vector if the vector.matrix product and the value of the matrix is known (only using left multiplication operations)

Given an unknown input vector $V= (v_1, v_2, v_3, v_4)$, a known $4\times 4$ matrix $A$ and a known vector-matrix product $M=[m_1,...,m_4]$. Can you discover $V$? Normally you would just take the ...
-1
votes
2answers
44 views

Determine whether a set of 4, 2x2 matrices form a base for M2.

I am having a hard time solving this question: Let $A,B,C,D,E$ be $2\times2$ matrices above R field. If $\{\,AE,BE,CE,DE\,\}$ linearly independent then $E$ must be an invertible matrix. it feels ...
3
votes
2answers
32 views

Show that the sequence of norms of inverses of a convergent sequence of matrices diverges to infinity.

This is a question I found while working on the book "Analysis in Euclidean Spaces" by Ken Hoffman. Suppose $(A_n)$ is a sequence of invertible matrices from $\mathbb{R}^{k \times k}$ that ...
2
votes
1answer
2k 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 ...
4
votes
5answers
1k views

How can I calculate a $4\times 4$ rotation matrix to match a 4d direction vector?

I have two 4d vectors, and need to calculate a $4\times 4$ rotation matrix to point from one to the other. edit - I'm getting an idea of how to do it conceptually: find the plane in which the vectors ...
0
votes
0answers
6 views

Problem on Principal Component Analysis (P.C.A.)

Let $X \; = \; (X_1, X_2, \ldots, X_m)^T$ and $Y \; = \; (Y_1, Y_2, \ldots, Y_n)^T$. Let, $S$ = pooled variance-covariance matrix obtained from $X$ and $Y$. Let, $\alpha$ = principal component ...
1
vote
0answers
19 views

Inverse of a triangular block matrix (sufficient and necessary conditions for the existence)

Consider the following theorem: Let $\mathrm{\mathbf{A}}\in\mathbb{C}^{n\times n}$ be an upper triangular block matrix with $2\times2$ blocks ($p=2$), i. e., ...
13
votes
1answer
381 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
2
votes
2answers
58 views

Eigenvalues and eigenvectors of the Householder matrix $H = I - \frac{2}{u^Tu} uu^T$

So during my first revision for the semester exams, I went through exercises in books/internet and I found 2-3 that caught my eye. One of them was the following: Let $u \in \mathbb R^n$ be a ...
4
votes
1answer
148 views
+50

Multiplication of unitary matrices to make symmetric off-diagonal elements zero

Context Starting with a unitary matrix $U$ of size $m \times m$, I have read of a way to obtain a diagonal matrix by sequentially multiplying $U$ from the right by unitary matrices $V$ of a certain ...
0
votes
0answers
25 views

How do I convert an x,y,z to an Q configuration?

I am trying to implement a tracking application for a robot arm, which purpose is relocate itself based on the position of an object seen from the tool point. The robot arm itself is a UR5, and at ...
3
votes
1answer
29 views

Showing a map is nilpotent

Let $\mathbf{A},\mathbf{B}\in\mathrm{M}_n(\mathbb{R})$ such that $\mathbf{A}$ invertible and diagonalisable, and $\mathbf{AB}=\lambda \mathbf{BA}$ for some $\lambda >1$. I want to show that ...
1
vote
2answers
18 views

For an invertible linear transformation $T$, is there a basis for which the representation of $T$ is an arbitrary invertible matrix $B$?

Assume that $V$ is a $n$-dimensional $k$-vector space where, $k$ is an algebraically closed field. Furthermore, assume that $T: V \rightarrow V$ is an endomorphism such that with respect to an ordered ...
10
votes
2answers
2k views

Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$

As the title says, am searching for a proof of If $A,B \in \mathbb{R}^{n\times n}$ and $AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B) \leq n$ I am doing this as preparation for an upcoming ...
6
votes
3answers
1k views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
0
votes
1answer
21 views

Is summation of some binary invertible matrices, invertible?

Let A and B be 2×2 matrices over $Z$. If A, A+B, A+2B, A+3B, A+4B are invertible, and all the elements of their inverses are integer. Show that A+5B is invertible and all its elements are integer. We ...
3
votes
2answers
60 views

Solution of $A^\top M A=M$ for all $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can't find out it there are other such matrices and if so, ...
0
votes
1answer
1k views

Difference between rotation and pure rotation

Hi i am trying to understand my teacher's assignment. I have 2 write 2 Matlab functions ...
-1
votes
0answers
14 views

Cardinality of Matrix set [on hold]

I need the Cardinality of the set of $2x2$ matrices with determinant $1$. I have no idea where to start yet.
0
votes
0answers
17 views

Can $0 \preceq A\preceq B$ and $span(A)\subseteq span(B)$ lead to $\sigma_i(A)\leq \sigma_i(B)$?

Can $0 \preceq A\preceq B$ and $span(A)\subseteq span(B)$ lead to $\sigma_i(A)\leq \sigma_i(B)$? In the question, $A\preceq B$ means that $B-A$ is a positive semi-definite matrix, $span(A)\subseteq ...
1
vote
1answer
27 views

Clever way to prove $\langle A,X\rangle=x^TAx$ with $X=xx^T$, $A\in S^n$?

How to prove $\langle A,X\rangle=x^TAx$ with $X=xx^T$, $A\in S^n$? (inner product of matrices) $xx^T$ is rank one. The following is one way to prove it: $$\langle A,X\rangle=\text {tr}(AX)$$ ...
0
votes
1answer
2k views

Find the values of $a,b,c$ such that a matrix has infinite, unique, and no solutions.

Find the values of $a$, $b$ and $c$ such that a matrix has infinite, unique, and no solutions. $$x+y=0$$ $$y+z=0$$ $$x+z=0$$ $$ax+by+cz=0$$ We can't use determinants so I turned the equations into ...