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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9 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
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1answer
22 views

Matrix multiplication to make all numbers in a 3x3 matrix negative

Let's say I have the matrix called Delta, $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ What would I have ...
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2answers
20 views

Operations on 3x3 matrix through matrix products

What would I have to multiply the following matrix ... $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ by so ...
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1answer
15 views

Determinant of block matrix with null row vector

I'm a bit confused on a problem. I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1x1) ...
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0answers
18 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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4answers
113 views

Can we say that, $\det(A+B) = \det(A) + \det(B) +\operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB)$.

Let $A,B \in M_n$. Is this true formula?$$\det(A+B) = \det(A) + \det(B) + \operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB).$$
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0answers
9 views

How to compute homography matrix H from four corresponding points

I am using 4 point correspondence to compute elements in Homography matrix $H$. \begin{align*} [x']={}& [h_1 h_2 h_3] [x] \\ [y']={}& [h_4 h_5 h_6] [y] \\ [(1)]=[h_7 h_8 h_9] [(1)] ...
2
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1answer
11 views

calculating the orientation of an object

If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to ...
1
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1answer
43 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
3
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1answer
29 views

linear transformation proof problem

So question is : For any $m\times n$ matrix $A$,let $T_A$ be the linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ defined by $T_A(x) = Ax$ for all $x \in \mathbb{R}^n$.Let $A$ and $B$ be ...
2
votes
2answers
32 views

Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: $ $[A]$ = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $ I have the eigenvalues: ...
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1answer
18 views

A question in limit matrix polynomial

Suppose ${A_j},\,{\Delta _j} \in {\mathbb C^{n \times n}},\quad\big(\,j = 0,\,1,\,2,\,\ldots,\,m\,\big)$ ${P_\Delta }\left(\lambda\right) = \left({A_m} + {\Delta _m}\right){\lambda ^m} + \, \cdots ...
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votes
0answers
6 views

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form $A = P^TLDL^TP$, where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
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0answers
21 views

Matrix Inequality for the identity and a traceless matrix

Given a traceless matrix $C$ $\in M_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$. This was cross-posted to ...
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0answers
10 views

Finding closest vector for all rows in a matrix

I have two matrices 1. D ($m \times n$) and 2. C ($k \times n$). Typically, $m \approx 10^4, n \approx 100, k \approx 100 $. For each row r in D, I need to find the index of the row in C that's ...
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2answers
28 views

Row swapping through matrix multiplication

Let's say I have a matrix \begin{bmatrix}a&b\\c&d\end{bmatrix} What would I have the multiply the matrix above by to obtain the following? **\begin{bmatrix}c&d\\a&b\end{bmatrix}
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2answers
29 views

Recursive formula to 3x3 matrix

I was given a recursive formula and I need to convert it into a $3\times3$ matrix. What is a general formula to do this? My recursion is in the form: $$R_{n+2} = 4R_{n+1} + 5R_n + 2R_{n-1}$$ Just ...
1
vote
3answers
56 views

Is the following set a group?

Let $ G= \begin{pmatrix} a & a\\ a & a\\ \end{pmatrix} $ where $a\in \Bbb R, a \neq0$. I need to show that $G$ is a group under matrix multiplication. The ...
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votes
1answer
24 views

Constructing a specific Rank-One Matrix

Given u $\in \mathbb{R}^{n}$ and v $\in \mathbb{R}^{m}$ with unit $L^{2}$ norm, i.e. $\|u\|_{2}$ = $\|v\|_{2}$ = 1. Construct a rank-one matrix B $\in \mathbb{R}^{mxn}$ such that $Bu = v$ and ...
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1answer
39 views

How does determinant expansion by different rows work?

I have almost always seen the determinant expanded by using the first row: $$ A = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} $$ Such as: $ |A| = ...
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vote
1answer
16 views

Order of $LU$ factorisation

Can someone tell me how to calculate the order of a) $LU$ decomposition as well as b)the gaussian elimination of a square matrix $A$? I am at a loss ... Given:: $A$ is a $n\times n$ matrix and ...
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1answer
36 views

RQ decomposition

Can someone explain me how we can compute RQ decomposition for a given matrix (say, $3 \times 4$). I know how to compute QR decomposition. I know the function in MATLAB which computes this RQ ...
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votes
1answer
34 views

Inner product vs. vector triad form

This program involves the matrix-vector computational primitive $y \leftarrow Ax$ where $x,y\in\mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$. A is taken to be dense and banded in the two parts of ...
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2answers
24 views

Matrix problem invovling orthogonal matrices

If $$P=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\text{ and } A=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$$ And $Q=PAP'$ then ...
0
votes
2answers
23 views

Evaluating condition for no roots using gauss jordan elimination

Find the number of values of $k$ for which the system of equations has no solution: $$(k+1)x+8y=4k$$ $$kx+(k+3)y=3k-1$$ This is the augmented matrix: $$\begin{bmatrix} k+1 & 8 & 4k ...
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0answers
25 views

Looking for algorithm that partition a matrix given a criterion with restriction on the number of operations

Context: Be a vector $V \in \mathbb{R}^p$, a matrix $X\in \mathbb{R}^{(p\times p)}$ and a second vector $B\in \mathbb{R}^p$ such that $V =X B$ I'm looking for a way to find a partition of the type ...
1
vote
1answer
37 views

Is $|A| < |B|$ if $A-B$ is positive definite?

I want to prove this. Say if $A-B$ is a positive definite matrix then can we find a relation between $\det(A)$ and $\det(B)$? e.g. is $|A| < |B|$.
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1answer
52 views

For any skew-symmetric $3×3$ matrix $A$, does there exist a symmetric $3×3$ matrix $B$ such that $AB+BA=O$

I got $2$ non zero matrices: $A$ and $B$. $A^t=-A$ and $B^t=B$. I try to understand if there is such matrix $B$ so the statment $AB + BA = O$ is true. I know that for $B=I$ it's not becuase $A$ will ...
0
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0answers
27 views

Matrix calculation - Why is my approach wrong? And what is a proper solution

let A ∈ Mat_m,n(F). Given that the linear equationsystem Ax = b, for b ∈ F_m, only has one solution; b = 0. Show that A is the zero matrix. So what I did was: Ax = b Ax - Ax = b - b (A-A)x = 0 ...
1
vote
1answer
26 views

Matrix equation implies invertibility

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix with positive entries $\lambda_i > 0$ (some of them might coincide). If we have the matrix equation $A D A^t = ...
0
votes
3answers
42 views

Understanding the method to find Eigenvectors

For a matrix: $ $[A]$ = \begin{bmatrix} 5 & 4 \\ 1 & 2 \\ \end{bmatrix} $ I have the eigenvalues: $\lambda = 6, 1$ Now for each value I need to find ...
0
votes
1answer
23 views

inner product of two random vectors

Two random vectors $\mathbf a$ and $\mathbf b$. Vector $\mathbf a$ has uncorrelated entries satisfying $\mathbb E [\mathbf a \mathbf a^{\rm H}]=\sigma^2{\mathbf I}$. Now I need to calculate ${\mathbb ...
0
votes
0answers
11 views

Reducing a rectangular matrix of large rationals to small rationals

I have a large matrix (~1000 by ~2000), whose entries are purely rational numbers, typically involving large fractions, that is numbers (much) larger than 10^8 in denominator/numerator. The original ...
2
votes
5answers
89 views

Are there singular matrices such that if we change any entry it will be non-singular?

Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix with real entries such that its determinant is zero, but if one changes any single entry one gets a matrix with non-zero ...
2
votes
2answers
54 views

If A and B are diagonalizable then so is AB

When we have to n×n matrices that can be made diagonal (maybe not in the same basis), is it true that the same works for their product?
0
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0answers
26 views

Singular matrices over prime fields

Show that if a given matrix A with integer coefficients over $\mathbb{F}_{p}$ is singular for infinitely many primes $p$, then it is for all primes.
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2answers
32 views

Find the matrix representation

The question I'm stuck on asks: Find the matrix representation of the differential operator D acting on the space of polynomials of degree at most $3$ with basis $(3, 1 + x, x − x^ 2 , 1 + x^ 3)$, ...
0
votes
1answer
15 views

Eigenvector with matrix amlost full with zeros

Hi i have weird problem with calculate eigenvector from simplest matrices. So have something like this: $A = \begin{bmatrix} \frac{1}{2} & 0 \\ 2 & \frac{1}{2} \end{bmatrix}$ Eigenvalues are ...
0
votes
2answers
27 views

Maximum 2x2 squares in given rectangle

I have a matrix of size nxm which consists of 0s and 1s..so i have to place 2x2 squares in matrix where there is 0. You cant place square where 1 is present. The question is maximum 2x2 squares that ...
0
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0answers
18 views

non-singular matrix block matrix over $\mathbb{Z}_p$

Let $p$ be a prime number and $A,B,C\in M_n(\mathbb{Z}_p)$ be nonsingular circulant matrices. How can I prove that this matrix $$\begin{bmatrix} A &B\\ B &C \end{bmatrix}$$ is nonsingular? I ...
3
votes
0answers
37 views

Maximum column sum norm of inverse matrix, $\|A^{-1}\|_1$

$A$ is an $N \times N$ nonsingular matrix with bounded maximum row sum norm and unbounded column sum norm, that is, $\|A\|_\infty = O(1)$, and $\|A\|_1=O(N^\alpha)$, where $0<\alpha\leq1$. ...
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vote
1answer
33 views

Find “almost inverse” of positive definite bilinear form

Let $A$ be a positive definite $d \times d$ matrix, and define $A(x,x)=x^TAx$. Let $x$ be a point such that $\vert x^T\xi\vert^2\leq \xi^T A\xi$ for all $\xi\in\mathbb{R}^d$. Does this somehow imply ...
3
votes
1answer
55 views

How to calculate the derivative of logarithm of a matrix?

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$ The derivative of ...
3
votes
2answers
38 views

For which values of $a,b$ is the matrix invertible?

I am trying to figure out the below question: 15. For which values of the constants $a$ and $b$ is the matrix $$A = \left[\begin{array}{cc} a & -b \\ b & a \end{array}\right]$$ ...
0
votes
0answers
14 views

How does one convert between rotation pseudo-vectors and rotation matrices for any number of dimensions? [on hold]

I already know the two and three dimensional cases but I want to know a generic formula. In two dimensions one just has one angle, $\theta$. And the rotation matrix is $\left[\begin{array}{cc} \cos ...
0
votes
2answers
18 views

The relationship between matrix rank and its characteristic polynomial coefficients

Given the matrix characteristic polynomial coefficients. Is there a quick way to determine the rank of the matrix?
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1answer
29 views

Derivative of matrix logarithm with respect to matrix

I saw in this post that $\frac{d}{dt}\text{logm}(Z(t)) = \frac{dZ(t)}{dt}(Z(t))^{-1}$ Is this true to say: $\frac{d}{{dU}}{\mathop{\rm logm}\nolimits} (A) = {A^{ - 1}}\frac{d}{{dU}}A$ where U is ...
0
votes
1answer
17 views

An exercise about basis for orthogonal subspace (solution check)

I believe what I did in this exercise is correct, but I'm wondering if there is a faster way to do this kind of computation. I'm practicing for an exam that requires me to be really fast solving ...
1
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0answers
38 views

Geting $A$ from $AA^{T}$ [duplicate]

I have a symmetric matrix $B$ (actually a covariance matrix of a set of variables) and I want to write it in the form of $AA^{T}$. How can I get $A$? Thanks.
2
votes
1answer
49 views

Find the kernel of the linear transformation

So the question asks: find the kernel of the linear transformation $T : \mathbb{R}^4 \to \mathbb{R}^3$ defined by $T(x) = Ax$ where $A$ is the matrix: $$\begin{bmatrix}1 & 0 &1 & 0\\0 ...