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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7
votes
1answer
75 views

What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$?

What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$? The best I can come up with is that $A=bi\cdot M$ for $b$ a non-zero real number and $M$ a non-zero idempotent ...
1
vote
0answers
117 views

Matrix exponentiation by Lagrange interpolation formula

Okay so I'm trying to get my head around calculating powers and functions of matrices. So given a matrix $A=PJP^{-1}$ suppose we wish to calculate $A^n$ using the Lagrange interpolation formula as ...
1
vote
2answers
900 views

Finding the determinant of a $4\times4$ matrix

How does one find the determinant of a $4\times 4$ matrix? I am using Cramer's rule to solve a system of linear equations but don't know how to find the determinant of a $4\times 4$ matrix. Our matrix ...
1
vote
0answers
31 views

What are incoherent matrices

What does incoherence means in terms of matrices? I am brushing on some compressive sampling theory and I did not find any easy to understand or straight forward answer about what does the word ...
3
votes
3answers
303 views

Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant,…but they are not similar.

Question: Find two $2 \times 2$ matrices $A$ and $B$ with the same rank, determinant, trace and characteristic polynomial, but they are not similar to each other. My thought: I come up with two ...
1
vote
2answers
326 views

Matrix Exponential of Identity Matrix

I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? ...
3
votes
1answer
147 views

Why do we define addition of matrices only when they have the same size

What happens if we define $$ \begin{pmatrix} 1 & 2 \\ 1 & 2 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 ...
1
vote
0answers
58 views

Can I calculate this sum using matrix multiplication?

I would like the following sum for the matrices $A,B$, both $n \times n$: \begin{align} \sum(A_{ij} \cdot B_{jk} \cdot A_{kz} \cdot B_{zf}) & \text{ for all } j,k,z,f \text{ such that } ...
4
votes
2answers
139 views

Show each eigenvalue of a companion matrix has geometric multiplicity $=1$.

Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector $\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where $A$ ...
16
votes
3answers
1k views

Ratio of largest eigenvalue to sum of eigenvalues — where to read about it?

Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio $$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$ is a measure of the "rank-one-ness" ...
0
votes
2answers
44 views

Linear maps, matrices, nullity and rank.

I am currently trying to solve this question in my first year linear algebra course: I understand that the assoc. matrix is the coefficients, eg for (a) [[1, -1];[5, 0]], but I'm not sure how to ...
3
votes
2answers
334 views

Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
1
vote
0answers
38 views

About the rank of (sub) matrices whose entries are roots of unity

Let $\Omega$ be a matrix with entries $a_{jk}=\omega^{jk}$, where $0\leq j,k\leq n-1$, and $\omega=e^{-2\pi i/N}$, with $N\in \mathbb{N}$, so $\Omega$ looks like $$ \Omega=\begin{pmatrix} 1 & 1 ...
2
votes
1answer
49 views

How to invert a matrix

I would like to disprove the following claim, that seems false to me, finding a counterexample. Let $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times k}$, for $k < n$. Let us assume that $rk(A) ...
2
votes
1answer
91 views

Rank of matrix a submatrix $B$ from $A$

Question: A submatrix $B$ consisting of "s" rows of $A$ is selected from an n-square matrix $A$ of rank $r_{A}$. prove that the rank of $B$ is equal to or greater than $r_{A}+s-n$. My thoughts: I ...
1
vote
1answer
35 views

$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...
1
vote
1answer
64 views

Linear Algebra True/Flase

Are these two statements true or false, if brief justification/counterexample could be given it would be appreciated. $(1)$ $I_{V}$ is an identity operator on vector space $V$, $\dim V=n$ and $A$ is a ...
0
votes
3answers
54 views

How to find rank of a matrices?

Here is the question given in my text book IF the rank of the matrix $\begin{bmatrix}-1 & 2 & 5\\2 & -4&a-4\\1&-2&a+1\end{bmatrix}$ is 1, then the value of a is: a) ...
1
vote
1answer
39 views

Is there a pseudo inverse $X$ such that $ABX=A$?

Question The title pretty much sums it up. I need to find a matrix $X$ such that: $A B X = A$, with $A\in R^{n\times n}$, $\text{rank}(A)=n$, $B\in \mathbb{R}^{n\times m}$ given. The matrix $X$ ...
1
vote
1answer
72 views

pseudo-inverse by SVD decomposition has not accurate results?

The goal is finding $\frac{{\partial f}}{{\partial {\bf{A}}}} = 0$ where $ f\left( {{\bf{A}},{\boldsymbol{\alpha }}} \right) = {\left( {{{\bf{p}}^{\bf{T}}}{{\bf{A}}^{\bf{T}}}{\boldsymbol{\alpha }} ...
0
votes
0answers
105 views

Transform one curve into another

I have been working on something for a while now, and I can't really get my head around it. I consider two curves with data points and want to determine the most optimal transform from one to another. ...
1
vote
0answers
37 views

Is there would be any matrices $A$ & $B$ where the relation $AB$-$BA$=$I$ holds? [duplicate]

Here , the actual question is to find any matrices $A$ and $B$ such that $AB-BA=I$ relation holds. but actually, I dont think that we could not find any such matrices. As, the diagonal elements of ...
0
votes
3answers
54 views

Calculate Matrix A from eigenvalues, but no given eigenvectors

Here is my question: Write down a nontriangular 3 by 3 matrix whose eigenvalues are 6, 9, 2. I understand that you can calulate Matrix A using the formula A=V$\Lambda$$V^-1$, but is there a way to ...
1
vote
1answer
44 views

Matrix of a given operator $A \otimes A$

Let $V$ be a 3-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $e_{1}$, $e_{2}$, $e_{3}$ ...
2
votes
1answer
177 views

Graph of a matrix

How to define the graph of a square matrix $\mathbf{G}$ with real entries? I know that given a graph $\Gamma(V, E)$, one can define its adjacency matrix $\mathbf{A}$. But given a matrix $\mathbf{G}$ ...
3
votes
1answer
161 views

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as with SVD?

Is there a faster way to calculate a pseudo-inverse of a matrix than using SVD that is as numerically stable as using SVD?
0
votes
1answer
2k views

How to find linearly independent columns in a matrix

For a general square matrix $A$, how do I find which columns are linearly dependent? When I say linear independent I mean not linearly dependent with any other column or any combination of other ...
0
votes
2answers
154 views

Similar matrix and diagonal matrix

What is the difference between a similar matrix and a diagonal matrix? According to my textbook, the definition for both is basically: B=P$^{-1}$AP. Say if there are three matrices: A, B and C. If A ...
0
votes
1answer
34 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
1
vote
2answers
129 views

Show that the order of the matrices must be even.

Let $A,B$, two matrices with the order of $n\times n$. Given that $AB + BA = 0$ and $A,B$ are invertible (meaning, there are $A^{-1}, B^{-1}$). Prove that $n$ must be even number. $$\eqalign{ ...
1
vote
0answers
117 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
1
vote
1answer
55 views

To prove that these matrices are invertible

Let $A$ and $B$ be $n \times n$ matrices such that $||I - AB|| < 1$. Prove that $A$ and $B$ are invertible, and $$A^{-1} = B \sum\limits_{k=0}^{\infty} (I - AB)^k \text{ and } B^{-1} = ...
2
votes
1answer
44 views

How do you handle this kind of probability?

What is the probability of selecting a singular matrix from $\Bbb{R}^{3\times 3}$? I calculated it to be zero based on their being approximately $9$ degrees of freedom to choose entries of $A$ such ...
0
votes
2answers
44 views

Matrix question help.

Consider $$X = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}.$$ Find a real matrix $A$ for which $A^2 = X$. I don't know how to answer this or where to start. ...
1
vote
2answers
53 views

How many $3 \times 3$ matrices are singluar?

How many $3 \times 3$ matrices are singluar? Describe the methodology used to achieve the result.
1
vote
0answers
50 views

What does it mean to compute a normal to a triangle in a “clockwise direction”

I am trying to understand how this works. I am given 3 points, each representing a vertex of a triangle. I must then "organise" the points and calculate the normal of the resulting triangle in a ...
0
votes
1answer
36 views

Problem about M-matrix

Is a symmetric M-matrix positive definite? I intuitively think this is not correct. Can someone prove this or provide a counter-exmaple? I really appreciate it.
2
votes
1answer
60 views

Commutative matrix proof

I've got the following question: $A \in M_{nn}(\mathbb{K})$ is a matrix and $AB=BA \forall B \in M_{nn}$. Proof that $ A=aI_n \forall a \in \mathbb{K}$. and one given solution starts with: $ % ...
0
votes
0answers
18 views

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow ...
7
votes
1answer
260 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
2
votes
1answer
145 views

Converting a PDE to a matrix form

I have to solve the problem $U_{xx}-U_{xy}+2U_y+Uyy-3U_{yx}+4U=0$ using diagonal matrix as described in this article page 44 section 3.2 But my problem is there the matrix A is symmetric matrix ...
0
votes
1answer
32 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
0
votes
3answers
60 views

For which values of $k$, we have $A = A^{-1}$?

I got this question in hw. Can anyone help me solve it? Let $ A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right) $ For which values of ...
1
vote
1answer
60 views

Differential equation of the form $y'=Ay+b(x)$ with $b(x)=(\sin{(\omega x)},0)$

I have a question regarding the following specific differential equation. $$y'=\left(\begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix}\right)y+\left(\begin{matrix} ...
1
vote
1answer
27 views

Positivity of certain matrix

Let $A=[[a_{ij}]]$ and $B=[[b_{ij}]]$ be two positive semi-definite matrices of same dimensions. Further they have a property that, if $a_{ij}=0$ then $b_{ij}=0$ (i.e. the nonzero entries appear in ...
1
vote
1answer
98 views

Nonsingular block matrix

Let us consider a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ and the block partitioning $$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & ...
0
votes
0answers
22 views

How to nullify one of the axises rotation in a rotation matrix?

Let's say that I've got a matrix with some rotation stored. Now I wan't to somehow make an Y rotation equals 0 (or rather make it equal to the starting moment without rotation). How would I do it? I ...
0
votes
1answer
54 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
0
votes
2answers
36 views

How to calculate the determinant when the diagonal is in terms of $k$?

Determine the value of $k$ so that the columns in this matrix are linearly dependent: $$\begin{bmatrix} k & -1/2 & -1/2\\ -1/2 & k & -1/2\\ -1/2 & -1/2 & k ...
3
votes
0answers
36 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...