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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1
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2answers
34 views

Matrix of orthogonal projection endomorphism

I can't understand this fact about orthogonal projections. Considering the projection endomorphism $p_W:V\rightarrow V$ which is the projection endomorphism on a vector subspace $W\subset V$. If an ...
1
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0answers
23 views

How do I show that the given matrix can be decomposed?

Suppose $P\subseteq\mathbb R^n$ is a polyhedron given by $m$ constraints $\langle a_i,x\rangle\leq b_i, i=1,2,...,m$ and let $w_1,w_2,...w_n$ be its vertices. Define $S=(s_{ij})$ by ...
0
votes
1answer
45 views

Real-Valued Symmetric Square Matrices and Min-Max

A real-valued symmetric square matrix is called positive definite if $(x,Ax)>0$ for all $x\neq0,$ where $(.,.)$ represents the scalar product. For a positive definite matrix determine ...
2
votes
1answer
21 views

rank of block triangular matrix and its relation to the rank of its diagonal blocks

Prove that $$rank\begin{pmatrix}A & X \\ 0 & B \\ \end{pmatrix}\ge rank(A) + rank(B)$$ where $$A,B\in \mathbb C^{m \times m}$$. I know the intuition behind it (i.e. maximal independent ...
40
votes
4answers
4k views

Must eigenvalues be numbers?

This is more a conceptual question than any other kind. As far as I know, one can define matrices over arbitrary fields, and so do linear algebra in different settings than in the typical ...
0
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3answers
41 views

Matrix of a linear transformation that associates a matrix to its transpose

Find the eigenvalues and eigenvectors of the transformation $T: \mathbb{R^{2x2}} \rightarrow \mathbb{R^{2x2}}$, which associates, to each $A \in \mathbb{R^{2x2}}$, its transpose, that is, ...
0
votes
0answers
12 views

Constructing a matrix for a 2D lattice

For part of a project I'm doing I need to construct matrices for different cases and solve them to find energy dispersion relations. I've done it for a 1D chain of atoms, but I'm having trouble making ...
2
votes
1answer
32 views

Finding the gradient in least squares

In Linear squares optimization I have A=\begin{pmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^k \\ 1 & t_2 & t_2^2 & \cdots & t_2^k \\ \vdots & \vdots& ...
1
vote
1answer
45 views

Can we always for an invertible matrix $M$ find real number $\alpha \neq 0$ such that $M+\alpha$ is invertible?

I do not know enough about matrices, maybe only enough to be able to create question like this one, but I would like to see an answer. Let $a_{ij}$ be some element of invertible $n\times n$ matrix ...
1
vote
1answer
47 views

If $A$ is non negative and has a positive eigenvector $ \Rightarrow $ A is diagonally similar to a non negative matrix

If $A\in M_n$ is non negative(all $a_{ij}\ge 0$), and has a positive eigenvector(all $x_i>0$), why is $ A$ diagonally similar to a nonnegative matrix, all of whose row sums are equal?
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votes
3answers
54 views

Symmetric $3 \times 3$ Matrix [closed]

A symmetric $3\times 3$ matrix $A$ with real number entries satisfies $A^3 = I$. What can you say about the eigenvalues of $A$ and their multiplicity? What can you deduce from that about $A$ itself? ...
1
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1answer
80 views

Proof of convexity in a quadratic function

I have the following quadratic objective function (almost variance function); where $f_n, i=1,...,n$ are $n$ function and $\overline f$ is the mean of $f_n$ for all $ i=1,...,n$ $$\min ...
0
votes
1answer
43 views

Finding a matrix knowing determinant

Suppose $\det A = 1$ and you know all the cofactors. How can you find $A$? So my attempt was taking the formula $A^{-1} = C^T/\det A$. If $\det A = 1$ then $A^{-1} = C^T$. And now we also know ...
1
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0answers
33 views

Can a matrix be orthogonal with respect to a positive definite $\phi$?

I've a doubt on othogonal matrices. I know that an orthogonal matrix is a matrix $O$ such that $O^{T}O=O^{-1}O=I$ and also that $O$ has on the columns and on the rows the coordinates of the vectors ...
0
votes
1answer
64 views

Find a subspace of dimension 2 from 4 vectors

first of all sorry if the title is not really specific but I thought about it and I didn't find something short which fits better my case. My problem is the following. I have a matrix ...
1
vote
3answers
47 views

Find the value of the following $n \times n$ determinantes

Find the value of the following $n \times n$ determinantes $$\begin{vmatrix} a_1+x & x & x & \ldots & x \\ x & a_2+x & x & \ldots & x \\ x & x & a_3+x & ...
0
votes
0answers
104 views

Tensor with multi-rank $(1,1,1)$

I want to Show that a $2 \times 2 \times 2$ tensor cannot have multi-rank $(1, 1, 2)$ and it has rank $1$ if and only if it has multi-rank $(1, 1, 1)$?
0
votes
1answer
26 views

$A\in M_n$ is nonzero and nonnegative , $A$ has a positive eigenvector $\mathop \Rightarrow \limits^? $ $\rho (A) > 0$

Suppose that $A\in M_n$ is nonzero and nonnegative (i.e: all $a_{ij}\ge 0$) . If $A$ has a positive eigenvector (i.e :all $x_i>0$), Why does $\rho (A) > 0$? (Note: $\rho (A) = \max \left\{ ...
1
vote
1answer
53 views

Integral of exponential with linear term

$$x \in \mathbb{R^n},$$ M is a positive symmetrical nonsingular nxn Matrix and j is an arbitrary vector in $$\mathbb{R}^n.$$ The following has to be calculated: $$Z(j) = \int_\mathbb{R^n} ...
4
votes
0answers
34 views

Get bounding rectangle segments of a rotated rectangle (matrix?)

My problem: I have: $x$, $y$ & $\alpha$ - the aspect ratio $o$:$p$ (red rectangle) I want to have $n$ & $m$ in dependancy of $x, y, \alpha, o, p$ I tried to figure it out with ...
1
vote
2answers
43 views

Meaning of formula

The exercise was: Draw the triangle with vertices $A = (2;2)$, $B = (-1;3)$, and $C = (0;0)$. By regarding it as half of a parallelogram, explain why its area equals $$ \mathrm{area}(ABC) = ...
0
votes
1answer
125 views

Derivative of Schatten p-norm

The nuclear norm is a special case of the Schatten p-norm. I know how to find out the derivative of nuclear norm, but What is the derivative expression of the matrix Schatten p-norm?
2
votes
4answers
79 views

Show that $A$ is an invertible matrix if $\left(A+I\right)^3=0$ and find $A^{-1}$

If $A\in M_{n\times n}\left(\mathbb{R}\right)$ is such that $ (A+I)^3=0$, show that $A$ is an invertible matrix and find the inverse of $A$. My idea was: ...
0
votes
1answer
68 views

Determine a determinant is divisible by 23 or not

Consider the fact that $25875, 46552, 41354, 48691, 95818$ are all divisible by $23$. Use this fact to determine if \begin{vmatrix} 2 &5 &8 &7 &5 \\ 4 &6 &5 &5 &2 \\ 4 ...
1
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0answers
19 views

Time Complexity Analysis of Matrix Operation

What is the Time Complexity of the Following operation X'(X transponse) ans XX' Could you explain me the cost of these operation in time complexity domain ? What ...
0
votes
1answer
50 views

Write matrix as a linear combination of polynomials

I can't figure out how to solve this:
3
votes
1answer
102 views

Finding symmetric commuting matrices $A,B,C,D \in M_n(1,-1)$ such that $ A^2+B^2+C^2+D^2=4nI_n $

I am trying to construct a Hadamard matrix of order 28 using Williamson's construction. But I am unable able to construct the necessary symmetric and commuting matrices. Definition: $H_n \in ...
2
votes
1answer
35 views

Reverse engineer a matrix multiplication

Here's a puzzle. I'm looking for ideas on how to research solutions. Given: Secret $n\times 1$ vector $x$ Public $m\times n$ matrix $B$ with $m \ll n$ (assume $B$ has rank $m$) Public product $b = ...
1
vote
1answer
49 views

Is it possible to estimate the sign of real part of eigenvalues of a 10 by 10 matrix only by observing all the entries?

I have a symbolic 10 by 10 matrix. It is not difficult to get the eigenvalue expressions by using Matlab. But the expressions of some eigenvalues are too long to be analyzed. I was wondering if there ...
1
vote
1answer
32 views

Normal and positive implies symmetric?

again a matrix question: Suppose I have a matrix that is normal (i.e. $A^TA=AA^T$, thus normal over the real numbers) such that all entries are non negative. Does this imply that the matrix is ...
1
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0answers
36 views

What does a eigenvalue equal to 0 represent?

The only relation I can think of is that the determinant of corresponding matrix A is 0 so A is not invertible. Could anyone suggest any other properties related to eigenspace, orthogonal ...
2
votes
0answers
41 views

Write linear transformation matrix in terms of basis

I'm having some trouble with a practice problem for my linear algebra class. The problem is as follows: Consider the matrix $A = \begin{bmatrix} 1 & 1 & 2 & 2\\2 & 2 & 5 & ...
0
votes
1answer
14 views

how to solve the system of a.x=b with Matrix?

Hi i'm studying linear algebra, first of all sorry for screen shots if it's not a good idea but I don't know how to write a matrix in StackExchange. So question gave me A Matrix and I have to find ...
3
votes
2answers
30 views

Powering a sum of two easily powered matrices

I am currently studying matrices and in order to understand them better I want to know why I can't do certain things in my calculations. This question is just about that. The task is to calculate ...
0
votes
0answers
35 views

Finding an matrix for an operator.

I was attempting to find a matrix for the function $x\frac{d}{dx}$ in the span of the set $\{1,x,x^2\}$ for the the standard dot product. Could someone guide me in how to do this?
1
vote
1answer
23 views

Finding a diagonalizing matrix associated with Jordan

Find the Jordan normal form $J$ of the upper triangular matrix $A = \begin{pmatrix}2 & 0 & 1 & 2 \\ 0 & 2 & 2 & 1 \\ 0 & 0 & 2 & 1\\ 0 &0 & 0& 3 ...
3
votes
2answers
26 views

A question on a certain block decomposition of semi-definite matrices.

Let $m,n\in\mathbb{N}$, with $m,n>1$. Suppose $K\in \mathbb{M}_{mn\times mn}(\mathbb{C})$ is positive semidefinite. We can always write $$K=\sum_{i,j=1}^m E_{i,j}\otimes K_{i,j},$$ for some ...
2
votes
0answers
54 views

Eigenvalues of triangular block matrix

I need to find an expression for determining the eigenvalues of this matrix block $$ Acc=\begin{bmatrix} A & I \\ L & 0 \\ \end{bmatrix} $$ where $A$ is a ...
2
votes
2answers
74 views

Finding the determinant of anti-diagonal matrix

How would one find the determinant of an anti-diagonal matrix ($n \times n$), without using eigenvalues and/or traces (those I haven't learned yet): My initial idea was to swap the first and n-th ...
0
votes
0answers
24 views

A matrix inequality involving square roots

Let $X$, $Y$ be positive definite matrices satisfying $$ \lambda^{-1}X\leq Y \leq \lambda X,\quad \lambda\geq 1, $$ do there exist inequalities that relate $Y^{1/2}AY^{1/2}$ to $X^{1/2}AX^{1/2}$, ...
1
vote
1answer
28 views

Decomposing a matrix as a product of non-square matrices

Suppose $X\in \mathbb{M}_{n}(\mathbb{C})$ is positive semi-definite. Let $m\in \mathbb{N}$ be some integer greater than one. Under what conditions can we find two matrices $A,B\in \mathbb{M}_{n\times ...
1
vote
0answers
51 views

Semigroup of matrices and expander Cayley graphs

I am interested in proving or disproving that certain Cayley graphs are expander. Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & ...
0
votes
0answers
32 views

Why does the equality hold?

For $A \in \mathbb{R}^{n \times n}$ with $A=A^{T}$ we set: $$\lambda:= \max \{ \langle x, A x \rangle: ||x||_2=1\} \\ \mu:= \min \{ \langle x,A x \rangle: ||x||_2=1\}$$ Then for $x \in ...
2
votes
1answer
17 views

Eigenvalue alteration counter-proof

If $A$ is a square $n\times n$ matrix, with $\lambda_1,\ldots,\lambda_n$ being the eigenvalues of $A$, $v_1$ being the eigenvector associated with eigenvalue $\lambda_1$, and $d$ the column vector of ...
2
votes
1answer
97 views

Covariance Matrix of mean-centered Random Variables

I read here that for n x d data matrix X, where X is mean-centered, V = $X^{T}*X$ is its covariance matrix. Why is that? As I understand the element $V_{i,j}$ of the covariance matrix is defined by ...
0
votes
0answers
26 views

Dot product and matrix transpose

In Least squares optimization, I have $A=\begin{pmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^k \\ 1 & t_2 & t_2^2 & \cdots & t_2^k \\ \vdots & ...
1
vote
1answer
35 views

Primitive matrix

A matrix $M\in\mathcal{M}_n(\mathbb{R})$ is said to be primitive if it exists an integer $k$ such as $M^k$ has all its coefficients strictly positive. My question is : If $M$ is primitive and if ...
1
vote
0answers
32 views

Conditions for the satisfiability of a matrix inequality

Let $X\geq 0$ be a positive definite matrix and $A$ a square matrix, do there exist conditions either on $A$ or $X$ (or both) such that $$ X-AXA^\top\geq 0,\quad \quad (*) $$ holds, where $\top$ ...
0
votes
0answers
32 views

Smith Normal Form and free basis

If I reduce a matrix to Smith Normal Form, how do I find the free basis of the smith normal form of the matrix?
1
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1answer
55 views

How do you solve a system of linear equations in modular arithmetic.

I'm finding a hard time trying to proceed with this cryptography problem: If i'm given such a system of linear equations: $3x+5y+7z\equiv3 (mod\ 16)$ $x+4y+13z\equiv5 (mod\ 16)$ $2x+7y+3z\equiv4 ...