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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0answers
9 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 = ...
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0answers
11 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
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0answers
12 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.
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0answers
19 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
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1answer
32 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 ...
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0answers
17 views

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

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

Distance between all rows in 2 matrices expressed as a matrix equation

I have two matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. All real numbers. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element ...
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0answers
16 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 ...
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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
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2answers
28 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 & ...
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0answers
9 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 ...
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0answers
31 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 ...
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1answer
16 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
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0answers
21 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
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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 ...
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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
10 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 ...
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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
21 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
20 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 ...
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0answers
22 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 ...
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votes
0answers
20 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
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1answer
24 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 ...
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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.
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1answer
20 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
16 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.
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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 ...
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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 $\leq 1$?
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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
25 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?
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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
40 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 $x_1, ..., x_n \in \mathbb{N}$, $\sum_{i,j} a_i a_j 2^{x_ix_j}\geqslant 0$
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0answers
29 views

does a closed form solution exist for this equation?

I have a cost function $J$, which depends on a projection matrix $W$, which is unknown. When I get the partial derivative $\frac{\partial J}{\partial W}$ the equation is: $\frac{\partial J}{\partial ...
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0answers
20 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
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1answer
19 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
144 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 $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
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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
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1answer
33 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 ...
1
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1answer
22 views

Quickest way to calculate $(I-M)^{-1}(I-M^{n+1})$

What is the quickest way, as computationally efficient, to calculate $(I-M)^{-1}(I-M^{n+1})$, where $M$ is a given $n \times n$ singular matrix of $0$'s and $1$'s? My experiments show that the matrix ...
0
votes
2answers
45 views

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, Prove that $|(A+A^T)(B+B^T)|=0$

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, prove that $\det[(A+A^T)(B+B^T)]=0\ \ $ with $ \ k\in \mathbb{N}$ I don't have ideas for this problem. Thanks !
1
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2answers
17 views

Implications on structure of $B$ when $rank(A-B) = s$ for a fixed $A$

Consider the case where $A \in \mathbb{R}^{n \times K}$ where $n > K$ and $\text{rank}(A) = K$. Suppose we know $$ \text{rank}(A - B) = s$$ for some $s < K$. What does this say about the ...
0
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0answers
18 views

How to prove that a matrix with specific property is invertible?

If we have a square matrix $$ M = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & ...
2
votes
1answer
53 views

Eigenspaces and jordan normal form

I have a question here regarding the jordan normal form of two matrices where the eigenspace is one is contained in the other. Let $A,B$ be two $nxn$ matrices s.t $AB=BA$. I firstly proved that the ...
0
votes
0answers
30 views

Sparse Matrices and Tridiagonalization.

Assume that we are given a sparse matrix,let it be 90*90(1000*1000), would you say that a vector with lots of zeros(let it be 90*1(1*1000),and 65(500) zeros are there),is a smart option to initialize ...
2
votes
0answers
32 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...