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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6
votes
1answer
82 views

$A^2+B^2=AB$ and $BA-AB$ is non-singular

The question is: Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular? From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this? ...
0
votes
1answer
10 views

Square root of a matrix as it relates to the identity

Prove that for any $2×2$ matrix $M$ which is “sufficiently close” to the identity matrix, there exists a matrix A such that $A^2 = M$, and that this matrix A is unique if $A$ isrequired to be ...
1
vote
3answers
85 views

Let $A$ be a $4\times4$ matrix with real entries and eigenvalues $1$, $-1$, $2$ and $-2$, then which of the following statements are true?

If $B$ is a matrix defined as $B=A^4-5A^2+5I$, where $I$ is a $4\times4$ identity matrix, then $\det(A+B)=0$ $\det(B)=0$ $\operatorname{tr}(A+B)=4$ From the given conditions, I could only ...
0
votes
1answer
447 views

Using Gauss elimination to check for linear dependence

I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or ...
5
votes
1answer
5k views

Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.

Let $A$ be an $n \times n$ matrix. $i)$Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$. $ii)$Prove that if the sum of each column of $A$ equals $s$, then $s$ is ...
0
votes
0answers
20 views

What is the rank of this matrix?

So we have $f: \mathbb{R}^n \to \mathbb{R}^{n-m},^\ h : \mathbb{R}^m\to \mathbb{R}^{n-m}$ and $f(x_1,...,x_n)=(x_1,...,x_{n-m})-h(x_1,...,x_m)$ Developing the scripture : ...
0
votes
1answer
17 views

Computing change of base matrix

I'm having trouble understanding how to solve the following exercise (or rather, what is it asking for): Find the change of basis matrix for the following basis B and D for $\mathbb{R}^2$. ...
0
votes
1answer
20 views

Let A be a $3\times3$ real orthogonal matrix. Prove that there exists a vector $w$ in $R^{3}$ such that $Aw=w$

Let A be a $3\times3$ real orthogonal matrix. Prove that there exists a vector w in $R^{3}$ such that $Aw=w$ or $Aw=-w$ . Tried: $(Aw)'Aw = w'A'Aw=w'w$$\implies Aw=w$ I think it is incorrect.
0
votes
0answers
14 views

Find a Matrix $C$ that will reduce matrix $A$ to the diagonal form, $C^{-1}AC$

Find a Matrix $C$ that will reduce matrix $A$ to the diagonal form by the transformation $C^{-1}AC$. I know that we are able to diagonalize a matrix and set A = $PDP^{-1}$ where P is the eigenvector ...
0
votes
1answer
14 views

A family of vectors is linearly independent.

Let $K$ be a field and $E$ be a $K$-vector space of dimension $n$. Let $\phi$ be an endomorphism of $E$. Let $(\lambda_1,\cdots,\lambda_n)$ be a family of distinct scalars and $(x_1,\cdots,x_n)$ be ...
0
votes
1answer
26 views

How can I tell if a matrix transformation is injective/surjective?

Determine whether or not $\mathbf v_1=(-2,0,0,2)$ or $\mathbf v_2=(-2,2,2,0)$ is in the kernel of the linear transformation $T:\mathbb R^4\to\mathbb R^3$ given by $T(\mathbf x)=A\mathbf x$ where ...
1
vote
1answer
15 views

Using simple matrix algebra to solve for a specific matrix (Beginner question)

The matrix $AB = C$ where $A$, $B$ and $C$ are all $2 \times 2$ non-singular matrices. How would I go about to solve for the Matrix $A$ and express it in terms of $B$ and $C$? There are two methods ...
0
votes
2answers
414 views

Minimize Frobenius norm with constraints

As a follow-up on my previous question, I would like to solve the following optimization problem: $\min \Vert MA-B \Vert_F^2-x^HMy\;\;s.t.\;\;M^HM=I$ where $A$ and $B$ are $N\times L$ complex ...
0
votes
1answer
10 views

Finding basis for column space of matrix

To find a basis for the column space of a matrix one finds the RREF of the matrix. The columns in the RREF are not a basis for the column space, but the same columns in the original matrix are a ...
1
vote
1answer
19 views

General Solution of System Of Equations (with 3 variables)

A system of equations is given as x + 4y +2z = 0 3x -2z = 4 3x -3y -4z = 5 The task is to find the general ...
1
vote
0answers
21 views

Show that if all row-sums of a square matrix $A$ are equal to $0$, then $A$ is singular [duplicate]

I need to show that if all row-sums of a square matrix $A$ are equal to $0$, then the matrix is singular. My idea was that to represent the situation, I can do as follows: $$A\vec{x} = \vec{0}$$ ...
1
vote
0answers
19 views

Vector on a sphere

I have for some time tried to understand the math behind explained in this post, but seem to not grasp. I think the way i visualize it might be incorrect, which make harder for me to grasp what is ...
1
vote
1answer
767 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
1
vote
1answer
18 views

When can we say a symmetric matrix is positive semi definite matrix?

Let $X$ be $v\times b$ matrix with $X1_{b}=r1_{v}$ and $X^T1_{v}=k1_{b}$ where $r,k$ are scalars. Then can we say that $I_v-\frac1{kr}XX^T$ is positive semi definite? I was thinking in the ...
1
vote
0answers
30 views

Linear Map of an ellipsoid in $\mathbb{R}^N$ into another ellipsoid in $\mathbb{R}^n$, with $n<N$

Starting from the closed set describing an ellipsoid in $\mathbb{R}^N$: $$\Omega_x = \{ x \in \mathbb{R}^N : (x-x_0)^T\Sigma_x^{-1}(x-x_0) \leq \varepsilon^2 \}$$ where $\Sigma_x \in ...
0
votes
0answers
9 views

solution of matrix equation

I was trying to solve the problem I have posted previously (here). and stuck up at the point where I need to find a simplified expression for $(\mathbf{I-DW})^{-1}$ Where $\mathbf{W}$ is a doubly ...
0
votes
0answers
7 views

Difference of Generalised inverse matrix nnd?

Suppose $A$ ($m\times n$) and $C$ ($k\times n$) be two matrices. Let $B^T=[A^T \ \ C^T]$ be $n\times (m+k)$ matrix. Suppose $\mathcal{C}(A^T)=\mathcal{C}(B^T)$. Is it true that for any $\ell \in ...
3
votes
1answer
59 views

Prove positive definite of a function

For $A,X,Q \in \mathbb{R}^{n \times n}$, define $h(X) = A X A^T + Q$ and $ h^j(X)=\underbrace{{h(h(}...h}_{j\text{ times}}(X)))$. If $X,Q$ are positive definite, $A\neq 0$ and for a certain integer ...
1
vote
0answers
35 views
+50

Change in eigenvalues by changing only one entry of a square matrix

Consider following square matrix $A$ of order $n$ $A=\begin{bmatrix} 0 & a_{12} & a_{13} & a_{14} & \cdots & a_{1n} \\ a_{21} & 0 & a_{23} & a_{24} & \cdots ...
2
votes
2answers
46 views

Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?

Suppose $A,B\in\mathbb{R}^{n\times n}$ are matrices such that $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$. I couldn't come up with a ...
2
votes
2answers
27 views

Under what condition on matrix $Q$ we have $tr(AQ)=tr(BQ)$

Let $A,B$ are similar matrices. Then, under what condition on matrix $Q$, we have $tr(AQ)=tr(BQ)$ $A$ and $B$ are similar matrices, so there exist an invertible matrix $P$ such that $$A=P^{-1}BP\\ ...
1
vote
1answer
10 views

Finding the change of basis matrix between bases defined by 2x2 matrices

Set M to be the set of all matrices of the form: $\bigl( \begin{smallmatrix} 0 & b \\ c & 0 \end{smallmatrix} \bigr)$, with b, c being real numbers. A basis for M (given) is $\mathcal E$ = ...
0
votes
1answer
11 views

What will be eigne vectors of 2x 2 symmetric Toeplitz

For a symmetric Teopliz 2x2 matrix I took following steps taking a matrix A = |2 1| |1 2| now their ...
2
votes
1answer
24 views

Finding the coordinate vector of a 2x2 matrix in a basis of 2, 2x2 matrices

Set M to be the set of all matrices of the form: $\bigl( \begin{smallmatrix} 0 & b \\ c & 0 \end{smallmatrix} \bigr)$, with b, c being real numbers. The basis for M (given) is $\epsilon$ = ...
0
votes
2answers
14 views

Determine reflection matrix over a line

I should determine reflection matrix over a line through the origin with direction vector $\vec{v}=\left(a,b\right) ^{T} $ I dont understand this really good and couldnt find anything helpful on ...
0
votes
1answer
42 views

Rational canonical form of the matrix $A$

Let the matrix \begin{equation} A=\begin{bmatrix} 2 & 1 & 2 \\ -2 & -1 & -4 \\\ 1 & 1 & 3 \end{bmatrix}. \end{equation} So far I found the characteristic polynomial ...
0
votes
0answers
19 views

Generalisation of Cramer's rule to matrices

I'm familiar with Cramer's rule for the system $Ax=b$ in that $$x_i=\frac{\det A_i}{\det A},$$ where $A_i$ is the matrix $A$ whose $i$-th column is replaced by $b$ and $\det A\neq 0$. In the general ...
-1
votes
0answers
29 views

Find a matrix p diagonalizes A and determine $p^{-1} A p$

Find a matrix $p$ diagonalizes A 3×3 matrix and determine $ p^{-1} A p $ $$ A = \begin{bmatrix} 2 & 0& -2 \\ 0& 3& 0 \\ 0& 0& 3 ...
2
votes
2answers
410 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 ...
0
votes
1answer
36 views

Solving equations using 3x3 determinants

Im trying to solve the following equations by use of determinants. I have scanned my work sheet (sorry for the mess) but i cant see where i am going wrong. The equations are at the top, following ...
2
votes
1answer
29 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A ...
1
vote
0answers
22 views

Transformation that preserves an increasing ratio between vectors

Consider two vectors $x=(x_1,x_2,x_3)$, $y= (y_1,y_2,y_3)$ such that all $x_i,y_i>0$ and \begin{align} \frac{y_1}{x_1}\le \frac{y_2}{x_2}\le \frac{y_3}{x_3} \end{align} Now consider an upper ...
0
votes
0answers
12 views

How to solve exponential matrix factorization with constrain: $UV^T>0$

recently I would like to optimize the following loss function: $$L=\sum_{ij}W_{ij}(X_{ij}-exp(-\sum_{l} U_{il}V_{jl}))^2$$ $$s.t. \sum_lU_{il}V_{jl} > 0$$ Where $W \in \mathbb{R}^{m \times n}, X ...
2
votes
2answers
36 views

Prove that $ABA^T$ is symmetric when $A$ and $B$ are symmetric matrices

I have been learning about matrix symmetry and came up with a question that I can't seem to prove. The idea is that the product of $ABA^T$ is a symmetric matrix. What I mainly have to go off of is ...
0
votes
1answer
53 views

How to represent tripartite graphs algebraically (as matrices)?

A bipartite graph can be represented by an adjacency matrix, or specifically, by a biadjacency matrix. Formally, let $G = (U, V, E)$ be a bipartite graph with parts $U = \{u_1, \ldots, u_r\}$ and $V ...
2
votes
0answers
15 views

Spectral norm of a matrix of cosines

I am considering the following matrix: $$ M_m = \begin{bmatrix} \cos\bigl(\tfrac{0\cdot 0}{m}\pi\bigr) & \cos\bigl(\tfrac{0\cdot 1}{m}\pi\bigr) & \dots & \cos\bigl(\tfrac{0\cdot ...
1
vote
0answers
23 views

What are the basis vectors of the cone of positive semi definite matrices?

I was wondering if we could find a set of basis vectors that span the cone of positive semidefinite matrices? I know this question is hard, but I would really appreciate if even someone can share ...
0
votes
1answer
21 views

Solve a generalized eigenvalue problem in LDA

http://www.facweb.iitkgp.ernet.in/~sudeshna/courses/ML06/lda.pdf Page 6. I don't quite understand how that can be solved... I have tried following general one $$det(S^{-1}_{w}S_B-JI)=0$$ But I am ...
0
votes
1answer
12 views

Weird transposing after dot product and transformation

I'm reading a paragraph in a book where a plane equation ($N\cdot Q + D = 0$, N being the normal and D the distance from the origin, Q any point which belongs to the plane) is transformed by a matrix ...
1
vote
3answers
23 views

Reduced row echelon form of matrix with trigonometric expressions

I'm trying to solve for the eigenvalues of and the eigenvectors of a rotation matrix (about the z-axis): $$A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & ...
0
votes
0answers
30 views

Derivative of L2 norm

I am reading a paper about image processing and I have a question. In the paper we have equations like below. $X_{C1} = 0.596X_R - 0.274X_G - 0.322X_B$. $X_{C2} = 0.211X_R - 0.523X_G + 0.312X_B$ ...
0
votes
1answer
41 views

Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq ...
0
votes
0answers
23 views

differentiation of a norm of matrix function

I need to differentiate the following function W.r.to $x$ $y=\|x (\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1} - b\|_2$ where $0<x<\frac{2}{max_i{|{v_2}_i|}}$,$\mathbf{v_1}\in ...
0
votes
1answer
72 views

Does this form of matrix have a name?

I'm looking for the name of this kind of $n$-by-$n$ matrix: $$\left(\begin{array}{cccc} -s_1 & b_{12} & b_{13} & b_{14} \\ b_{21} & -s_2 & b_{23} & b_{24} \\ b_{31} & ...
1
vote
2answers
67 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...