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
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 $(-6,0)$,...
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
37 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
98 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
30 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
186 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
33 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
86 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 $\det (A) \ne 0$?
1
vote
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
16 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
61 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&...
1
vote
0answers
17 views

Relation between row-space and column-space vectors

Let $A$ be any $n$ by $m$ matrix. $V$ is an orthonormal vector in column-space of $A$. $U$ is an orthonormal vector in row-space of $A$. Now, why is the following relation True? $$AV=U\Sigma$$ , ...
0
votes
1answer
35 views

Any “interesting” theorems for element-wise matrix product?

From the point of view of linear algebra, the "natural" multiplication operation for matrices is the usual matrix product, and there are lots of theorems involving this product---e.g. the result $det(...
0
votes
1answer
40 views

inequality between operator norm and infinity norm

For a matrix $A$, is there any relation between operator norm (https://en.wikipedia.org/wiki/Matrix_norm#Induced_norm) and infinity norm (defined as the maximum of the absolute value of all the ...
1
vote
0answers
45 views

closeness of matrices

I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. ...
1
vote
2answers
58 views

When a Markov chain converges to a steady state, what kind of convergence is it?

Let $A$ be a transition matrix, the steady state distribution $x$ satisfies the distribution $Ax = x$. One can prove that under certain circumstances, $$\lim_{n\rightarrow\infty}A^n q=x$$ where $q$ is ...
1
vote
1answer
28 views

Transition matrix has independent eigenvectors

If $ A_{n \times n} $ is a transition matrix $ - $ a positive matrix in which the sum of all entries in each row or columns equals $ 1 - $ is it true that $ A $ must always have $ n $ linearly ...
0
votes
1answer
28 views

For which value of $k$ does the matrix $A$ have one real eigenvalue of multiplicity $2$?

For which value of $k$ does the matrix $$A = \begin{bmatrix}-6&k\\-1&-2\end{bmatrix}$$ have one real eigenvalue of multiplicity $2$? So I understand that if the discriminant is $0$ such that ...
3
votes
1answer
71 views

Algebraic or Analytic Proof of a Polynomial Identity

Let $m$, $n$, and $r$ be integers with $0\leq r \leq \min\{m,n\}$. Define $$f_{m,n,r}(q):=\left(\prod_{j=1}^r\,\left(q^m-q^{j-1}\right)\right)\,\left(\sum_{\substack{{j_1,\ldots,j_r\in\mathbb{Z}_{\...
-1
votes
2answers
53 views

Determinant of a $3 \times 3$ matrix [closed]

I have a matrix of order $3 \times 3$. When I take its determinant it give 1700 from all the rows and columns except row 2. I don`t know whats going on $$\begin{bmatrix}1&20&0\\0&0&10\...
2
votes
2answers
59 views

Number of $k\times n$ matrices of rank $k$

How can I determine the number of $k\times n$ matrices with entries in $\mathbb F_p$ with rank $k$ (of course $k<n$) The formula if $k=n$ is $(p^n-1)(p^n-p)\dots(p^n-p^{n-1})$, now how can I ...
0
votes
1answer
19 views

Solve out specific variables from underdetermined linear system

I have an underdetermined linear system (more variables than equations) as: $ \mathbf{Ax = b} $ and I am interested in solving it a far as possible. The caveat here is that I have my priority list ...
0
votes
0answers
14 views

Rotation of basis vectors to new orthogonal axes

I have a set of orthogonal axes described as three vectors, which relate to the principal axes of an ellipsoids. My question is how do I calculate a rotation matrix, or Euler angles, which rotates an ...