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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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
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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?
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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 ...
1
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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.
5
votes
1answer
318 views

Rank of matrix as a difference of ranks

If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank ...
1
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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 $\Bbb R^n$ ...
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4answers
2k views

Rank of product of a matrix and its transpose [duplicate]

How do we prove that $rank(A) = rank(AA^T) = rank(A^TA)$ ? Is it always true? @ Najib Idrissi, timmbob, apnorton, pizza, Davide Giraudo If the people who actively mark questions related to 3 years ...
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0answers
17 views

$Mod 3$ or $mod 5$ for for an $N_5(3, 5)$ is a $[5, 2, 4]5$ Reed Solomon code.

I am trying to reduce a matrix to the form $G = [I|-B^T]$ but I just cannot get my solution to match the solution in my notes. I am reducing it Mod 3 as it is part of a coding theory question in ...
3
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3answers
463 views

Is $AA^T$ a positive-definite symmetric matrix?

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Is it true that $AA^{T}$ is positive-definite? Clearly $AA^{T}$ is symmetric. I have shown that a symmetric matrix $S\in M_{n\times n}(\mathbb{R})$ ...
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?
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2answers
148 views

Trace of matrices inequality

If I have two matrices, $\mathbf{A}$ which is symmetric and postive definite, and $\mathbf{B}$ symmetric, positive definite, and all entries in $\mathbf{B}$ are between 0 and 1, with the diagonal ...
1
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1answer
48 views

Smith Normal Forms over Fields and PIDs

I need to reduce the following matrices into the Smith Normal form over the field $(\mathbb{Z}/2\mathbb{Z})[x]$: $$M_{1} = \left ( \begin{array}{ccc} x & 1 & 0 \\ 0 & x & 1 \\ 0 ...
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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
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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 ...
1
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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
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1answer
611 views

Smith Normal Form

Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$ $$\begin{pmatrix}   (x+a)(x+b) & 0 & 0 &0 \\  0 & (x+c)(x+d) & 0 & 0 \\   0 ...
4
votes
1answer
44 views

Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?

Let $A \in {M_n}$ and $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
2
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1answer
23 views

Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
7
votes
6answers
872 views

is matrix transpose a linear transformation?

this was the question posed to me. does there exist a matrix $A$ for which $AM$ = $M^T$ for every $M$.the answer to this is obviously no as i can vary the dimension of $M$. but now this lead me to ...
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
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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 $$ ...
0
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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 ...
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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 ...
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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, ...
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 $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
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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} + ...
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
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1answer
67 views

Diagonalisation of a unitary matrix

Let $A \in \mathrm{U}(n) \subset \mathbb{C}^{n \times n}$ a unitary matrix. Show that: $\exists ~ S\in \mathrm{U}(n)$ so that $\bar{S^t}AS=D:=\begin{pmatrix}\lambda_1&&0\\&\ddots & ...
0
votes
1answer
24 views

Diagonalising an infinite-dimensional Hermitian square matrix

I have a quantum state which takes the following form: $$\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert.$$ This is an infinite Hermitian matrix ...
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 ...
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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? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
1
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2answers
39 views

Prove boundedness of the matrix series

Suppose $A$ is a square matrix, such that all eigenvalues of $A$ has norm strictly less than $1$, can I say $\sum_{i=k_0}^kA^{k-i}$ is bounded for all large enough $k_0$ and $k$? From some other ...
1
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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 $a_j$. ...
3
votes
2answers
85 views
1
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0answers
25 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
1
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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
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1answer
29 views

Linear Algebra-invariant subspaces

Suppose $V$ is a real vector space and $T\in \mathcal L (V)$ has no (real) eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.
0
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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.
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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
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 ...
2
votes
0answers
35 views

Inequality with eigenvalues

Let matrix $ X $ is Hermitian and denote $ \lambda_1(X) \ge \lambda_2(X) \ge \ldots \ge \lambda_n(X) $ eigenvalues of matrix $ X $. Prove that $ \lambda_i(A + B) \le \lambda_i(A) + \lambda_1(B) $ I ...
0
votes
1answer
21 views

find Direction vector

i got this problem ( very trivial I guess) $39x -51y =15$ $-52x + 68 = -20$ I've done the Gauss reduction and got this, Matrix: \begin{pmatrix} 1 & \frac{-17}{13} & \frac{5}{13} \\ 0 ...
0
votes
1answer
31 views

What does $\sigma$ mean in this context?

This is a problem taken straight from my Numerical Methods course : Prove that : $\sigma (A^{-1})$ = { $\frac{1} {\lambda_1} ,\frac{1} {\lambda_1}, ... ,\frac{1} {\lambda_n} $ } . However, nothing ...
1
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0answers
31 views

Differentiating a matrix

Let $$f(x) = \left[\begin{array}{ccccc} 6 &-5 &-2 &1 &7\cr -7 &0 &-2 &2 &-3\cr -3 &0 &0 &-9 &-8\cr x &6 &-3 &-3 &1\cr -3 &0 ...
3
votes
1answer
32 views

Finding a kernel and an image of $T^2$

Let $T$ be a linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^4$ that is defined by: $$T\begin{pmatrix}x\\y\\z\\u\end{pmatrix}=\begin{pmatrix}0\\z\\y\\x\end{pmatrix}$$ Find the kernel and image ...
3
votes
2answers
37 views

If $A$ can be written as a sum of nilpotent matrices why $trcA=0$?

Let $A \in {M_n}$. If $A$ can be written as a sum of two nilpotent matrices, why $trcA=0$?
3
votes
0answers
40 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
0
votes
1answer
14 views

Determinant with one parameter, how to deal with this?

Let $t\in \mathbb R$ be a parameter, and $$|A(t)|= \begin{vmatrix} a_{11}+t &a_{12}+t &\cdots &a_{1n}+t\\ a_{21}+t &a_{22}+t &\cdots &a_{2n}+t\\ \vdots &\vdots ...
0
votes
0answers
14 views

Estimate the upper bound of the spectral norm a block matrix

I actually want to estimate the upper bound of the following matrix: $\Phi(k,t) = \prod_{s=2}^{k-t+1} \left[\begin{array}{cc}a(k-s)\tilde{W}+(b(k-s)+2a(k-s))I_N & -b(k-s)\tilde{W}-b(k-s)I_N \\I_N ...
2
votes
0answers
14 views

Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...