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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2
votes
1answer
27 views

how to prove if [T]b is diagonal then there is a scalar “a” such that T(v)=av

hey i was trying to prove the next proposition: given T:V->V for every Basis B, if the matrix [T]B is diagonal, then there is a scalar "a" for every v in V such that T(v)=av this is what i managed ...
2
votes
3answers
31 views

How does this reduced matrix indicate that the vectors are linearly independent?

I know that a set of vectors is linearly independent when a linear combination of them equal to zero is only satisfied by coefficients that are all zero. For this particular question, we have a ...
3
votes
1answer
38 views

Name of the LU decomposition algorithm

On the wikipedia page of LU decomposition there is an algorithm that produce the decomposition. It is called Doolittle algorithm. I'm really interested who is Doolittle? Or from where the name comes ...
1
vote
1answer
14 views

Tensor Notation Upper and Lower Indices

I want to ask what the difference between the tensors $T_i^{\; j}$ , $T_j^{\; i}$ , $T_{\; i}^{ j}$ , and $T_{\;i}^{j}$ are. In particular I am asking about the matrix representations of these tensors ...
1
vote
2answers
44 views

$A$ is normal matrix and has distinct eigenvalue, and $AB=0$. why $B$ is normal matrix?

Let $A,B \in {M_n}$ . suppose $A$ is normal matrix and has distinct eigenvalue, and $AB=0$. why $B$ is normal matrix?
0
votes
0answers
32 views

Does the inequality $v'Au\le u'Du$ hold under given conditions on the vectors $u,v$ and matrices $A,D$?

Matrix $A$ with $A_{ij}\geq 0$, $D$ is a diagonal matrix with $D_{ii}=\sum_{i=1}^nA_{ij}$. $u,v$ are two vectors with proper dimensions, and $\|v\|\leq\|u\|$, and the inequality is: \begin{equation*} ...
2
votes
3answers
39 views

Show that the rank of $A =\begin{bmatrix}B&C\\D&E\end{bmatrix}$ equals $r$ if and only of $DB^{−1}C = E$

Let $k$ and $n$ be positive integers and let $F$ be a field. Let $A =\begin{bmatrix}B&C\\D&E\end{bmatrix}$ be a matrix in $\mathscr{M}_{k×n}(F)$, where $B$ is a nonsingular matrix in ...
0
votes
0answers
18 views

Diagonal decomposition, square root and eigenvector / eigenvalue of a matrix

I have encountered a problem of finding eigenvector and eigen value of a matrix of type $$ A = \dfrac{1}{2} \begin{pmatrix} 4&1&-2\\ -4&1&6\\ 2&0&-2 \end{pmatrix} $$ Also I ...
0
votes
1answer
32 views

Basis of a Z-module

I think I might know how to start this problem but I'm not sure how to finish. Here is the statement: Determine a basis for the ℤ-module of integer solutions to the following system of equations: ...
0
votes
1answer
8 views

tridiagonal matrix with a corner entry from upper diagonal

I am trying a construct a matlab code such that it will solve an almost tridiagonal matrix. The input I want to put in is the main diagonal (a), the upper diagonal (b) and the lower diagonal and the ...
1
vote
0answers
23 views

Iterative method for a positive definite matrix [on hold]

Given A-> a positive definite matrix, if W(x)= 1/2(Ax,x)-(b,x) is a functional form and x-> x + a_i * v_i is the i-th step iteration( v_i is the ith eigen vector of A) 1. how do I find the step size ...
-1
votes
0answers
22 views

if $A$ is normal matrix and has distinct eigenvalue, and $AB=0$ why $B$ is normal matrix?. [on hold]

Let $A,B \in {M_n}$ . if $A$ is normal matrix and has distinct eigenvalue, and $AB=0$ why $B$ is normal matrix?
0
votes
3answers
32 views

Finding inverse of matrix with trig functions

$$\begin{bmatrix} \cos(30^\circ) & 0 \\ \sin(30^\circ) & 1 \end{bmatrix}$$ I am fine with finding the inverse of a standard $2 \times 2$ matrix but I am struggling to find the inverse of this ...
0
votes
0answers
15 views

Bandwidth of matrix through QR factorization [on hold]

If matrix $A$ is symmetric and has bandwidth $p$ and has QR factorization as $A=QR$, what can one say about the bandwidth of $RQ$? Are they the same and why?
0
votes
2answers
20 views

Find basis so matrix is in Jordan Canonical Form

$M = \left(\begin{array}{ccc}0 & -3 & -2 \\1 & 3 & 1 \\1 & 2 & 3\end{array}\right)$ I want to find a basis $B$ such that matrix for $M$ w.r.t $B$ has the form: ...
0
votes
2answers
23 views

Is it possible to have a system of equations that all equal 0, and not have each unknown's value be 0?

I'm doing about a 2 hour long homework assignment where by hand I must construct a 10x10 matrix representing a system of equations. Based on the pattern I'm seeing, I can tell all of the equations ...
1
vote
2answers
75 views

Why does $\mathrm{Rank}(A{A^*} - {A^*}A) \ne 1$?

Given $A \in M_n$, why does $\mathrm{Rank}(A{A^*} - {A^*}A) \neq 1$?
4
votes
1answer
25 views

What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
0
votes
1answer
58 views

$A$ and $B$ are normal matrices and $AB=0$. why is $BA=0$? [on hold]

Let $A,B \in {M_n}$ be normal,and $AB=0$.why is $BA=0$?
4
votes
1answer
83 views
+50

Is Frobenius norm induced by 2 vector norms?

Let in the space $V$ defined norm $ ||\cdot||_V $ and in the space $W$ defined norm $ ||\cdot||_W $ Then consider operator norm induced by 2 vector norms $ ||\cdot||_V $ and $ ||\cdot||_W $ $ ||A|| ...
1
vote
1answer
12 views

proving equation of an invertible matrix

as far as i can tell the following sentence is true but what are the steps to actually prove it? ...
1
vote
2answers
50 views

Matrix $AB = 0$ , so $A$ and $B$ are not invertible

I am trying to show that if a matrix $AB = 0$ , then the matrices $A$ and $B$ are not invertible. Edit: Could we show the same thing with A and B =/ 0?
0
votes
1answer
16 views

Invertible matrix equation

I am trying to prove OR to rule out the following sentence and i'm kind of stuck. if A,B are Invertible matrix, then A+B is also an Invertible matrix? what are the steps to prove OR to rule it ...
2
votes
1answer
27 views

Symmetricity and orthogonality

Can a 3 or more dimensional orthogonal matrix be symmetric ? I am learning linear algebra and I couldn't seem to figure it out. I understand an Identity matrix or any column matrix with either 1 or ...
1
vote
1answer
11 views

Is upper Hessenberg form preserved through similarity transformation

Suppose $X$ is non-singular and $M$ is upper Hessenberg. Is $X^{-1}MX$ also upper Hessenberg.
1
vote
0answers
74 views

Pseudoinverse - Interpretation

An $m$-dimensional (column) vector $y$ is defined as follows: $Ay=x+v$, where $A$ is an $m*n$ matrix with $m<n$ (and full row rank), $x$ is an $m$-dimensional column vector of constants and $v$ ...
1
vote
1answer
43 views

How can I maintain linear independence through a commutator?

Consider a Lie algebra $\mathcal{L}$, a linearly independent generating set $\mathcal{G}$, and an element $X \in \mathcal{L}$. Edit: Note that $\mathcal{G}$ is not necessarily a basis; the generation ...
1
vote
1answer
25 views

Proving isometry and continuity from a positive definite symmetric real matrix

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\epsilon$ be the Euclidean metric on $\Bbb R^n$ ...
7
votes
1answer
60 views

Singularity of a positive linear combination of rank one matrices

Given a set of rank one matrices $A_1,..,A_n$, we need to find out if there exists $x \in \mathbb R^n$ with $x\gg 0$ (i.e, positive) such that $$ \sum_{i=1}^n x_i A_i = x_1 A_1 + .... + x_n A_n $$ ...
0
votes
0answers
45 views

What is the fastest algorithm for 4x4 matrix multiplication?

I was wondering wich is the faster algorithm for multiplication of 2 4x4 matrices. I read about Strassen but before implementing it (as is costly) I want to be sure I'm not leaving better ones ...
0
votes
0answers
17 views

How to find the number of transitions, after which the stationary distribution could be found in Markov chain?

Say I have the initial state space vector S = [1 0 0]. and I know both the transition matrix, P and final stationary distribution, S' = [0.3 0.5 0.2]. If I was asked to calculate after how many ...
8
votes
3answers
1k views

Determinant of a 5 × 5 matrix

I have a little problem with a determinant. Let $A = (a_{ij}) \in \mathbb{R}^{(n, n)}, n \ge 4$ with $$a_{ij} = \begin{cases} x \quad \mbox{for } \,i = 2, \,\, j \ge 4,\\ d \quad \mbox{for } ...
3
votes
2answers
65 views

(Linear algebra) if $A$ is normal matrix then, eigenvectors of $ A$ are orthogonal.

I know that the eigenvectors of a unitary matrix are orthogonal. Then is that also true for a normal matrix? How do I prove?
0
votes
1answer
23 views

Matrix with given row and column sums

Let $N$ and $K$ be two given integer numbers different from zero. Let $S_n$ with $n=1,...,N$ and $C_k$ with $k=1,...,K$ strictly positive integer numbers such that $$ ...
3
votes
2answers
37 views

Derivative of Matrix Exponential as Integral

I saw this "standard" identity in a physics paper and I was wondering how to prove it \begin{align*} \frac{d}{dx} e^{A+xB}\bigg|_{x = 0} = e^A\int_0^1 e^{A\tau}B e^{-A\tau}\,d\tau \end{align*} I tried ...
0
votes
0answers
17 views

Determining Counts of Discrete Objects Using Linear Algebra

I'm teaching myself linear algebra and was able to solve the following question using trial and error, but--how would one setup and solve a question like this using Linear Algebra? I have 32 bills ...
0
votes
0answers
27 views

How to find eigen vector for an eigen value in generalized eigen value problem

I have a generalised eigen value problem of the form $A$x = λ$B$x. I have computed the eigen value (say λ1) I am interested in using Eigen library(C++). However, because the library does not support ...
0
votes
0answers
36 views

Calculate distance between known intersecting points [on hold]

I have been working on this problem for awhile now and I think I just need a few fresh minds to help me out. I have 4 lines that intersect and form a shape. This is part of a much larger problem, ...
0
votes
0answers
28 views

What is my error in this matrix / least squares derivation?

I'm doing a simple problem in linear algebra. It is clear that I have done something wrong, but I honestly can't see what it is. let, $y = Ax$, $y_{ls} = Ax_{ls}$ where A is skinny and full rank, ...
5
votes
1answer
67 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
0
votes
0answers
30 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
1
vote
0answers
42 views

When is the product of two arbitray matrices symmetric?

Let $\mathbf{A}$ be a real $n \times m$ matrix. Let $\mathbf{B}$ be a real $m \times n$ matrix. How to solve the following matrix equation? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
0
votes
3answers
58 views

Basic way to show for $n\times n$ matrices $A$ and $B$, that $(AB)^{-1} = (B^{-1})(A^{-1})$

In looking at matrix inverses, I know the following works (I is the identity matrix): If $AB$ are nxn matrices and are invertible, then $(AB)C = I$, and therefore $C = (AB)^{-1}$. I can show that ...
1
vote
1answer
17 views

One hypothesis concerning Hamming distance matrix

Suppose $a_1, a_2, \ldots, a_m$ are different strings of the same length n. And let $V = [v_1, v_2, \ldots, v_n]$ be a matrix such that $V_{i, j}$ is a Hamming distance between $a_i$ and $a_j$. ...
1
vote
1answer
11 views

What is the cofactor of an element that is zero in a matrix?

Does the cofactor of an element in a matrix that is zero always equal to zero?
0
votes
1answer
38 views

Tutte matrix - Determinant

I'm trying to understand the proof of the "magic theorem" about the Tutte matrix which states: Let $T$ be the Tutte matrix of $G(V, E)$. Then, $$\det(T) = 0 \quad\Longleftrightarrow\quad G ...
0
votes
1answer
30 views

Determinant of lower triangular matrix

Does a lower triangular matrix have a determinant that is equal to the product of the elements in the diagonal similar to an upper triangular matrix.
0
votes
0answers
13 views

Computing covariance matrix in PCA

I am implementing PCA in matlab and I have to compute the covariance matrix. I am using 'cov' command from matlab to compute the covariance matrix. But it is very slow and takes a lot of time to ...
1
vote
1answer
30 views

Smith Normal Form and quotient $\mathbb{Z}^{3}/M \mathbb{Z}^{3}$

I am learning modules and the Smith Normal Form, but I got stuck in the following: I found the Smith Normal Form of $$M = \begin{pmatrix} 21 & 0 & 1 \\ 8& 4 & 1\\ 3& 8 & 1 ...
1
vote
1answer
37 views

Finding determinant of a 4x4 matrix

I am trying to find the determinant of this matrix but was told by my teacher that we wouldn't need to find the determinant of more than $3\times 3$ matrices so I am guessing there is a way of solving ...