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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3
votes
1answer
32 views

Real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$.

so I'm supposed to let $A$ be a square real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. And I'm supposed to show that $A=\lambda I$ for a constant ...
2
votes
0answers
18 views

Generators of $\Gamma_0(N)$

Let $\textbf{T}:=\bigl(\begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \bigr)$, $\textbf{S}:=\bigl(\begin{smallmatrix} 0&1/\sqrt{N}\\ -\sqrt{N}&0 \end{smallmatrix} \bigr)$ and $H$ the ...
1
vote
1answer
31 views

What is the derivative of this?

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?
0
votes
1answer
15 views

Inverse of matrix sum

I found on the Wikipedia page "Determinant" the following property: For any invertible $m \times m$ matrix $X$, $\det(X + AB) = \det(X) \det(I_m + BX^{-1}A)$. Is this true? If so, how is this ...
0
votes
0answers
8 views

Matrix function to express pair-wise distances of rows in $X, Y$

There are two real matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element $d_{ij}$ ...
0
votes
1answer
19 views

System of linear equation with one parameter

I'm trying to understand and solve a linear equation but i'm not sure how to go about it next, I was trying to reduce it with row operations but I can't seem to get all zero's under the first 'pivot' ...
1
vote
1answer
18 views

Proving full column rank of a matrix

Let $x$ be a $K\times 1$ vector of random variables satisfying that $E[xx']$ is nonsingular. For some given integers $M\geq 1$ and $L\leq K$, let $z_1,\ldots,z_M$ be $L\times 1$ column vectors ...
0
votes
0answers
7 views

Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
1
vote
1answer
28 views

Eigenvalues of a Product of two matrices A and B inside trace operator expressed in terms of any eigenvalue of A or B?

This question has been in asked in a few varieties here but not in this one. If we have a real, symmetric, positive-definite matrix $A$ and a real, symmetric, positive-definite matrix $B$ and we know ...
2
votes
1answer
12 views

Logic supporting column operations on matrices

In matrices, we justify row operations by drawing parallels with solving a system of equations i.e.: 1.Interchanging rows = Interchanging equations \ 2.Adding one multiple of a row to another = ...
1
vote
1answer
17 views

Does negative definiteness imply anything about ALL principal minors?

Unfortunately I haven't received any response for my previous question, so I'm trying to solve it in a different way. I know that iff matrix $H$ is negative definite, its leading principal minors ...
0
votes
0answers
15 views

How to understand the meaning of 'Oblivious' in Oblivious Subspace Embedding?

For the definitions of Oblivious Subspace Embedding and Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf.
4
votes
0answers
30 views

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
0
votes
1answer
47 views

Proving Submultiplicativity on a Matrix Norm

Let $||A||=(\sum_{i=1}^{n}\sum_{j=1}^{n}{a_{ij}^p})^{1/p}$, and let p=2. Then prove that $\|AB\|\le \|A\|\|B\|$ I have looked at numerous proofs for this, and I don't see one that satisfies me ...
0
votes
0answers
19 views

R in QR decomposition always upper triangular? [on hold]

Why is the matrix R in a QR decomposition always an upper triangular matrix?
0
votes
0answers
18 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
-2
votes
0answers
20 views

Linearly independent columns [on hold]

Let $M$ be a square upper triangular matrix with non-zero diagonal entries. Prove that the column vectors of $M$ form a linearly independent subset of $\mathbb{R}^n$. Please somebody answer and soon.
2
votes
2answers
36 views

Skew symmetric 4x4 matrix of full-rank

I have come across the fact that a 4x4 skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 & ...
-3
votes
0answers
10 views

HOW TO PLOT DAG (DIRECTED ACYCLIC GRAPH) in BNT toolbox for matlab.

I have used markov chain monte carlo (MCMC) in BNT toolbox for matlab, from which i have got one output "sampled_graphs " which is cell array. Now how to plot DAG (Directed acyclic graph ) from ...
0
votes
0answers
33 views

explaining the pattern

I have been given the following math puzzle: you are given a matrix that is filled by the following rule: every cell i,j is evaluated by taking the lowest non-negative number that is not present in ...
1
vote
1answer
18 views

Divide matrix using left division

In matlab, I defined a=[1;2;3] b=[4;5;6] both a and b are not square matrix. and execute a\b will return ...
0
votes
0answers
22 views

Is there a relation to determine condition (positive or negative definite) of C, if C = A+B and A, B are positive and negative definite?

I have a question: Matrix A and B are positive and negative definite, respectively. Is there a relation to determine whether C is positive or negative definite, if C = A+B?
0
votes
1answer
33 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...
-1
votes
0answers
13 views

Transforming roll along a path

I'm working on a script to transform a bunch of planes in 3d space. The source data is a bunch of points in 3d space that first form an L shape and then the movement path. Point 2 is the starting ...
0
votes
1answer
15 views

Rank of block matrix

Given a $q\times n$ matrix $E$ whose rank is $n$. Imagine that every element $[e_{ij}]$ of $E$ is replaced by a $m\times p$ matrix $F_{ij}$, whose rank is $p$. And in general, each $F_{ij}$ is ...
0
votes
0answers
11 views

Proof for Determinants using Laplace and induction.

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ I need to prove that det(A)=det(B). I thought induction might be one solution, but I don't know how to apply the Laplace ...
0
votes
0answers
22 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
0
votes
1answer
21 views

Given a square matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues?

Given a matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues? If instead the matrix had its nonzero entry component at ...
1
vote
0answers
23 views

How to determine if a set represents a line, plane or hyperplane?

How do you approach a question that gives you a set and asked to determine if it represents a line, plane or hyperplane? The Question: https://www.dropbox.com/s/0gscqur18kqg3ma/SpanningQuestion.PNG ...
2
votes
0answers
22 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
1
vote
1answer
25 views

Solve the following matrix equation $X'X=A$

I have square matrices $X$,$A$ and $X'X-A=0$. $A$ is given and is positive definite and I need to get matrix $X$. I know $X$ is not unique since $TX$ such that $T'T=I$ will satisfy. My problem is ...
-1
votes
1answer
32 views

Prove the equality of two determinants. [on hold]

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ Proof that $det(A)=det(B)$? Thanks in advance.
1
vote
1answer
22 views

Triangularization of a matrix.

so I need to find an invertible matrix $P$ such that $P^{-1}AP$ is upper-triangular, where $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$ So I found that the eigenvalue is $1$ which ...
0
votes
1answer
17 views

Linear algebra - projection matrix - inverse matrix

I am not sure how to prove this one: Let $A$ be a projection matrix so that $A^2=A$ and $A$ is not equal to zero. Find the inverse matrix of $I+cA$. Thanks.
1
vote
2answers
53 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
0
votes
1answer
23 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
0
votes
1answer
35 views

$\operatorname{rank}(A) = $max number of rows of submatrix $B$; Proof

I don't understand how to proof the following: The rank of a matrix $A \in M$ ($m \times n$, Field) equals the maximum number of rows of a square submatrix $B$ of $A$ with $\det (B) \neq 0$. The ...
0
votes
0answers
17 views

Operator norm of a matrix less than or equal to one

Do all matrices of operator norm $\leq 1$ have the sum of the absolute values of their rows $\leq 1$?
1
vote
3answers
174 views

To show two matrices are conjugate to each other

Given two matrices A and B $$ A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 ...
3
votes
2answers
26 views

Eigenvalues of a transpose multiplication

Say I have a matrix $\mathbf B \in \mathbb R^{m\times n}$. Is it correct to say that the eigenvalues of $\mathbf B^T\cdot\mathbf B$ are always positive?
1
vote
0answers
22 views

What does matrix decomposition really mean?

Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & ...
2
votes
1answer
41 views

Is $2^{xy}$ a positive definite kernel?

Is $2^{xy}$ a positive definite kernel on $\mathbb{N}$? i.e. for all $a_1, ..., a_n \in \mathbb{R}$, for all $x_1, ..., x_n \in \mathbb{N}$, $\sum_{i,j} a_i a_j 2^{x_ix_j}\geqslant 0$
1
vote
0answers
21 views

Gauss-Jordan elimination in the form of (A|I)

So Gauss-Jordan elimination can be performed through the form of $(A|I)$ where $I$ is the identity matrix. We carry out row elementary operations as usual until the matrix becomes the form $(I|B)$, ...
1
vote
1answer
21 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
3
votes
3answers
145 views

“Orthogonal” Rectangular Matrix

Is it possible to have a matrix $\mathbf B \in \mathbb R^{m\times n}$ such that it satisfies: $$\mathbf B^T\cdot\mathbf B = \mathbf I_n$$ Where $\mathbf I_n$ is the $n\times n$ identity matrix. Or ...
2
votes
1answer
28 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.
0
votes
2answers
35 views

Change of Basis for $2\times2$ matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
-1
votes
1answer
36 views

how to creat a matlab program? [on hold]

How can I creat a tridiagonal matrix in matlab if the elements of matrix is again a matrix instead of a scalar.n how to solve them using crouts methods.I am new to matlab.
1
vote
1answer
35 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
0
votes
1answer
16 views

Find the standard matrix of T given T is a linear transformation

$T:\mathbb{R}^2\to \mathbb{R}^2$ first performs a horizontal sheer that transforms $e_2$ into $e_2 + 2e_1$ (leaving $e_1$ unchanged) and then reflects points through the line $x_2 = -x_1.$ I am ...