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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0
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0answers
16 views

How can I take a distance matrix and construct a coordinate representation from it

Say I have a distance matrix M of rank n, where the distance between the ith and jth point is M[i,j]. the diagonal of such a matrix will be 0. How can I convert the distance matrix M to coordinates ...
2
votes
0answers
32 views

Dot product of two vectors as the eigenvalue of a special matrix [duplicate]

I just noticed that for any two Cartesian vectors their dot product is precisely the only non-zero eigenvalue (if such exists) of the following matrix: $$\vec{a}=(a_1,a_2,a_3,\dots)$$ $$\vec{b}=(b_1,...
-2
votes
0answers
25 views

Do I calculate the determinant of a Jacobian Matrix the same way as a normal symmetrical Matrix?

As the title says: Do I calculate the determinant of a Jacobian Matrix the same way as a normal symmetrical Matrix? Or is there another way to do it?
2
votes
1answer
34 views

Derivation of gradient for non negative matrix factorization

I am looking at a paper for non-negative matrix factorization and can't seem to figure out the derivation for the gradient. The function is as follows: $f(W,H) = \frac{1}{2}||V-WH ||^2_F$ Where V ...
2
votes
3answers
83 views

Derivative of projection's norm squared with respect to a matrix

Background: Let $M^{n\times k}(\mathbb{R})$ denote the $n\times k$ matrices with real entries. For any smooth function $f: M^{n\times k}(\mathbb{R}) \to \mathbb{R}$, define the derivative $\frac{\...
2
votes
2answers
58 views

Decompose a matrix into diagonal term and low-rank approximation

For a matrix $A$ the Singular Values Decomposition allows getting the closest low-rank approximation $$A_K=\sum_i^K\sigma_i \vec{v}_i \vec{u}_i^T$$ so that $\|A-A_k\|_F$ is minimal. I'd like to do ...
-1
votes
1answer
29 views

About Rigid Matrices [closed]

I want to know the meaning of Rigid matrices and Rigidity of matrices contained in this definition "The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be ...
4
votes
2answers
75 views

The relation between axes of 3D rotations

Let's suppose we have two rotations about two different axes represented by vectors $v_1$ and $v_2$: $R_1(v_1, \theta_1)$, $R_2(v_2,\theta_2)$. It's relatively easy to prove that composition of ...
-4
votes
0answers
17 views

What is the Diagonal of a non-square matrix? [closed]

What is considered as the diagonal of non square matrix. Is it the set of elements of $M_{ij}$ where $i=j$?
2
votes
0answers
64 views

Is there a concept of “Cross determinant”?

Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$ Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could ...
1
vote
1answer
27 views

Norm of multiplication and multiplication of norms

It is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$. Is the following true for all $p \in \mathbb{R}$: $$\|u \cdot v \|_p \le \|u \|_p \cdot \|...
2
votes
0answers
19 views

The expected distortion of a linear transformation (continued)

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the "average distortion" caused by the action of $A$ on vectors. Consider the uniform distribution on $\mathbb{S}...
-1
votes
1answer
35 views

Range space of matrices over $\mathbb{Z}$

Let A and B be $m \times n$ matrices over $\mathbb{Z}$ such that $B=PAQ$ for some invertible matrices P and Q. Then can we tell that Range space of A is same as that of the range space of B when A ...
1
vote
2answers
21 views

Linear Equation by Elimination

What multiple of equation $1$ should be subtracted from equation $2$? $Eq. (1): 2x-4y=6$ $Eq. (2): -x+5y=0$ After the elimination, solve the triangular system. If the right side changes to $(-6,0)$,...
-2
votes
0answers
22 views

Linear Equations by Elimination [closed]

What multiple L21 of equation 1 should be subtracted from equation 2? 2x+3y=1 10x+9y=11 After that step, solve the triangular system by back substitution, y before x. Verify that x times (2,10) ...
3
votes
1answer
26 views

Solving the quadratic formula to determine stability of a system

I am trying to solve the $2\times 2$ matrix $$\begin{bmatrix} 0 &1 \\ -k &-b \end{bmatrix}$$ for a relationship between the variables $k$ and $b$ to determine when a system is stable. ...
0
votes
1answer
21 views

Checking positive definiteness of some matrix

Let $B$ be a bounded self-adjoint operator on the Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle)$ with $0 \not \in \sigma(B)$ and further let $\rho \in \mathbb{R}$ be strictly positive ...
1
vote
1answer
36 views

Condition for Linear Dependence

Let $\mathbf{x}\neq \mathbf{0}$ and $\mathbf{y}\neq \mathbf{0}$ be $n \times 1$ vectors, $\mathbf{A}\neq \mathbf{0}$ and $\mathbf{B}\neq \mathbf{0}$ be $m \times n$ matrices with $m>n$, and, for ...
-1
votes
1answer
97 views

Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$. [duplicate]

Let $A\in M_{n \times n}$ such that rank of $A$ is $1$. Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$.
1
vote
1answer
20 views

Basis consisting of vectors with non-negative entries only

For a given linear space $X\subseteq{\bf R}^n$ of dimension $k$, can we always find a basis $b_0,\dots,b_{k-1}\in X$ consisting of vectors with non-negative entries? If no, what is the smallest $\ell$...
2
votes
1answer
44 views

What are the hyperbolic rotation matrices in 3 and 4 dimensions?

So the hyperbola-preserving transformation in 2 dimensional space is given by the matrix \begin{pmatrix} \cosh(\phi) & \sinh(\phi) \\ \sinh(\phi) & \cosh(\phi) \end{pmatrix} I'm wondering ...
2
votes
0answers
28 views

Specific decomposition of quadratic 2x2 matrix

Consider the matrix $A = \begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix}$. Prove that there is only one decomposition A = B + C with $B,C \in \mathbb{R}^{2x2}$ that fulfill the following ...
2
votes
1answer
29 views

Combining Column- and Row-wise meanings of a matrix

A matrix can be thought of in terms of columns (then it represents the basis vectors of an coordinate system) or in terms of rows (then it represents a set of linear equations). How can we combine ...
3
votes
2answers
33 views

$S$ is a subspace of $V$, then does $S$ perp contain $V$ perp?

My question is very simple. If $S$ is a subspace of the vector space $V$, would that make $V^{\perp}$ contained by $S^\perp$? I am asked to prove this theorem, but I couldn't move a pencil :(
2
votes
1answer
185 views

is this matrix invertible

Is the following matrix invertible? $\left[ \begin{matrix} \sum\limits_{x=1}^{n}{1} & \sum\limits_{x=1}^{n}{x} & \sum\limits_{x=1}^{n}{{{x}^{2}}} & \cdots & \sum\limits_{x=1}^{n}{{...
1
vote
1answer
32 views

Why is $<T\vec x,\vec y>=<\vec x,T^*\vec y>$ for hermitian matrix T?

In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: $<\vec x, L\vec y>=<L^*\vec x,\vec y>$. I know that I can pull out ...
0
votes
1answer
30 views

Matrix derivative (chain rule application)

Let $x$, $y$ by vectors s.t. $x=f(y)$ and let $B$ be a constant matrix. What is $\frac{\partial x'Bx}{\partial y}$? The partial derivative $\frac{\partial x'Bx}{\partial x}=2Bx$ and we need to use ...
-4
votes
2answers
84 views

What is meant by $\mathbb R^m$ in this context?

I'm studying some introductory Linear Algebra text. So far, it's explained $\mathbb R^2$, $\mathbb R^3$, $\mathbb R^n$ — all understood. Then $\mathbb R^m$ came out of the blue with no background ...
3
votes
1answer
41 views

Let $\det (A + \alpha I) = 0$ for all $\alpha \in \mathbb{C}\backslash \left\{ 0 \right\}$. Can we say that $\det (A) \ne 0$?

Let $A \in {M_n}$ and $\det (A + \alpha I) = 0$ for all $\alpha \in \mathbb{C}\backslash \left\{ 0 \right\}$. Can we say that $\det (A) \ne 0$?
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0answers
24 views

Can $\sum_{|I|=k}\det(G_I)^2$ be represented by the eigenvalues of $G$?

In the question, $G\in R^{n\times n}$ is the symmetric positive semi-definite (SPSD) matrix, $\det(\cdot)$ is the determinant of the matrix, $G_I$ is a principal submatrix of the SPSD matrix $G$, and $...
3
votes
2answers
50 views

Does subtracting a positive semi-definite diagonal matrix from a Hurwitz matrix keep it Hurwitz?

I am having a linear algebra problem here. I will be grateful if someone can help me. Let $A\in \mathbb{R}^{n\times n}$ be Hurwitz and diagonizable, and let $B$ be a diagonal matrix whose diagonal ...
1
vote
1answer
15 views

Proof: $\underset{\|q\|=1}{\max} q^TAq = \lambda_{\max}$ with $q$ the corresponding eigenvector ($A$ symmetric)

This problem is quite old and there should be similar problems. I know the following technique: \begin{equation} \begin{aligned} q^TAq=q^TU\Lambda U^Tq=(U^Tq)^T\Lambda (U^Tq) \end{aligned} \...
0
votes
1answer
27 views

Square matrix with independent columns

So my question is can we conclude that a square matrix with independent columns would be a basis for $$R^{n=m}$$ right? no matter what right?
0
votes
0answers
26 views

matrix multiplication result value range

Here is the initial question: About the output value range of LeGall 5/3 wavelet Today I found actually the transform can be seen as a matrix multiplication. It is easy to calculate the wavelet ...
0
votes
1answer
29 views

About LU decomposition

Question asks me to give an LU composition of the matrix below $$ \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \\ \end{bmatrix} $$ I ended up with the following U = $...
0
votes
2answers
27 views

Image of subspace under the matrix linear transformation

Consider the linear transformation $\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ whose matrix in relation to the canonical base is: $[T] = \begin{bmatrix} 1 & 2 & -1 \\ 0 & 2 &...
1
vote
1answer
20 views

Finding the rank of an endomorphism

Recently I tried to prove a statement I know should be easy, but for some reason I just can't prove it. The statement is: given a $9 \times 9$ matrix $N$ sucht that $N^3 = 0$ and that $rk(N^2)$ = 3, ...
0
votes
1answer
30 views

Geometric interpretation of a linear system

Solve the following system of linear equations in terms of parameter $a\in\mathbb R$ and explain geometric interpretation of this system: $ax+y+z=1,2x+2ay+2z=3, x+y+az=1$. By Cronecker Capelli's ...
2
votes
1answer
15 views

The true definition of invariant functions on Matrix algebra

According to terminologies in "Invariant theory" a true definition for an invariant function $f:M_{n}(\mathbb{R})\to \mathbb{R}$ is the following: Definition 1: A continuous function $f$ is ...
3
votes
3answers
89 views

Differentiating $\mbox{tr} (ABA^TC)$ w.r.t. $A$

Why is $\nabla_A \mbox{tr} (ABA^TC) = CAB + C^TAB^T$? Here $A, B, C, D$ are all $n \times n$ matrices. $$\nabla_A f(A) = \left[\begin{matrix} \frac{\partial f}{\partial A_{11}}... \frac{\partial f}{...
0
votes
0answers
22 views

Check if the echelon form is correct

I am taking a new course on linear algebra online, and I am trying to convert matrices into echlon forms. I need to ask if there's any textbook method using which we can check if the row ...
0
votes
0answers
11 views

How to explain a sum of two mahalanobis projection?

I have to explain the use of the sum of two mahalanobis matrix the sum is done on the L component of the Mahalanobis matrix where $M=L^TL$ so i have $L=L_1+L_2$ and I formulated the following : $$(\...
0
votes
2answers
26 views

Linear Matrix Inequality of a matrix $T$ with $\| T \|_2 < \rho$

Assume $T \in \mathbb{R}^{d\times d}$. Let $\rho(T)$ be the spectral radius of $T$ and $\rho \ge 0$. Prove that $\rho(T) < \rho$ if and only if there exists a $P \succ 0$ satisfying $T^TPT - \rho^...
2
votes
2answers
55 views

Problem calculating the sine of a matrix

Given the matrix $A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}$, I want to calculate the sine $\sin(A)$. I do so by diagonalizing A and plugging it in the ...
0
votes
1answer
36 views

Multiple linear combinations

Lets say I have a vector $v$ and $n$ vectors $u_1,\; ... \;, u_n$ Is there a fast way to know if there are more than 1 or just 1 possibility to express $v$ as combination of $u$'s ? The information ...
1
vote
1answer
24 views

Existence of solution to underdetermined linear system with variable coefficient matrix.

I'm trying to think through a network flow problem, and while I could probably shuffle this into a form that a linear programming method would work, it feels like there ought to be something more ...
3
votes
3answers
59 views

Find vectors that span the kernel of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$

I have the following matrix: \begin{bmatrix}1&2\\3&4\end{bmatrix} and I'd like to find the vectors that span the kernel. The book I'm reading isn't helping me understand this concept at ...
0
votes
0answers
43 views

Eigenvectors of an approximated symmetric matrix

A $3 \times 3$ symmetric matrix has the form $$S=\begin{bmatrix} x & y & z \\ y & w & 0 \\ z & 0 & u \end{bmatrix}$$ While finding eigenvalues, I had to approximate the ...
-1
votes
1answer
22 views

Definite Integral of Kronecker product of matrices

How to prove: $$\int_k^{k+1}\int_k^{k+1} (A⊗B) dxdy = \int_k^{k+1}\int_k^{k+1} A dx⊗\int_k^{k+1}\int_k^{k+1} Bdy $$, where k is an integer and A and B are matrices consisting of variables x and ...
3
votes
3answers
60 views

Finding the inverse matrix

I have these matrices: Find the inverse matrices: \begin{bmatrix} 1 & 1 & 0& 0&\dots & 0& 0\\0 & 1 & 1& 0&\dots & 0& 0 \\0 & 0 & 1& 1&...