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
47 views

Signs of real part eigenvalues for nonsymmetric matrix.

Since the search of the eigenvalues is in general not "simple", equally valid, is the method of reducing with moves of Gauss, that preserve the determinant, (add to multiple rows of other rows, move ...
2
votes
2answers
33 views

An “apparent” contradiction for eigenvalues signs of $A=\left( \begin{array}{cc} a & -a \\ a-1 & 1-a \\ \end{array} \right)$.

The following matrix $$A=\left( \begin{array}{cc} a & -a \\ a-1 & 1-a \\ \end{array} \right)$$ has eigenvalues $\lambda=0,1$ $\forall a\in\mathbb R$. Therefore $\lambda\geq 0$. ...
0
votes
1answer
32 views

Use well-known relation: $\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O(\epsilon^2).$ for evaluate $\det(I + M)$.

Let $d_1$, $d_2\in[0,1]$. Use well-known relation: $$\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O(\epsilon^2).$$ for evaluate $$\det(I + M)$$ where $I$ is the $2\times2$ identity ...
1
vote
1answer
50 views

Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $v$

Let $V$ be the plane with equation $x_1 + 4x_2 + 2x_3 = 0$ in $\mathbb{R}^3$. Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $v = \begin{bmatrix} -12 \\ 4 ...
0
votes
1answer
28 views

If columns/rows of an $n\times n$ matrix $M$ are linearly independent what is the rank of $M$?

1.) I think I can answer the case when the rows are linearly independent vectors: Since the rows of the matrix $M$ are linearly independent, we cannot create an all $0$ row in the matrix therefore ...
0
votes
0answers
25 views

One of the following two matrix is symmetric positive definite?

Recently, I com across a questions, which can be given as follows, If there are two symmetric positive semi-definite matrix $W$ and $T$, but they satisfy the following condition: $null(W)\cap ...
2
votes
1answer
22 views

A construction of a Hadamard matrix

Let $H_n$ be a $2^n \times 2^n$ matrix indexed by all subsets of $[n] = \{1,\ldots,n\}$ and let the entry at the intersection of the row and column indexed by the sets $X$ and $Y$ be $$(-1)^{ |X \cap ...
4
votes
1answer
30 views

Show that trace is a unique linear functional [duplicate]

If $W$ is the space of $n \times n$ matrices over a field, and $f$ is a linear functional on $W$ such that $f(AB)=f(BA)$ for each $A,B$ in $W$, and $f(I)=n$, then $f$ is the trace function. I ...
0
votes
1answer
61 views

invertibility of $A^{-1} + B^{-1}$ [duplicate]

Let $A$ and $B$ be two invertible $n\times n$ real matrices. Assume that $A+B$ is invertible. Show that $A^{-1} + B^{-1}$ is also invertible.
-1
votes
0answers
19 views

How to transform this problem to a matrix optimization problem?

I wonder how can I loose the following set of equations to an optimization problem ? Suppose given three real vectors $w_0$, $f$ and $\delta$, and a positive entry vector $c$, such that for every ...
1
vote
1answer
32 views

Suppose $A$ is an $n \times n$ matrix. Prove that similarity transformation is reflexive. ie $A=P^{-1}AP $

I know a correct answer to this question is to just assume that $P$ is the identity matrix. But, is this argument BELOW correct. $$A=A \Rightarrow AP=AP\Rightarrow A=P^{-1}AP $$
0
votes
0answers
15 views

Impose linear constraints

I am implementing Han and Kanade's [1] perspective factorization method. I reached the point where I need to impose some constraints on the Motion matrix so it becomes valid. The matrix I want to ...
0
votes
1answer
28 views

Inequality about Frobenius norms for matrices [closed]

For any square matrix A, but not necessarily symmetric, what are some ways to prove the inequality $$ \|A^2\|_F^2\leq\|A^TA\|_F^2, $$ where $\|B\|_F^2=tr(B^TB)$ is the Frobenius norm of matrix $B$ ?
0
votes
0answers
22 views

Find convex efficient columns in Matrix

Consider a path-incidence matrix $A$ of a graph, where vertices are e.g. machines, paths are alternative production paths for a given product and entries $a_{ij}$ denote the workcontent for machine ...
2
votes
2answers
63 views

Calculate determinant of $ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $.

I have a block matrix $$ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $$ where $A$ is a $(n-1)\times (n-1)$ matrix, $\vec d,\ \vec c$ are two ...
0
votes
1answer
38 views

Computation of transformation matrix for jordan normal form: how to choose eigenvectors

During this semester at university we we're introduced to the jordan normal form of a matrix. While we never wrote down an explicit algorithm of how to find the matrix $B$, such that $B^{-1}AB$ is a ...
0
votes
1answer
35 views

What is called the following (matrix) operator?

Let $\mathbf{A}=\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1m} \\ A_{21} & A_{22} & \cdots & A_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ A_{m1} & A_{m2} ...
2
votes
1answer
49 views

Matrix and Abelian groups question

Let $A$ be a Matrix: $$ A=\begin{pmatrix} 1 & 2\\ 4 & 1 \end{pmatrix} $$ Let $f\colon v\to Av$ be a homomorphism from $Z^2$ to $Z^2$. Find a base $(v_1,v_2)$ to $Z^2$ and $2$ integers ...
2
votes
3answers
63 views

$A$ is a real orthogonal matrix, prove that $ (I+A)^{-1}(I-A)$ is skew-symmetric

$A$ is a real orthogonal matrix and $(I + A)$ is non-singular. Prove that $ (I+A)^{-1}(I-A)$ is a skew-symmetric matrix. Attempt: ...
3
votes
1answer
25 views

$n$ dimensional determinant using recurrence relations

Find determinant $$D_n(a,b,c)= \begin{vmatrix} a & b & 0 & 0 & \cdots & 0 & 0 & 0 \\ c & a & b & 0 & \cdots & 0 & 0 & 0 ...
0
votes
2answers
34 views

Show that $G = SL_3(F_p)$ acts transitively on $H = F_p^3\setminus \lbrace0\rbrace$

I need to show that $G = SL_3(F_p)$ acts transitively on $H = F_p^3\setminus \lbrace0\rbrace$. That means to show that, for all $s,t \in H$, there is $g \in G$ such that $gt = s$. I tried to make ...
1
vote
1answer
26 views

Dimension of $Range(A)$ and $Range(A^2)$

Which of the following matrices satisfies the property "Dimension of $Range(A)$ and $Range(A^2)$ is 2 and 1 respectively." $$A = \begin{bmatrix} 0 & 0 & 1 & 1\\ 0 & 0 & 1 & ...
0
votes
0answers
19 views

multiplication between a matrix and a givens rotation

I want to multiply a matrix A with a givens rotation G. As a reference to this very important link:Click here, they explained in pages 13 and 14 how this multiplication can be achieved. In this PDF, ...
1
vote
1answer
31 views

Calculating a stochastic matrix with multiple states

I am struggling with how to calculate the values of a Markov matrix which has multiple states. For example, Imagine an unfair 6 sided dice. The chance of rolling a 1,2,3,4,5 or 6 is 0.3, 0.25, 0.2, ...
0
votes
1answer
30 views

Graph Theory : Strongly regular graph

A simple graph G which is neither empty nor complete is said to be strongly regular with parameters $(v,k,λ,μ)$ if: v(G)=v; G is k-regular; any two adjacent vertices of G have λ common neighbours, ...
1
vote
1answer
29 views

How can point group symmetry operations be used to reduce the number of independent crystal properties?

How can point group lattice symmetry operations be applied to reduce the full second-rank elasticity tensor (in Voigt notation) from: to, for example, in the cubic case, this: A reference would ...
-1
votes
1answer
25 views

Super Simple Proof of Cofactor Expansion

Is it possible to provide a super simple proof that cofactor expansion gives a determinant value no matter which row or column of the matrix you expand upon? E.g., super simply prove that $$\det(A) ...
0
votes
1answer
19 views

Is there a matrix product which results in this relation?

Let $\pmb{a} = \left[a_1\;a_2\;\dots\;a_n\right], \pmb{b} = \left[b_1\;b_2\;\dots\;b_m\right]$. Then $$K = \left( \begin{array}{ccc} a_1b_1 & a_1b_2 & \cdots & a_1b_m \\ a_2b_1 & ...
2
votes
3answers
119 views

If $AB\not=BA,$ then do $A$ and $B$ have to be matrices

Is this solvable? Or are there other things that fit the bill for $A$ and $B$?
1
vote
0answers
11 views

Matrix representation of function concatenation using other basis than polynomials.

I have now familiarized myself with the Carleman-matrices which represent function composition of polynomials (actually taylor series terms) and built some of my own. I noticed that for any finite ...
-3
votes
1answer
34 views

Entries of a matrix?

The wikipedia article on matrices states: "..is a rectangular array—of numbers, symbols, or expressions,.." https://en.wikipedia.org/wiki/Matrix_(mathematics) Do you agree with that? Are the ...
2
votes
0answers
50 views

Fastest way to find linearly independent columns of a matrix

Given a rectangular matrix $X$ of size $n\times m$ with $m>n$, what is the fastest way to find the linearly independent coloums. Robust methods like SVD or RRQR decompostion have complexity of ...
1
vote
1answer
24 views

Function composition as representable by matrices?

I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and ...
1
vote
1answer
27 views

Calculating determinant of a matrix product

Let $M =\begin{pmatrix} 1 & 0 & ... & 0 \\ 0 & 1 & ...& 0 \\ ... & ... & ... & ...\\ 0 & 0 & 0 & 1 \\ x_1 & ...
0
votes
0answers
32 views

4x4 Matrix with homogeneous coordinates

I learn for a linear Algebra exam and I have the task: "What is the $4\times 4$ matrix , a rotation about the $\pi/3$ describes in homogeneous coordinates about the axis? What is the ...
3
votes
3answers
122 views

Find all constants where a matrix is symmetric

I have a matrix like below: $$M = \begin{bmatrix} 2 && a-2b+c && 2a+b+c \\ 3 && 5 && -2 \\ 0 && a+c && 7\end{bmatrix}$$ In order for the matrix to be ...
3
votes
1answer
95 views

Why is the determinant of $\sum_{i=1}^n A_i^2$ non-negative?

Let $A_i$ be an $n\times n$ matrix in $\mathbb{R}$ and $\{A_i\}_{i=1}^k$ are pairwise commutative: $A_iA_j = A_jA_i$. How to show $det(\sum_{i=1}^k A_i^2)\geq 0$? We may consider this question in ...
1
vote
1answer
64 views

Prove that $\det(E^{\mathsf{T}})=\det(E)$, where $E$ is an $n \times n$ elementary matrix.

Prove that $\det(E^{\mathsf{T}})=\det(E)$, where $E$ is an $n \times n$ elementary matrix. I am not familiar with eigenvalues. I only know that the cofactor expansion can be done along any row ...
12
votes
3answers
346 views

Is there a name for matrix product with reversed indices?

The typical matrix product is as follows: $$ (\mathbf{A}\mathbf{B})_{ij} = \sum_{k=1}^m A_{ik}B_{kj}\,. $$ Is there a name or characterization for one such as $$(\mathbf{A}\mathbf{B})_{ij} = ...
0
votes
0answers
53 views

Transition matrix proof

Let $P=\begin{bmatrix}1-a&a\\b&1-b\end{bmatrix}$, with $0<a,b<1$. Show that ...
0
votes
2answers
31 views

Eigenvalues of a non-symmetric matrix

Let $M \in \mathbb{R}^{n \times n}$ be a symmetric, positive definite matrix, such that $M \preccurlyeq I$ (i.e., $I-M$ is positive semi-definite). Prove or disprove that the matrix $$ ...
0
votes
0answers
12 views

Using multilateration with weighted nodes in determining location

I'm using multilateration to determine the coordinates of a point, given distance estimates to $n$ fixed anchors (or nodes). Since distance values are only estimates, I'm actually calculating the best ...
-1
votes
0answers
27 views

Linear Algebra conversion?

I'm trying to create the following matrix via programming but I can't seem to get the equation right (https://upload.wikimedia.org/math/f/b/a/fbaee547c3c65ad3d48112502363378a.png). $$R = ...
0
votes
0answers
15 views

LU factorization: right-looking algorithm

I know the LU factorization,and I have found several block methods for LU factorization,one is called right-looking LU factorization but I am not know this very well,so can anybody tell me how ...
0
votes
0answers
22 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$\begin{align} \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C ...
0
votes
0answers
17 views

Mapping sphere surface to a vector space such that distances are preserved?

I have a unit radius sphere (say in 3D) centered in origin. Thus the shortest distance between two points on the sphere is the geo-desic. Is there a transformation (linear or non-linear) on the points ...
7
votes
2answers
171 views

Mapping sphere surface to a vector space such that distances are preserved?

I have a unit radius sphere (say in 3D) centered on the origin. Thus the shortest distance between two points on the sphere is the geodesic. Is there a transformation (linear or non-linear) on the ...
4
votes
3answers
158 views

problem of a positive definite matrix

Let $H$ be a positive definite matrix and $I$ the identity matrix. If $k_1,k_2>0$, can we conclude that $k_1H-k_2I$ is positive definite if $k_1\gg k_2$?
2
votes
0answers
40 views

Inverse of sum of 3 matrices

I need a way to compute the inverse of the sum of three matrices: $(A + BB^T + \beta I)^{-1} $ where $I$ is identity and $\beta$ is a constant. I am not very familiar with linear algebra, but a ...
0
votes
0answers
22 views

Matrix Factorization with Arbitrary Dimensions

Continuation of a previous question here. Suppose I have a $n\times m$ matrix $A$. I choose some $k$, and want to find a factorization $A=XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$. In ...