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), ...

learn more… | top users | synonyms (2)

0
votes
0answers
14 views

What is the mathematically operation referred to where you put a matrix of lower dimension into a matrix of higher dimension

Suppose I have a vector $A = [a_1, a_2, a_3]$; now let's construct another vector $B =[A, 1]$ What is this process of putting a smaller matrix into a larger one generally referred to? Embedding?
1
vote
0answers
25 views

Determine all t ∈ ℝ for which At is diagonal

so I have this matrix: At= 2+t 4 2+t 2+t t-2 0 -6+t -2-t -t+2 -4 -t+2 -2-t 0 0 0 2t I must determine all t ∈ ℝ for which At is ...
0
votes
1answer
31 views

Eigenvector Problem

Given a matrix $X$, let $eigvec(X)$ be its eigenvector associated with the largest eigenvalue. Is there a relationship among $eigvec(X+X^T)$, $eigvec(X)$ and $eigvec(X^T)$? In other words, can I use ...
1
vote
1answer
27 views

Finding the corresponding Perron eigenvalue

Find the Perron root and the corresponding Perron eigenvector of A. $\begin{bmatrix} 0 &1 &1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ I figured out the Perron root which happens to ...
3
votes
0answers
30 views

Transpose of the adjacency matrix

As homework I had to do an adjacency matrix for the following graph: My solution was the following: $$ \begin{bmatrix} 0&0&1&0&0 \\ 1&0&0&1&0 \\ ...
0
votes
0answers
14 views

$A \times B^{-1}$ has irreducible characteristic polynomial when $A,B$ are random integer matrices — simple proof?

Let $A,B$ be $d\times d$ integer matrices with each entry drawn uniformly from $[0,2^n)$, and define the rational matrix $C = A \times B^{-1}$. Is there a simple way to prove that $C$'s characteristic ...
0
votes
0answers
23 views

Linear maps and subspaces

The set-up for my question is this, let $k \le n$, let $E \subseteq \mathbf{R}^n$ be a $k$-dimensional subspace. Let $I \subseteq \{1,\ldots, n\}$ such that $|I| = k$, then we can define coordinate ...
0
votes
1answer
15 views

Inversing badly-conditioned square matrix: methodology

I have a badly-conditioned square matrix. I need to inverse it. For inversing, currently I'm doing the following steps: I take the badly-conditioned matrix with size of $n$ by $n$ By reduced row ...
0
votes
1answer
33 views

matrix vs vector span {} linear algebra

I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean? =Span= Can it only be made of ...
0
votes
1answer
18 views

Prove that the LDU factoriztion is unique [on hold]

How would one prove that the LDU factorization of a matrix is unique?
2
votes
1answer
21 views

If $R$ is a commutative simple ring with identity , then is any matrix ring $M_n(R)$ also simple? [duplicate]

If $R$ is a commutative simple ring with identity , then is any matrix ring $M_n(R)$ over $R$ of matrices of size $n$ also simple ?
2
votes
1answer
15 views

Generators of $Sp(2n)$

Let $\sigma =\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$. Define $J_{2n} = \underbrace{\sigma \oplus \cdots \oplus \sigma}_{\text{$n$ copy}}$. We define a $2n \times 2n$ real matrix matrix ...
1
vote
0answers
22 views

Is there a formula for the sum of absolute eigenvalues in terms of matrix elements?

Given a symmetric matrix $X \in \mathbb{R}^{n \times n}$. We know the following: trace$(X) = \sum_{i=1}^n x_{ii} = \sum_{i=1}^n \lambda_i$ where $x_{ii}$ is the $i$th element on the diagonal of $X$, ...
0
votes
2answers
53 views

Find a positive definite matrix B such that $B^2=A$. [on hold]

Find a positive definite matrix B such that $B^2=A$, where $$A=\begin{pmatrix} 2&-1\\ -1&2 \end{pmatrix}$$
0
votes
1answer
17 views

How to prove that $B$ is positive definite when $\|A-B\|\leq\lambda_\min(A)$ for some positive definite $A$?

Denote by $\mathbb R^{n \times n}$ the vector space of $n \times n$ matrices with real entries. For $A \in \mathbb R^{n \times n}$, the notation $A\succ 0$ means that $A$ is symmetric and positive ...
-1
votes
0answers
7 views

Need to find the coefficient matrix [on hold]

I have $f([1 0 0]^t) = [ 3 -2 -1]^t f([0 1 1]^t) = [ 1 1 1]^t f([1 1 1]^t) = [ 1 2 3]^t$ if $f(x) = AX$ for any vector where $x$ belongs to $\mathbb R^3$. Find Coefficient matrix A $^t$ stands for ...
4
votes
2answers
261 views

Why does Gaussian elimination not preserve similarity of a matrix?

I am trying to understand reduction of an unsymmetric real square matrix to Hessenberg form from Numerical Recipes Vol. 3. In it, the author states that one does not use Gaussian elimination for ...
0
votes
0answers
15 views

If through row reduction of a square matrix you can produce a row of zeros does that automatically make that matrix singular?

If through simple row operations such as adding one row to another I can produce a row of entirely zero does this mean that the matrix will be singular? For example: $$\left\{ \begin{matrix} 1 & ...
2
votes
1answer
23 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
3
votes
1answer
42 views

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

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
1answer
26 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
38 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
2answers
21 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
13 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
20 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
19 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
1answer
11 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
30 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 = ...
2
votes
1answer
20 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
32 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
52 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
20 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.
3
votes
2answers
54 views

Skew symmetric $4\times 4$ matrix of full-rank

I have come across the fact that a $4\times 4$ 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
12 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
34 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
19 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
18 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
22 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
23 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
26 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 ...