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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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1answer
14 views

How is this matrix expression equivalent?

I have two matrix expression that are supposed to be the same which I don't get. First one $\begin{pmatrix}N_sE & \\N_pE &\end{pmatrix} \begin{pmatrix} w\end{pmatrix} = \begin{pmatrix} -...
1
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0answers
44 views

Householder transformations to upper triangular form

Let $A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$. How to transform this matrix with Householder transformations to an upper ...
0
votes
1answer
42 views

$n \times n$ matrix associated with $F(X)=-X$

From S.L Linear Algebra: Find the matrix associated with the following linear map: $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ given by $F(X)=-X$ Considering that dimension of vector spaces ...
0
votes
3answers
48 views

Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$ J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix}, $$ where $s$ is the eigenvalue (with ...
0
votes
1answer
16 views

Effect of squaring and the identity-matrix on eigenvalues and the characteristic polynomial

I am new to Linear Algebra, and would like some feedback regarding the following question: True or false? Let $A$ be a square matrix over $R$ If 3 is an eigenvalue of $A$, then 10 is an eigenvalue ...
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2answers
24 views

Markov matrices with no ones on off-diagonals don't have -1 as an eigen value

Every time I saw a Markov matrix with an Eigen value of -1, it had some 1's on its off-diagonals. The most obvious example is a simple permutation matrix: $$M = \left(\begin{array}{ccc}0&1\\1&...
0
votes
1answer
26 views

Row swapping not working as expected for determinant

Suppose I have determinant A such that $$ A= \begin {vmatrix} 1&0&1 \\ 5&-1&0 \\ 1&0&0 \end {vmatrix} $$ I conduct row operation $ R_1 <-> R_3 $ So I should have $$ A ...
0
votes
1answer
20 views

Basis functions and positive definiteness

Suppose I have a set of functions $g_i(x)$ which form a basis on some interval $[a,b]$. My question is whether the matrix $$ A_{ij} = \int_a^b g_i(x) g_j(x) dx $$ is positive definite? My approach so ...
0
votes
1answer
33 views

Lipschitz constant of a matrix

I am studying the Lipschitz continuity and trying to solve the following question: If a function $f(x)= Ax$ is defined for $x \in \mathbb{R}^2$ with $A= \begin{bmatrix} a & b \\ c ...
1
vote
1answer
31 views

Is there anything special with a 3x3 matrix where the 3rd row is 0 0 1?

I'm coding using p5.js and I'm looking at this method https://p5js.org/reference/#/p5/applyMatrix Using that method, I can multiply my current matrix with any matrix of the form: $$ \begin{...
6
votes
1answer
46 views

Representation of negative Quantum entropy in terms of eigenvalues, i.e., $\text{Tr}(M\log M -M)=\sum_{i=1}^{n}(\lambda_i\log(\lambda_i)-\lambda_i)$?

Negative Quantum entropy or Negative Von Nuemann entropy is defined as $f(M)=\text{Tr}(M\log M -M)$. Where $M$ is a positive definite matrix in $\mathbb{S}_+^n$, $\log$ is natural matrix logarithm ...
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1answer
34 views

Determinant of a large symmetric block matrix

Consider a given matrix $Q \in \text{Mat}_N(\mathbb{R})$, which is invertible, and $n \geq 1$. I am looking for the determinant of the symmetric block matrix $I_n(Q)$ of total size $nN \times nN$: $$...
3
votes
4answers
109 views

How to show the derivative of $f(M)=\text{Tr}(M\log (M) -M)$ is $\log (M)$?

Let $M$ be a positive definite matrix in $\mathbb{S}_+^n$. Let $\log$ be natural matrix logarithm which $\log(M)$ is defined as $\log(M)=\sum_{i=1}^{n}\log(\lambda_i)v_iv_i^T$ where $(\lambda_i,v_i)$ ...
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2answers
19 views

Linear Equations with three variables [on hold]

Suppose $p(x)=ax^2+bx+c$ is a polynomial where $a, b$, and $c$ are fixed but unknown constants. Setup and solve a $3 \times 3$ system of linear equations to find $a,b$, and $c$ if we know that $p(1)=4,...
0
votes
0answers
30 views

Borel subgroup of $SL_2(\mathbb{Z})$

As the title indicates, I want to ask what is the Borel subgroup of $SL_2(\mathbb{Z})$? I believe I read about it in one of James Milne's notes. But now I cannot find it. That is why I want to ask.
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0answers
5 views

How to avoid scaling issues when rotating a child matrix, if its parent matrix has a nonuniform scale?

If a parent transform has a nonuniform scale, then I try to rotate its child transform locally, the child is sheared weirdly. How can I avoid that?
1
vote
1answer
37 views

Why taking integral of both sides of matrix inequality is allowed?

How to show if $\nabla^2 f(x) \succeq \alpha I$, then the function is $\alpha$-strongly convex? In my optimization notes I have $$\nabla^2 f(x) \succeq \alpha I \rightarrow \alpha\text{-strongly ...
1
vote
4answers
87 views

How to prove that $(AB)^t = B^tA^t$

The proof given in my book (and I came up with as well) is: However, the part that throws me off is line #3 where they do $\Sigma A_{jk} B_{ki} = \Sigma B_{ki} A_{jk}$ I understand that ...
0
votes
1answer
24 views

Question about matrix equation and it's simplification

For example, we had such an equation (where A is a matrix, and we need to find X, also where T is transpose) - A = $(2 \cdot X)^T$ So, we will have something like this? According to the sources ...
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0answers
9 views

Converting a transformation from world frame of reference to local?

I have a matrix $W$ that is the local-to-world transformation (Rotation/Scale/Position) of a point $P$ in 3D space. I also have a rotation quaternion $R$ and a translation vector $T$ that further ...
2
votes
1answer
43 views

Some questions about simplification of the equation with matrices

I have such an equation $(X^{-1} \cdot 2 \cdot A)^{-1} - B = 4X $ Where $B$ and $A$ are matrices. So, I did such a simplification. \begin{align} & \frac 12 A^{-1}X - B = 4X\\ \iff & \frac 12 ...
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votes
2answers
51 views

Diagonalizing matrix with fractions [on hold]

I'm revising for an exam in linear algebra, and I've found myself stuck on this one specific exercise. I'm supposed to decide a matrix $P$ and a diagonal matrix $D$ from my matrix $H$ (which I'll ...
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votes
1answer
28 views

Null space of a rotation matrix

If we have a rotation matrix of the kind: link to the rotation matrix how do i compute the null space of this matrix? I know that to obtain the null space we need to write the matrix in echelon ...
2
votes
1answer
31 views

Prove that for $\pi_g(\alpha)=\alpha g$, $\pi$ is a group action

Let $R$ a principal ideal domain and let $G$ the group of all the invertible members of $M_2(R)$. Let $\Omega :=R^2$. For all $g\in G$ let $\pi_g:\Omega\to\Omega$ defined by $\alpha \pi_g=\alpha g$ ...
2
votes
0answers
58 views

How do I invert this matrix?

Given two vectors $$\vec{v} = \begin{pmatrix} v_1\\ \vdots\\ v_n \end{pmatrix} , \vec{w} = \begin{pmatrix} w_1\\ \vdots\\ w_n \end{pmatrix} \in \mathbb{R}^n$$ such that ...
0
votes
1answer
22 views

Matrix units in commutator subgroup

Let $m \ge 2$ be an integer, $R = M_{m}(K)$ be the algebra of all $m\times m$ matrices over $K$ an infinite field, if $L = [R,R]$ and $a,p \in R$ are non-zero elements. Then, if $[au, pu] = 0$ for all ...
0
votes
0answers
20 views

Conjugacy classes in special linear groups over $\mathbb{Z}/p^k$

Let $n$ be an integer and $p^k$ a prime power. Question: Is the number of conjugacy classes of elements in $\mathrm{SL}_n(\mathbb{Z}/p^k)$ known? Remark: An answer for $n=2$ would already help me.
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0answers
13 views

Expression for maximum column index of every row in matrix

Let $H$ is $(k \times j)$ matrix. How can one describe "for every maximum column of each row in $H$"? Can I write $\underset{j}{\text{argmax}} (H_{kj}), {k = 0,\ldots,K}$? I feel something is wrong ...
1
vote
1answer
27 views

Augmented matrix in equation form

I recently learned about using an augmented matrix to find the inverse of a matrix by putting the entire identity matrix on the right side of an augmented matrix and getting the left side to reduced ...
0
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0answers
12 views

Why does finding the partial derivative solve this payoff matrix?

So I'm taking this math course and I decided to do all the homework the first day of class, and as I was pulling an all-nighter doing it, I got to the topic of game theory and payoff matrices. I never ...
0
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0answers
7 views

Applying a Sigma-Point Kalman Filter to State of Charge Estimation

I've found a research paper that would allow me to implement a Kalman filter to estimate state of charge for a LiPo battery, but cannot make sense of some of the symbols as I'm a first year. Can ...
0
votes
1answer
15 views

Computational complexity of finding determinants

What is the computational complexity of finding the determinant of a matrix in this form? \begin{bmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1,n-1} & x_{1,n} \\ x_{...
3
votes
1answer
32 views

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true?

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true? $(A)$ trace of $A^2$ is positive $(B)$ $A$ has non zero eigenvalue. $(C)$ All entries of $A^2$ can't be ...
4
votes
1answer
36 views

Is there any inequality between 2-norm condition number and Frobenius norm condition number for rectangular matrix?

What I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$n - ...
6
votes
1answer
69 views

Fast way to Invert ADA' when D is a diagonal matrix that changes each iteration?

So I have a statistical learning algorithm in which D is a diagonal matrix that changes each iteration while A stays the same. I'm looking for a fast way to invert ADA' each iteration which ends up ...
1
vote
0answers
23 views

Gauss-Seidel: I can't get diagonally dominant equations

I've got the question: Use the Gauss-Seidel method to solve: $8x_1 - 16x_2 = 6$ $4x_1 + 8x_2 = 0$ I know that the equations have to be diagonally dominant, but I can't see how to determine this. I ...
1
vote
2answers
38 views

Can $\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$ be solved for $\lambda$?

In this post I suggested that the expression $$ \| \left( X'X + \lambda I \right) ^{-1} X'y \| = t $$ couldn't be easily solved for $\lambda$, because you need to "invert" the norm. But in general ...
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0answers
22 views

What kind of matrices can be visualized by graphing the column vectors?

In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis ...
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1answer
20 views
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2answers
19 views

Let $v \in R^{k}$ ,with $v^{T} v \neq 0$ . Let $P=I-2 \frac{v v^{T}}{v^{T}v}$, where $I$ is the $k X k $ identity matrix.

Let $v \in R^{k}$ ,with $v^{T} v \neq 0$ . Let $P=I-2 \frac{v v^{T}}{v^{T}v}$, where $I$ is the $k \times k $ identity matrix. Then prove that eigenvalue of $P$ are $1,-1$ and $P^{2}=I$ $P^{2}=(I-2 \...
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votes
2answers
45 views

Matrix equations, simplify them.

I have some equations and I don't know am I doing the simplification right. Can someone check it? For example, we have an equation $AX = 4X + B$, where $A$ and $B$ are matrices. So, what I have done ...
1
vote
1answer
22 views

Inverting variable size block matrix

Given a block matrix $M$ $$\bf M=\left(\begin{array}{cccc} a\ \mathbb{I}_{2} & \boldsymbol{\boldsymbol{A}}_{12} & \boldsymbol{A}_{13} & \boldsymbol{0_{(2,3)}}\\ \\ \boldsymbol{A_{21}} &...
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votes
0answers
23 views

Differential equation for a vector : condition to be conservative

Assume there is a vector $\mathbf{N}(t)$ of elements $N_i$ and of dimension $n$ and there are matrices $\mathbf{B}_i$ of dimension $n\times n$, and that : $\begin{equation}\left\{ \begin{split}& \...
1
vote
0answers
38 views

Finding the the partial derivatives of a transformed surface, in the original space

I am trying to calculate the from the surface described by a function $P(x, y, z, w): \mathbb{R}^4 \to \mathbb{R}^4$. The surface is really only dependent on $x$ and $z$ but for computational reasons, ...
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votes
0answers
12 views

Inverse of matrix expansion with negative exponents

The answer to this question shows that if I have a real nonsingular matrix $M$, such that its Taylor expansion in $\epsilon$ is $$M(x+\epsilon)= \sum_{n=0}^\infty M_n(x) \epsilon^n $$ its inverse ...
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0answers
7 views

Why are points on opposite sides of the invariant line translated in different directions when being sheared?

Urgent help needed: The image shows a figure being sheared with a shear factor of 1 and the x-axis being the invariant line. A transformation matrix is used for this to happen. Why is it that the ...
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0answers
23 views

showing that Symplectic matrix is diagonalizable [on hold]

please how to Show that any symplectic matrix is ​​diagonalizable by using the fact that every unit matrix is ​​diagonalisable.
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0answers
6 views

On $QR$ factorization of upper Hessenberg matrices with real entries

Let $A$ be an $n\times n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements : (1) There ...
0
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0answers
37 views

Numerator layout for derivatives and the chain rule

We have three matrices $\mathbf{W_2}$, $\mathbf{W_1}$ and $\mathbf{h}$ (technically a column vector): $$ \mathbf{W_1} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \;\;\;\;\;\;\;\;...