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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1
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0answers
12 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
7 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
10 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
18 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
13 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
0answers
17 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
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 ...
2
votes
2answers
34 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
0answers
10 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
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
21 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
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
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 ...
0
votes
1answer
35 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 ...
0
votes
0answers
26 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. $X_{C1} = 0.596X_R - 0.274X_G - 0.322X_B$. $X_{C2} = 0.211X_R - 0.523X_G + 0.312X_B$ ...
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 & ...
-1
votes
1answer
25 views

find the set of all positive integers n for which there is a real matrix A of dimension n×n such that $A^{−1}=−A$.

Need to find the set of all positive integers $n$ for which there is a real matrix $A$ of dimension $n\times n$ such that $A^{−1}=−A$. Tried: let $\lambda$ be an eigen value of A then we have ...
0
votes
0answers
18 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 ...
-5
votes
0answers
18 views

Matrices & Trigonometry [on hold]

Please refer fig A & Fig B.In fig A - ABC & A'B'C' are equal triangles & In Fig B - ABCD & A'B'C'D' are equal. I want to find X,Y theta offset of C' from C in fig A & of D' from ...
0
votes
2answers
15 views

Show $Du(\mathbf{x})=[Dv(Q^T\mathbf{x})]Q^T=[Dv(\mathbf{x}')]Q^T$ and $Hu(\mathbf{x})=Q[Hv(\mathbf{x}')]Q^T=Q[Hv(\mathbf{x}')]Q^{-1}$

Let $u:\mathbf{R}^2\to\mathbf{R}$ and assume that all of the second-order partial derivatives of $u$ are continuous on $\mathbf{R}^2$. For each $\mathbf{x}\in\mathbf{R}^2$, regard $\mathbf{x}$ as ...
3
votes
2answers
54 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
2answers
9 views

Set density of random matrix in Sage

I'm using Sage to calculate a bunch of matrix operations over GF(2), using the code below to randomly generate an invertible matrix: ...
1
vote
2answers
22 views

Find the determinant of the linear transformation in $\Bbb R^{2\times 2}$

$L(A)=A^T$ from $\Bbb R^{2\times 2}$ to $\Bbb R^{2\times 2}$ I know $\det(A^T) = \det(A)$ if $A$ is a square matrix. I also know that a basis for $\Bbb R^{2\times 2}$ is ...
0
votes
1answer
19 views

Possible quick solution of SVD of covariance matrix of Xv, where v may change, while X does not.

I am current trying to work on one algorithm, that for Iteration $t$, I need to calculate the SVD of $(X\text{diag}(v^t))^T(X\text{diag}(v^t))$. This could be very slow if $X$ is of high dimension. ...
2
votes
1answer
37 views

Weird characteristic polynomial question

Let $F_A:\,\mathrm{M}_2(\mathbb{C})\to\mathrm{M}_2(\mathbb{C})$ be defined by $\mathrm{M}\mapsto \mathrm{MA}+\mathrm{AM}$. I am doing a question which asks me to write the characteristic polynomial of ...
2
votes
1answer
33 views

How do I determine if matrix A is diagonalizable?

I am trying to figure out how to determine the diagonalizability of the following matrix: $A=\begin{pmatrix} 1 &0 &0 &0 \\ 2&1 & -3 & -2\\ 3& 0 & 0 &-9 \\ ...
0
votes
1answer
14 views

Finding the number of matrices given certain conditions

Given that $$B = \begin{pmatrix}2&4\\3&2\end{pmatrix} + 13A_1$$ where $A_1\in M_2(\mathbb{Z}_2)$. We want to find all the possible choices of $B$ such that $\gcd{(\det{B},26)}=1$ and ...
-1
votes
0answers
32 views

If $AB = BA$ and $AC = CA$, prove that $BC = CB$ [on hold]

Suppose $A, B, C$ are three $n \times n$ matrices such that $A$ has $n$ distinct eigenvalues. If $AB = BA$ and $AC = CA$. Then we want to show that $BC = CB$. Thank you.
0
votes
1answer
23 views

Linear Algebra: If a is a $5\times 8$ matrix which of the following are true?

Can someone tell me which of the following are correct and explain why that is true? To me they are all false, but I am unsure.
0
votes
0answers
8 views

How to define this space? (matrix of coordinates)

We will let $F$ denote an arbitrary field such as the real numbers $R$ or the complex numbers $C$. For any positive integer $n$, the space of all $n$-tuples of elements of $F$ forms an ...
0
votes
0answers
12 views

Advantages and Disadvantages of Gaussian Elimination Over Iterative Methods.

I am having a hard time understanding the advantages and disadvantages of using Gaussian Elimination over other Iterative Methods such as the Jacobi iteration and what are the advantages of using ...
0
votes
0answers
7 views

Equations of determinants of matrix and adjoints of order 2

A be a square matrix of order 2 with |A|$\ne$0 such that |A+|A|adj(A)|=0,then the value of |A-|A|adj(A)| is : My attempt I took the matrix as $$ \begin{bmatrix} a & b \\ ...
0
votes
1answer
13 views

Lowering a non-zero weight vector gives a non-zero vector (representation of $\mathfrak{sl}(2)$)

In Lie algebras we study $\mathfrak{sl}(2)$ (the complex span of the usual matrices $X,Y,H$ where $X$ and $Y$ are the raising and lowering operators respectively). The defining commutator relations ...
0
votes
0answers
24 views

Finding the Generalized Eigenspace

Given is the matrix, \begin{bmatrix}0&0&-2&0&0\\0&0&1&0&0\\1&1&2&0&0\\-1&-1&-2&-1&-2\\1&1&2&1&2\\\end{bmatrix} Find ...
1
vote
0answers
18 views

Probability of n distinct eigenvalues

For a randomly generated $n$ by $n$ matrix, is the probability that it has $n$ distinct eigenvalues equal to $1$? I have a feeling it must be. But, if that's the case, why do we concern ourselves so ...
1
vote
0answers
106 views

Construct a matrix of polynomials to optimize condition-like score

I'm a physicist currently working on my PhD. Within my studies, my colleagues & I encountered a (strictly mathematical) problem that baffles us (and anyone else we've talked to so far) and is also ...
4
votes
1answer
38 views

$\det(I+A)$= sum of all principal minors of $A$

I'm having a hard time proving or finding a proof for the following result. It should follow from an application of the Laplace expansion. Let $n\in\mathbb{N}$, $[n]=\{1,\dots,n\}$, and ...
0
votes
1answer
52 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 ...
0
votes
0answers
8 views

How to maximize generalized Rayleigh ratio

The generalized Rayleigh ratio is defined by $$R(\vec{x})=\frac{\vec{x}^TA\vec{x}}{\vec{x}^TB\vec{x}}.$$ The vector $\vec{x}$ is a $p \times1$ unitvector, the matricies $A$ and $B$ have the size $p ...
0
votes
0answers
23 views

matrix transposition during Euclidean distance

I'm trying to follow along with an online class that's using matrix vectorization for image comparison. Although I have the solution, I don't understand one of the matrix transformations. This is ...
4
votes
3answers
336 views

Solve for unknown matrix

Let $A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$ and let $B = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$ Solve $A X = B$ for a matrix $X$ My guess is that i: ...
-1
votes
1answer
21 views

If $A$ is Hermitian, then $A^+$ is Hermitian. [on hold]

If $A$ is Hermitian, then $A^+$ is Hermitian. We know that $A=A^*$ and we wish to show that $A^+=A^*$. I'm not sure how to manipulate this, maybe using the SVD identies?
0
votes
0answers
10 views

Gell-Mann Matrices with dot and cross product

We know that, given $\vec{a}$ and $\vec{b}$, then $(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma})=(\vec{a}\cdot\vec{b})\mathbb{I}+i(\vec{a}\times\vec{b})\cdot\vec{\sigma}$. $\sigma_i$ are the ...
0
votes
1answer
37 views

What are singular value of $A$?

Let $ A = \left( {\begin{array}{*{20}{c}} {x + (\frac{3}{4} + y)i}&1&1\\ 0&{(x - \frac{5}{4}) + iy}&1\\ 0&0&{(x + \frac{3}{4}) + iy} \end{array}} \right)$, and $x,y\in ...
0
votes
0answers
26 views

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 ...
0
votes
1answer
13 views

Matrix transformations on objects

I am trying to solve the following question: I have created the scaling, translation and rotation matrices that I feel will transform the left figure to the figure on the right: Scaling $$ ...
3
votes
0answers
38 views

ALL Orthogonality preserving linear maps from $\mathbb R^n$ to $\mathbb R^n$?

That is we have a linear transformation, i.e. an $ n\times n $ matrix $A$, such that for every pair of vectors $ v $ and $ w $ we have $$ \langle v,w\rangle=0 \ \ \ \implies \ \ \ \ \ \langle ...
0
votes
2answers
23 views

Given a Matrix A, prove that 1/9A is an orthogonal matrix.

$$Let A= \begin{pmatrix} 4 & -7 & 4 \\ -1 & 4 & 8 \\ -8 & -4 & 1 \\ \end{pmatrix} $$ The problem is to prove that $1/9A$ is an ...
6
votes
1answer
75 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
2answers
43 views

Derivative of n x n Invertible Matrix

For an invertible $n$ x $n$ matrix $A$, define $f(A):=A^{-2}$. Calculate the derivative $D\space f(A)$. (i.e. give $D\space f(A)B$ for arbitrary $B$.) I'm not super sure how to go about this?