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
16 views

Prove that a real matrix is a matroid

Problem $A$ real matrix, size $m\times n$ $M$ some structure, possible matroid $E(M)$ set of all columns of A (we're considering them vectors) $I(M)$ set of all linearly independent columns of A ...
0
votes
2answers
9 views

Prove a covariance matrix is positive semidefinite

Given a random vector c with zero mean, the covariance matrix $\Sigma = E[cc^T]$. The following steps were given to prove that it is positive semidefinite. $u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = ...
1
vote
0answers
41 views

Realation between Matrices ?!

The answer to the following question could be trivial. Let $A_1, A_2$ be symmetric $n\times n$ matrices, $x=(x_1,\ldots,x_n)\in \mathbb{R}^n$. If the maximum is taken for over ($\|x\|=1,\, and ...
0
votes
1answer
10 views

Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
0
votes
1answer
19 views

How polynomials are represented in matrix form for Univariate Polynomial.

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
0
votes
1answer
53 views

Wolfram|Alpha refuses to find the inverse of a large 6x6 matrix.

Just to be clear, this isn't a question on how to find the inverse of a matrix, I just don't want to find the inverse by hand (I hope you see why). $$ \begin{pmatrix} 1 & 2006 ...
-1
votes
0answers
16 views

show i-th projection is a linear transformation

For $i ∈ {1,2,...,m}$, define $\pi : F_m → F$ by $\pi(x_1,x_2,...,x_m) = x_i$ (the $i$-th projection). (a) Show that it is a linear transformation. (b) If $T : F_m → F$ is a linear transformation ...
0
votes
0answers
5 views

Why does “up to scale” make homograph matrix lose one freedom?

Can anyone explain "if H is up to scale, then dof(H)=8" in the following discussion? degree of freedom of Homography matrix Thank you!!!
1
vote
0answers
20 views

Derivative of an Euclidean-Vector norm.

Consider: x a $N \times 1$ vector , with elements $x_i$ b a $N \times 1$ vector , with elements $b_i$ A a $M \times N$ matrix , with elements $a_{ij}$ ( Symmetric matrix - Block Circulant ) As we ...
1
vote
1answer
16 views

Sylow's theorem for group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$

Let $G=SL(2,\mathbb{F_3})$ - group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$. (a) Find the order of $G$. I think it is $24$ but not sure how to verify it. (b) ...
0
votes
0answers
6 views

Factor the matrix (scalar $\times A$) into permutations of $A$

Here's an example of $A . B = scalar \times C$, done with magic squares. The last square does not have a consecutive range of digits. Drop the magic square requirement. In $2\times2$ matrices we ...
0
votes
0answers
13 views

Real Symmetric Positive Definite Matrix [on hold]

Assume $H = A + Bi$ is a positive $m \times m$ Hermitian matrix, where $A, B \in R^{m \times m}$. How can we show that $C = \begin{bmatrix} A & -B \\ B & A \end{bmatrix}$ is a real ...
1
vote
2answers
41 views

Ring of matrices has no nontrivial ideals [duplicate]

It is a theorem that a commutative ring is a field if and only if it has no nontrivial ideals. Clearly this does not hold in the noncommutative case. I am trying to show for instance that the ring of ...
0
votes
0answers
16 views

Distance Geometry Problem (DGP) Programming Language Recommendation

We have been studying DGPs in clinic recently and I was hoping I might be able to get recommendations for computing languages in the processing of large network solutions. Specific computations ...
0
votes
1answer
35 views

Linear Algebra. Is this question realte to combination and factorials?

I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks.
0
votes
1answer
16 views

Determinant of block matrix with off-diagonal blocks conjugate of each other.

I am working on finding the determinant of the following block matrix $$ \begin{pmatrix} C & D \\ D^* & C \\ \end{pmatrix}, $$ where $C$ and $D$ are $4 \times 4$ matrices with complex entries ...
0
votes
0answers
26 views

How to solve system of equilibrium probability state equations

I have started studying markov chains where i have these statistical equilibrium probability state equations.These equations are solved for a particular case $s_1=4,a_1=5,s_2=2, a_2=1$ and a 15*15 ...
-1
votes
0answers
28 views

Let $\operatorname{rank} A=1$ then there are, $x,y\in \mathbb{C}^n$ such that $A=xy^T$ [on hold]

Let $A\in M_n$ and $\operatorname{rank} A=1$. Are there $x,y\in \mathbb{C}^n$ such that $A=xy^T$?
2
votes
2answers
35 views

Positive definite matrix meaning in human language? “Definite”?

I have to consult Wikipedia every time to re-learn what is positive (semi) definite. So that I am sure I will be able to decompose it further in some ways. Wikipedia Now I am trying to truly ...
5
votes
0answers
23 views

Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear ...
-2
votes
1answer
41 views

How would I find the equation of a graph using matrices? [on hold]

Assuming the graph below is a 5th degree polynomial, how would I go about finding its equation using matrices? Edit: So if I had the data points: $A(2006, 531.37), B(2013, 484.13), C(2028, ...
1
vote
0answers
26 views

Eigenvalues of the subtraction of a gram matrix and a psd rank $1$ matrix.

If $V$ is $p \times r$ matrix ($r<p$) with rank $r$, then $G=V^TV$ is a $r \times r$ gram matrix with rank $r$. Let $x$ be a $r \times 1$ vector, and $xx^T$ is a psd rank $1$ matrix. It's already ...
1
vote
1answer
39 views

Attempt to solve a matrix (counterbalancing) problem computationally gives “spooky” result: why?

This question is posted on the mathematics section of stackexchange because my uneducated guess is that the answer involves some basic mathematical principles, possibly in the domain of linear ...
0
votes
0answers
21 views

If two linear systems are equivalent, they have the same size augmented matrix. [on hold]

If two linear systems are equivalent, they have the same size augmented matrix? It is false but do any one know why for this?
1
vote
1answer
22 views

Multiplying inverse matrices easily.

Okay, this is more of a confirmation question than anything: I have been given two matrices $A^{-1}$ and $B^{-1}$. Then the inverses of these are: $A$ and $B$. I need to calculate $(AB^{T})^{-1}$. ...
0
votes
1answer
19 views

Alternative methods to solve DLP for $GL_{3}(\mathbb{F}_2)$

Is there (or rather what is) a more elegant/efficient way to solve the DLP for $g^x=h$ in $GL_3(\mathbb{F}_2)$ where $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 ...
1
vote
1answer
13 views

Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$. I know that if two matrices are row equivalent, we can ...
0
votes
1answer
19 views

Can square matrices be represented as the union of vectors and some other set?

I believe all invertible matrices can be representable as $A = |A| \, \mathrm{adj}\left(A\right)^{-1}$ (a rotation part times a scaling part.) All invertible matrices can then be mapped to vectors ...
1
vote
2answers
31 views

Matrices and diagonalization.

I could verify that $P$ statement is false by just calculating the determinant but couldn't answer $Q$ statement. Any clue about $Q$??
0
votes
2answers
32 views

It is true that $rank(xy^T)=1$? [on hold]

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
-2
votes
0answers
24 views

Gaussian Elimination vs matrix inversion [on hold]

Why Gaussian Elimination is better than matrix inversion in therms of FLOPS? Also how LU decomposition improves the shifted inverse power method?
2
votes
1answer
14 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
0
votes
0answers
22 views

Eigen vectors and determinant of a block matrix

I have two questions regarding matrix $A$. The matrix $A$ can be partitioned into four tridiagonal matrices $A_1$, $A_2$, $A_3$ and $A_4$. $$A=\begin{pmatrix} A_1&A_2\\A_3&A_4 ...
-3
votes
0answers
16 views

Unitarily equivalent Triangular matrices [on hold]

Could anyone help me to prove the following problem? Suppose $(x_1,x_2,\dots,x_n)$ is a permutation of $(y_1,y_2,\dots,y_n)$, then any triangular matrix with diagonal entries $(x_1,x_2,\dots,x_n)$ is ...
1
vote
1answer
49 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
0
votes
2answers
33 views

Null space is an Invariant subspace

Let $\lambda$ be an eigenvalue of a square matrix $A$. Show the null space of $(A-\lambda I)^j$ is an A-Invariant subspace of $\mathbb{C}^n$ for all positive integers $j$. Proof without requiring ...
2
votes
0answers
28 views

SVD of a Matrix Product

Suppose we have a matrix $A$ with dimensions $m$ by $n$ and a column-wise permutation matrix $R$ (re-orders columns) with dimensions $n$ by $n$. Then we have a matrix $X$ which is constructed as $X ...
1
vote
1answer
44 views

inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \geq ...
0
votes
1answer
30 views

What is meant by In-Place Matrix Inversion?

I come across the term "In Place Matrix Inversion" a lot in numerical libraries like NumPy and ND4J. What does it mean ? How is it different from the normal matrix inversion ? What are the advantages ...
0
votes
1answer
31 views

Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
0
votes
1answer
43 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
5
votes
3answers
133 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
0
votes
1answer
37 views

Is matrix $A^i A^j = A^j A^i$

I want to know if $$A^i A^j = A^j A^i$$ holds or not. It seems like an obvious, but I am wondering if there is a more formal proof
0
votes
2answers
54 views

How can I show that for matrix $A$ , $A^t A $ is not equal to $ A A^t $ in general?

How can I show that for matrix $A$ , $A^t A \neq A A^t $ $A^t$ means the transpose of $A$. That is the entire question and I have no idea how to begin... please help!
0
votes
0answers
28 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
0
votes
0answers
64 views

maximum frequencies of numbers in a matrix

I have a matrix A of size n*n.Consider a new matric M : M[i][j]=max of frequencies of numbers occuring in ith row and jth column(A[i][j]) counted once. I have a ...
2
votes
2answers
38 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
0
votes
2answers
34 views

How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
1
vote
1answer
22 views

choosing a square matrix to have a product with one 1 und other 0's

Let $A$ be a $m\times n$ real matrix with maximal rank. Let $i\in\{1,\dots,m\}$, $j\in\{1,\dots,n\}$. I'm curious if it is possible (for any choice of $i,j$) to find a square matrix $B$ such that ...
0
votes
1answer
29 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...