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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0
votes
3answers
45 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
25 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
13 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
92 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
55 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
votes
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
17 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
28 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
votes
0answers
20 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
38 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 $0<\alpha\leq1$. ...
1
vote
1answer
37 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
56 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
40 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
16 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 ...
1
vote
1answer
31 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
vote
0answers
39 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 ...
-3
votes
0answers
65 views

Square Root Matrix [on hold]

I got an equation like this: $$ M_1 = M_2^{1/2}, $$ where $$ M_2= \begin{pmatrix} 1&0&1\\ 0&1&0\\ 0&1&1 \end{pmatrix}. $$ My question is, how to square root matrix $M_2$ and ...
2
votes
1answer
38 views

Calculate complex eigenvector

Hi i have problem i hope that someone can make this for me more clear: So i have matrix $A = \begin{bmatrix} -2 & 1 \\ -2 & 0 \\ \end{bmatrix}$ I have to calculate eigenvector as matrix $P$ ...
0
votes
2answers
31 views

Can the cross product of two non-invertible matrices be invertible?

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible? Just used to help understand a Linear Transformation assignment question, don't ...
1
vote
2answers
46 views

condition number of matrix plus constant times identity

I saw this post on the eigenvalues of a matrix plus a constant times the identity matrix. Say $A$ is an $n\times n$ matrix (real and non-singular) with eigenvalues $\lambda_1,\ldots,\lambda_n$, then ...
0
votes
1answer
58 views

An exercise from vector spaces

I do not even know how to start. Find a symmetric matrix $A \in \mathbb R^{3\times 3}$ with the following properties. Both $[1,2,2]^T$ and $[2,1,-2]^T$ are eigenvectors. It has three distinct ...
1
vote
0answers
16 views

Proof that space of correlation matrices is compact

An $n\times n$ real symmetric matrix is a correlation matrix, if it is positive-semidefinite and all its diagonal entries equal 1. According to most references it is easy to see that the space of ...
2
votes
1answer
43 views

Condition number for computing $x$?

The question is: Consider the linear system $\left( {\begin{array}{*{20}{c}} 1&\alpha \\ a&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = ...
0
votes
2answers
41 views

Does Ax=b have a solution for every vector b in $R^3$

Let $A$= $\begin{bmatrix}3 & 1 & -1\\0 & 4 &0\\6 &3&-2\end{bmatrix}$ and $x$= $\begin{bmatrix}x1\\x2\\x3\end{bmatrix}$ and $b$= $\begin{bmatrix}0\\4\\1 \end{bmatrix}$ Does ...
0
votes
1answer
15 views

Efficient Row Sum of Factorized Matrix

I am currently computing the row sums of a reduced rank factored matrix by reconstructing a row subset of the original (approximated) matrix. The matrix was factored using SVD: A -> U, S, V -> U, SxV ...
-6
votes
1answer
47 views

The sum of invertible matrices is also invertible? [on hold]

The sum of invertible matrices is also invertible ? THANKKSS!!!
0
votes
0answers
24 views

How to prove that a matrix lie group is (or is not) simple

I am having some difficulties in proving that some matrix lie groups are or are not simple. I am wondering what I should ask myself to solve this problem quickly. Should I look for normal subgroups? ...
1
vote
1answer
21 views

Changing the Form of this Factorisation

I'm brushing up on some high school maths and I'm currently revisiting determinants, specifically the factorisation of determinants. I'm working my way through a problem set and I keep getting stuck ...
1
vote
1answer
47 views

Matrix-vector product of a banded matrix

Suppose $A\in\mathbb{R}^{n\times n}$ is a banded matrix, i.e., a matrix with all of its nonzero elements on the main diagonal, i.e., $\alpha_{i,i}\neq 0$, the first superdiagonal, i.e., ...
0
votes
0answers
14 views

integral of trace function

How can I compute the below integral? $$ \int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu $$ in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix. I have ...
0
votes
0answers
36 views

The matrix with entries $M_{ij} = \frac1{t_i+t_j}$ is positive semidefinite

Prove matrix $M$, $n \times n$ is positive semidefinite if for $t_1 ,\dots,t_n > 0$: $M_{ij} = \frac1{t_i+t_j}$
2
votes
1answer
22 views

Does A+B symetric, A anti symetric, B symetric

I got $A^t = -A$ (A antisymetric) and B symetric: $B^t = B$. I need to know if $(A + B)^2$ is symetric. I couldn't find a formula which describe it. In addition, I know that A and B are non zero and ...
-3
votes
0answers
20 views

non negative matrix factorisation [on hold]

i am working on a project involving the use of non negative matrix (NMF) for the separation of a mixture of audio signals.can please give me the mathematical explanation of NMF so that i can ...
-5
votes
0answers
26 views

How to create a system of equations [closed]

How to create a system of equations delete
1
vote
0answers
17 views

What are the different ways of performing Triangular matrix-vector multiplication?

Suppose we have $$\left[\begin{array}{cccc} x_1 & 0 & 0 & 0 \\ x_2 & x_1 & 0 & 0 \\ x_3 & x_2 & x_1 & 0 \\ x_4 & x_3 & x_2 & x_1 \end{array}\right] ...
0
votes
1answer
31 views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
0
votes
0answers
37 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
0
votes
2answers
24 views

Cardinality of the set $S$ where $S=\{T:\Bbb{R}^3\to \Bbb{R}^3\mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)\}$

Let $S=\{\,T\colon \mathbb{R}^3\to \mathbb{R}^3\mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)\,\}$. Then $S$ is A. a singleton set B. a finite set containing ...
3
votes
1answer
25 views

Proving that $V = U \oplus W$ where $W$ and $U$ are sets of eigenvectors of $S: V \to V$

Let $V$ be a finite dimensional real vector space, $S : V \to V$ be a linear map such that $S^2 = I$. Show that $V = U \oplus W$ where $U = \{u \in V : Su = u\}$ and $W = \{ w \in V : Sw = -w\}$. ...
1
vote
0answers
27 views

Calculate Rotation and Translation Matrix to align elements of input matrix A to Target matrix B in 2d?

I have a matrix in 2D space; the matrix contains elements which I would like to translate into the center of the matrix. Then, I would like to rotate these elements (I mean the positions of the ...
1
vote
2answers
66 views

advanced solutions for an elementary problem!

Let $A\in M_3(\mathbb{R})$ be a matrix of rank one. Suppose that the first row of $A$ is an eigen vector of $A$. I want to show that $A$ is symmetric. My attemp: Actually its simple, for example ...
0
votes
1answer
27 views

Guass Jordan Elimination Matrix Problem

I followed the method as below: $ $[A]$ = \begin{bmatrix} 8 & 4 & 3 & | & 1 & 0 &0\\ 2 & 1 & 1 & | & 0 & 1 &0 \\ 1 ...
0
votes
1answer
21 views

Sum of component projection matrices

Show that if $X$ $=$ [$X_1$ $X_2]$ and $X_1'X_2 = 0$, then $P = P_1 + P_2$, where $P$ is defined as $X(X'X)^{-1}X'$, the projection matrix. Don't quite know where to start. I tried evaluating it by ...
0
votes
1answer
33 views

Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong \mathbb{R}/2\pi\mathbb{R}$ .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong ...
0
votes
1answer
20 views

vector dot product

I have the multiple choice questions (from a past exam, not for marks don't worry) that states: If $u$ and $v$ are vectors such that $\| u+v \| = 2$ and $\| u-v \|= \sqrt{8}$, then the dot product of ...
0
votes
3answers
37 views

Is it possible to solve for values in a matrix such that all rows and columns have equal sum?

Is it possible to solve for values in a grid such that all rows have the same sum and all columns have the same sum where values in the table can be any real number? meaning: ...
0
votes
1answer
19 views

Basis of a centralizer

If we consider $A$ to be a $2 \times 2$ matrix, the $2 \times 2$ matrix of ones $J_2=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, and the centralizer $\{A \mid AJ=JA\}$, is there a way to ...
1
vote
2answers
57 views

If I know that a matrix $G = (X^{T}X)^{-1}$, how can I recover what $X$ is?

If I have a matrix $G$ where I know that $G=(X^{T}X)^{-1}$, is there a way to find $X$? Specifically, I would like to find $G$ where $G$ is: $$G = \begin{bmatrix} 0.125 & 0 & 0 & 0 & ...