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
1answer
63 views

Finding orthogonal matrix that maps one vector to another

Let $w, v \in \mathbb{R}^k$ be two known vectors such that $||w|| = ||v||$ ($|| . ||$ is the usual Euclidean norm). My questions are related with the problem of finding $Q$ orthogonal such that $v = Q ...
0
votes
1answer
20 views

Rewrite an expression in terms of basis vectors

Given any vector k $\epsilon$ $R^{3}$ consider k= $\sum_{j=1}^{3}$ $c_{j}u_{j}$ where $u_{1}$,$u_{2}$,$u_{3}$ are the orthonormal basis vectors (I don't know how to make them bold sorry about that, ...
4
votes
1answer
166 views

How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
0
votes
1answer
43 views

Matrix Differentiation using Kronecker operator issue

Let X an $n\times n$ variable matrix and given vectors and matrices $p_1$ ($1\times n$), $p_2$ ($n\times 1$), $\Omega$ ($n\times n$). What is the derivative of the function $f(X)=p_{1}X^{-1}\Omega ...
24
votes
3answers
2k views

Checkboard matrix, brand new or old?

Ok so what I found was a square matrix of order $n×n$ where $n$ follows $2m+1$ and $m$ is a natural number the pattern these matrices follow is as follows: for a $3×3$ matrix: $$ A = \left( ...
1
vote
2answers
27 views

limit of a function with a matrix exponential

I spent too many time trying to solve this problem...and finals are coming. Please help me! I just can't see a method to do this demonstration: "For an $A_{n \times n}$ matrix, demonstrate that a ...
0
votes
1answer
25 views

One eigenvalue and eigensystem

Matrix $A \in \mathbb{K}^{n,n}$ has one engenvalue $\lambda \in \mathbb{K}$ and its engensystem $V_{\lambda}$ has dimension that equals to $n$. How to show that $A = \lambda I_{n}$?
2
votes
1answer
44 views

proof on similarity of matrices

Could you please help me with the following problem? Let $A$ be an $n$$\times$$n$ complex matrix. Prove that $A$ is similar to $B$, which is an $n$ $\times$ $n$ real matrix, if and only if $A$ is ...
0
votes
1answer
31 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
1
vote
0answers
16 views

Applications of Matrix in simplifying algebra [closed]

Inversions (and the Mobius Transformation, though it belongs to complex numbers) are pretty good tools in simplifying an algebric mess. What other tools exist apart from this and how may we use them? ...
2
votes
2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
0
votes
1answer
22 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
2
votes
1answer
54 views

Example of a non singular square matrix such that $A+A^{-1} = 0$

Is there any example of a non singular square matrix $A$ such that $A+A^{-1} = 0$? Are they any specific type of matrices or can these be found under any category of matrices (such as symmetric, ...
1
vote
1answer
31 views

Matrix multiplication computation

Any tips how to solve this? $$ \left[ \begin{matrix}1 & 2 & 0 \\ -2 & -5 & 1 \\ 11&15&5 \end{matrix}\right] \times \mathbf{X} \times \left[ \begin{matrix} -4&5&1\\ ...
1
vote
1answer
18 views

Vector notation question

Just a short question regarding notation: If this matrix represents a vector and I want to solve it for $t=2$, may I write it as follows: $ \left( \begin{array}{ccc} vt\\ vt-gt\\ \end{array} ...
0
votes
0answers
7 views

Composition of a rotation and a homothetic transformation of different centers?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$. Furthermore let $h_{\lambda,S}$ be the homothetic transformation of center $S\neq \Omega$ and ratio $\lambda$. What ...
1
vote
1answer
36 views

the relation between positive definite matrix and its ellipsoid

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Then, we associate with each $A \in S^n_{++}$ ...
2
votes
1answer
52 views

Do linear operators $A$, $B$ satisfying $A = B+BAB$ commute?

I have two linear continuous operators $A$, $B$ on Banach space $X$ (for example, square matrices), satisfying the equation $$ A = B + BAB, $$ and such that the continuous inverses $(\mathrm{Id} ...
2
votes
0answers
19 views

Transformations invariant wrt. $L_1$ norm.

$A$ is a real matrix of size $n \times k$, where $k \leq n$. $A$ has independent columns. Characterize the class of matrices $M \in \mathbb{R}^{k \times k}$ such that: $\forall x \in \mathbb{R}^k.\; ...
1
vote
0answers
29 views

Good resource to learn geometric interpretations of matrices [duplicate]

I need a good resource to learn matrices and all its properties through geometry. I feel geometry gives an insight into many matrix operations and a good resource will be useful for many students who ...
0
votes
1answer
20 views

Find all solutions to $Bx =[7, -10, 7, 0]^T$

$$ B=\left[ \begin{array} k1 & 0 & 2 & 1\\ -3 & 2 &-1 & 5\\ 2 & -1 & 1 & 4\\ 0 & 3 & 2 & 4\\ \end{array} \right] ...
0
votes
2answers
52 views

Commutative matrices

Knowing that $AB=BA$, find the matrices that commute with the matrix \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} I have assumed that multiplying matrix $\begin{pmatrix} a & b ...
5
votes
1answer
35 views

How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...
0
votes
1answer
23 views

conjugate of matrix multiplication [closed]

I have been trying hard to find what the conjugate is of the product of the matrices $A$ and $B$, but I did not have luck. Could you please help me? Thank you very much in advance.
0
votes
0answers
17 views

The conditions on marginals that guarantees a certain class of measures

$x$ and $y$ are $m\times n$ matrices. $a, B,C$ are $m\times 1$ matrices. $b, A$ are $n\times 1$ matrices. $$\sum a_i=1, 0\leq a_i\leq 1, \forall i$$ $$\sum b_j=1, 0\leq b_j\leq 1, \forall j$$ ...
0
votes
1answer
43 views

$A\succeq B\succeq 0$ and $C\succeq 0$ imply $AC-BC\succeq 0$?

If $A$, $B$ and $C$ are positive semidefinite real matrices such that $A\succeq B$, do we have: $$ AC\succeq BC\quad\text{and}\quad CA\succeq CB.\tag{i} $$ Here, $A$, $B$, and $C$ share the ...
0
votes
1answer
17 views

Matrices and linear maps: finding image and kernel of a linear map

Let $$V= \left\langle\begin{pmatrix}0 & 1 & 0 \\0 & 1 &0\end{pmatrix}, \begin{pmatrix}0 & 0 & -1 \\0 & 1 &0\end{pmatrix}, \begin{pmatrix}2 & 0 & 1 \\0 & 1 ...
2
votes
0answers
17 views

Hadamard products VS Matrix product

Some notation before introducing the question. We discretize an interval $[0,L]$ in uniform subintervals, each with length $\Delta x$ and we assume that vectors living over such a grid satisfies ...
3
votes
0answers
31 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
0
votes
1answer
24 views

Finding matrix A from eigenvalues and ODE

I have a differential equation $$x'=Ax+\left(\begin{matrix} 0 \\ e^t \\ 0 \\ \end{matrix}\right), x(0)=\left(\begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix}\right)$$ which I have solved to get ...
-1
votes
1answer
27 views

Number of submatrices of sum K

I have an array $A[]$ of N elements ($N<=1000$, $-1000<=A[i]<=1000$). We define a Matrix M such that $M[i,j]= A[i]*A[j]$. In the resulting matrix $M$, we have to count the number of ...
7
votes
3answers
290 views

Finding matrix exponential

I am trying to compute the matrix exponential for $$A=\left( \begin{array}{ccc} 1 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & -1 ...
0
votes
1answer
22 views

Rayleigh quotient inequality. ($|\lambda-q^*Aq|\leqslant2||A||_{2}||v-q||_{2}$ )

Given $A\in\mathbb{C}^{n\times n}$ and $(\lambda,v)$ an eigenpair verifying $\|v\|_{2}=1$, $q\in\mathbb{C}^n$ a unitary vector. Show that its Rayleigh quotient verifies ...
2
votes
2answers
64 views

Block Matrix Determinant Proof

I am trying to solve the determinant of a Block matrix $$\begin{bmatrix}A-Ia&B\\B &A-Ib \end{bmatrix}$$ where a and b are integers and I is an identity matrix, A and B are square. ...
1
vote
0answers
116 views

Can gradient descent solve this problem $\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2$?

How can I find the (approximate) solution to the following problem: $$\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2,$$ where $Var(.)$ denotes the variance? $A$ is matrix and $b$ and $x$ are ...
0
votes
1answer
39 views

Clarification for a linear algebra problem stated: Find All Solutions to $AX = B$

I am working on a Linear Algebra HW problem which goes like: Find all solutions $X = \left[\begin{matrix} x & y \\ z & w \end{matrix}\right]$ to the matrix equations $AX = B$ when $A = ...
0
votes
1answer
16 views

Finding base for a set of vectors

Given these sets of vectors: $$ T=\{(2,1,-1),(1,0,-1),(5,1,-4)\} $$ $$ S=\{(1,2,1),(1,1,2),(3,4,5)\} $$ 1) Find a base for the subspaces: $Sp(S)$, $Sp(T)$, $Sp(S\cup T)$ 2) Describe the vectors ...
2
votes
2answers
38 views

linear algebra problem in matrices

I have no idea how to approch this, any help will be greatly appreciated: Given: Matrix A of order $(k\times n)$ Matrix B of order $(n\times k)$ with $k\neq n$, prove that its not possible for ...
-1
votes
0answers
24 views

Show A is not similar to a Diagonal Matrix

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $A = \begin{bmatrix} 4 & 0 & 0 &0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & -2 & -3 \\ 0 & -1 ...
1
vote
1answer
22 views

The $5\times 5$ matrix whose $(i,j)$th entry is $1$ if $j$ is a multiple of $i$, and $0$ otherwise

The $5\times 5$ matrix whose $(i,j)$th entry is $1$ if $j$ is a multiple of $i$, and $0$ otherwise. I know that I’m supposed to show the work I’ve done, but I just have no idea what to do with this. ...
0
votes
2answers
18 views

Find the dim of the solutions for $Ax=0$

Let $A$ be a matrix: $$ A=\begin{pmatrix} 1 & 1 & -5 & -6 & 1 \\ 2 & 1 & -7 & -7 & 1 \\ 1 & 2 & -8 & -11 & 5 \\ ...
1
vote
1answer
20 views

Finding eigenvalues for a vectorspace such that the matrixrepresentation is a diagonal matrix

Problem: Let $T$ be a linear operator on the vectorspace $V = M_{2 \times 2}(\mathbb{R})$ and let $T\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & b \\ c & a ...
0
votes
2answers
21 views

Convergence rate of the power method for finding eigenvectors

Let $M$ be a real-valued square matrix with an eigenvector $w$ strictly larger (in absolute value of the corresponding eigenvalue $\lambda$) than all others, and let $v$ be any vector not orthogonal ...
3
votes
3answers
42 views

Derivative of $tr((AX)^tAX)$

I'm trying to calculate the derivative (with respect to the matrix $X$) of the function $f(X) = tr((AX)^t(AX))$, Chain's rule gives that $\nabla_X(f(X))=\nabla_X(tr(AX))\nabla_x(AX)$ However I'm ...
6
votes
0answers
48 views
+50

$L^2$ Bounds for Markov Chains.

Consider a non-negative, square stochastic $n \times n$ matrix $P$ (rows sum to one, $P$ is ergodic). We are interested in characterizing the set of $n \times n$ invertible matrices $A$ such that we ...
0
votes
3answers
85 views

Find all matrices where the matrix is its own inverse and the determinant is 1

I need to find all the matrices: $$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ such that $$ad-bc=1$$ and $$A^{-1}=A$$ How would I go about doing this? I know that $AA=I^2$, ...
5
votes
1answer
144 views
+100

Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
0
votes
0answers
10 views

how to visualize this statement: Matrix M falls in a Ball-set.

So the question is simple: Assume you are told that a matrix M has the following property: $\|M\|_2<1$, i.e. it falls in unitary ball. When we say it is inside a ball set, if you imagine a ...
3
votes
2answers
65 views

Street Fighter: is the game balanced?

Suppose that $A$ is a matrix that describes the matchup information of any pair of Street Fighter characters e.g., considering $3$ characters, assume that the first row/collumn is associated with a ...
0
votes
1answer
27 views

What method is used to find the determinant of this $4 \times 4$ matrix?

This is a pre-solved example in my book, I don't understand how they solved it. What method is used? Find the determinant of $A = \begin{bmatrix} 0 & 1 & 0 & 2\\[0.3em] -1 ...