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), ...

learn more… | top users | synonyms (2)

0
votes
1answer
20 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 $$ ...
2
votes
2answers
31 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
15 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
21 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
25 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, ...
0
votes
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} = ...
5
votes
1answer
63 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
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} + ...
1
vote
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? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
1
vote
3answers
50 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
12 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?
0
votes
1answer
31 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
26 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
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 ...
0
votes
0answers
16 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 cannot find the Smith Normal Form of : $M = \begin{pmatrix} 21 & 0 & 1 \\ 8& 4 & 1\\ 3& 8 ...
1
vote
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
31 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.
0
votes
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 ...
1
vote
0answers
28 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
37 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
32 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
82 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 ${C^2} = \lambda I$?
0
votes
0answers
11 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
30 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
32 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
17 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
15 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
56 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 ...
0
votes
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
10 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
37 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 ...
1
vote
1answer
19 views

Finding eigenvectors through triangularization

I have an exam tomorrow and am working through notes. We derived the following stochastic matrix: $$P=\left[ \begin{matrix} 0.8 & 0.5 & 0 & 0\\ 0.2 & 0.5 & 0 & 0 \\ 0 & 0 ...
0
votes
2answers
37 views

Finding complex eigenvalues

For the matrix \begin{pmatrix}1/2 & 1 & 3/4\\2/3 & 0 & 0\\0 & 1/3 & 0\end{pmatrix} Find the eigenvalues and corresponding eigenvectors. I did this with an online calculator and ...
0
votes
0answers
23 views

Vector multiplication with both subscript and superscript, in training algorithm for McCollugh-Pitts neurons?

Can someone here help me understand what's going on with the Vis, with both the subscript and superscript notation (as included in the images supplied below)? Is it Einstein Notation? I'm almost ...
2
votes
1answer
30 views

Find orthogonal Q given eigenvalue and eigenvector?

Given some upper Hessenberg matrix $H \in R^{n \text{x} n}$, i know how to find an orthogonal matrix which is a product of Givens rotations such that $P^THP$ is also upper Hessenberg, but I'm not sure ...
0
votes
0answers
6 views

Fast method or direct generation of random upper triangular matrix using integer-restricted gauss elimination

I'd like to generate non-singular random upper triangular matrices of the following form: $$ \left[\begin{matrix} 1 & r_1 \cos{\alpha_1} & r_2 \cos{\alpha_2} & r_3 \cos{\alpha_3} & ...
2
votes
1answer
18 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? ...
1
vote
1answer
18 views

Vector times symmetric matrix

For most of you here, this is probably quite basic. As for a symmetric matrix $A$ the first row equals the first column, multiplying the matrix with a column vector $b$ equals multiplying the ...
0
votes
1answer
12 views

What is the connection between $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ and $V(\vec x) = \frac {1}{2} x^TPx $?

For example, $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ has an equivalent representation $V(\vec x) = \frac {1}{2} x^TPx $ where $P$ is some matrix Can someone make this connection clearer for me ...
1
vote
2answers
30 views

Linear Applied Algebra | Verify the vectors

Question: My response: Math has never been a strength, particularly proofs, so I would appreciate any and every help. I am just not sure if I am following a proper procedure for the above ...
1
vote
1answer
36 views

Applied Linear Algebra | Linear Dependent Matrix

Question: My response: Am I solving the above question correctly? Or am I on the wrong path? Thank you for your help.
0
votes
2answers
26 views

Applied Linear Algebra |

Question: Determine whether or not any column in the matrix is a linear combination of other columns. Provide a general method for answering the same question for any n x n matrix A. My response: ...