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
11 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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0answers
15 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.
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1answer
5 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.
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0answers
7 views

Interpretation of a pseudoinverse in case of a random vector

An $n$-dimensional (column) vector $y$ is defined as follows: $Ay=x+v$, where $A$ is an $mxn$ matrix with $m<n$ and full row rank, $x$ is an $n$-dimensional column vector of ...
0
votes
1answer
10 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 $X \in \mathcal{L}$. What are the conditions on $X$ such that $\{[X,g_i]\; \big| \; g_i ...
1
vote
1answer
9 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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0answers
6 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 $$ \sum_{i=1}^n x_i A_i = x_1 A_1 + .... + x_n A_n $$ ...
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0answers
35 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 ...
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0answers
9 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 ...
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votes
3answers
316 views

Determinant of a 5x5 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
55 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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1answer
22 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
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2answers
34 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 ...
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0answers
16 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
24 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
28 views

Calculate distance between known intersecting points

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, ...
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0answers
17 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 $x_{ls} = ...
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1answer
65 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
26 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} + ...
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0answers
36 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? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
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votes
3answers
54 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
14 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 $a_j$. ...
0
votes
1answer
9 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?
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1answer
33 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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votes
1answer
27 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
12 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
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1answer
26 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 ...
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1answer
36 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
33 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
26 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.
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1answer
30 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 ...
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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 ...
4
votes
1answer
40 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?
3
votes
1answer
28 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
33 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
2answers
83 views

Let $A,B \in {M_2}$ and $C=AB-BA$. Why is ${C^2} = \lambda I$ true?

Let $A,B \in {M_2}$ and $C=AB-BA$. Why does ${C^2} = \lambda I$?
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0answers
12 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 ...
0
votes
1answer
13 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 ...
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 $$ ...
0
votes
1answer
20 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 ...
3
votes
0answers
36 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 ...
1
vote
2answers
34 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 ...
0
votes
1answer
18 views

Product of a vector, a matrix and a vector

Given are two n-dimensional column vectors a and c, and an nxn-matrix B. Transpose is indicated by ', for example a' is the transpose of a, c' is the transpose of c, and B' is the transpose of B. ...
0
votes
1answer
16 views

Principal Component analysis by eigenvalue decomposition.

I do know how to perform PCA by using SVD but I am unaware about how to use eigenvalue decomposition of X(transpose)*X matrix. I found a paper online which explains the approach to perform PCA by ...
2
votes
3answers
59 views

Clockwise rotation of $3\times3$ matrix?

I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me. Find the $3\times3$ matrix which represents a rotation clockwise through ...
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0answers
24 views

Basis of orthogonal complement subspace [duplicate]

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
1
vote
0answers
11 views

Unique linear combination in matrix with skew-symmetric condition

Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. ...
0
votes
1answer
35 views

Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
2
votes
1answer
16 views

Unique linear combination in matrix with column sum $0$?

Let $A$ be an $n\times n$ matrix with real entries such that the numbers in each column sum to $0$, and all diagonal entries are non-zero. So, $A$ is non-invertible, and some linear combination of ...
2
votes
1answer
38 views

Is there an easier way to find the inverse of a 3x3 matrix?

I know the normal process is to do row operations to transform the matrix to get the identity matrix and then apply the same row operations in the identity matrix to get the inverse. But this process ...