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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2
votes
1answer
50 views

Is this a theorem (is it correct?)?

My instruction notes have specified a theorem of matrix transpose that be there two compatible matrices $A$ and $B$ in respect of their sums and products, then: $(AB)^T =A^TB^T$ So I set on to ...
0
votes
0answers
164 views

Looking for a simple algorithm to scale/resize a matrix, or an image.

I am looking for a simple algorithm to scale a matrix of any size. Given matrix A of dimensions [w1,h1]. Given a scaling factor (or resizing factor) SF, which is a real number (not necessarily ...
0
votes
1answer
76 views

Nonsingular Z-Matrix <==> Nonsingular M-Matrix?

Here I'm considering only M-matrices that are also Z-matrices, so all the off diagonal elements are negative in all matrices I consider. If I have a Z-Matrix with real, positive eigenvalues, is it ...
0
votes
0answers
45 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
7
votes
3answers
343 views

Maximum number of linearly independent anti commuting matrices

Given n x n matrices, my book says the maximum size of a set of linearly independent mutually anti commuting matrices is $n^2-1$. I don't understand why this is true. Would appreciate any tips to ...
4
votes
1answer
92 views

Left and right eigenvectors perpendicular to each other

I just read in a textbook on numerical methods that you can always have that the right eigenvectors of a matrix can be taken as orthonormal to the left eigenvectors for a diagonalisable matrix. This ...
0
votes
2answers
43 views

Show that the map $\varphi:S \rightarrow S$ defined by $\varphi((m,n)) = (am + cn, bm + dn)$ is a bijection.

Let $\begin{bmatrix} a & b \\ c & d\end{bmatrix} \in \mathrm{SL}_2(\mathbb{Z})$. Let $S$ be the set $(\mathbb{Z} \times \mathbb{Z}) \setminus \{(0,0)\}$. Show that the map $\varphi: S ...
3
votes
2answers
80 views

$X$, find $A$ such that $A^m=X$

I encountered a problem as folows: Show a $3\times 3$ real matrx $A$, such that $$A^4=\left(\begin{array}{ccc}3&0&0\\0&3&1\\0&0&0\end{array}\right)$$ well, this problem ...
0
votes
1answer
90 views

finding an inner product so that matrix is self-adjoint

Given the endomorphisms $$A= \left( \begin{matrix} 1 & 2i & 3i\\ 0 & 2 & i\\ 0 & 0 & 3 \end{matrix} \right) \in Mat(3 \times 3, \mathbb C), $$ and $$B= \left( ...
1
vote
0answers
51 views

iterates of generalized matrix system

This may be somewhat of an underspecified question but I'll nonetheless give it a try. In the context of applied work, I've recently come across systems of the form $$ ...
0
votes
1answer
51 views

How to find a upper triangular matrix similar to A.

Given a matrix A how do you find a matrix Q such that Q inverse A Q is upper traingular, in complex. I know that it is possible and the proof uses a induction argument but i don't see how to find such ...
2
votes
1answer
52 views

Find a Hermitian Matrix

We are given two column matrices A and B. Can we find a Hermitian matrix $H$ such that $ A_{4 \times 1} = H_{4 \times 4} B_{4\times 1} $ ? We tried to solve it by multiplying a $1\times4$ row matrix ...
2
votes
1answer
111 views

Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
0
votes
1answer
29 views

Matrix double modulo multiplication to get identity

I have to multiply to matrices A and B which can consist of numbers 0,2,3,4,5,6 to get an identity matrix, however multiplication happens with moduli after every ...
1
vote
0answers
97 views

About Jordan-Chevalley decomposition

I have this problem: Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$. I saw that this ...
1
vote
0answers
69 views

can I normalized the tensor rank in this way?

Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)? For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has ...
7
votes
2answers
1k views

Are matrices rank 2 tensors?

I know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular or ...
1
vote
0answers
25 views

Which is the starting and ending basis? - Matrix of a linear transformation [Lay P294 Q 5.4.28]

Denote some arbitrary linear transformation as $L.$ When a question asks "to find a matrix of $L$ with respect to S and T", does this denote $[L]_{T \leftarrow S}$ or $[L]_{S \leftarrow T}$ ? How can ...
1
vote
2answers
25 views

Evaluating possible eigenvalues of a given equation

Let $A$ be a $3\times3$ matrix such that $A^{2} = 4A - 4I$ Evalute the possible eigenvalues of $A$. I have tried to multiply the equation by eigenvector $x$ and use the property of $Ax = \lambda x$ ...
2
votes
0answers
31 views

Bound on $\|RR^T\|$ for random matrix $R$

Let $R$ be a $n \times k$ matrix ($n<k$) where each entry of $R$ is drawn independently from $N(0,1)$. I wanted to know, what is a bound, possibly probabilistic, on $\| RR^T\|$? Any pointers or ...
2
votes
1answer
31 views

Symmetric matrix as a sum

After thinking about this question, I am wondering: is it true that any symmetric $n\times n$ matrix $A$ can be written as: $$\mathbf{A}=\sum_{i=1}^n \lambda_i \mathbf{B}_i$$ where ...
1
vote
6answers
72 views

Proving that all eigenvalues are $0$ bar one.

A symmetric $n\times n$ matrix is given by $\mathbf{A}=\lambda \mathbf{ee}^{\text{T}}$ where $\mathbf{e}$ is a unit vector. Show that $\mathbf{A}$ has an eigenvector of $\mathbf{e}$ with ...
0
votes
2answers
160 views

Gaussian Elimination and diagonalization

The standard procedure to diagonalize a square matrix $A$: 1) Find eigenvalues, eigenvectors and possibly generalized eigenvectors by solving $(A-\lambda I)\xi=0$. 2) Form a Linear transformation ...
1
vote
1answer
193 views

How many ways are there to represent a monomial order, defined by $>$, by term order via matrices?

During the lecture, my professor brought up the list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the ...
2
votes
2answers
52 views

Generate random symmetric positive-definite matrix

Is there a simple way to generate a random matrix that is symmetric and positive-definite? The symmetry seems like it could be achieved by generating a matrix $M$ with independent random entries and ...
2
votes
1answer
89 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
0
votes
1answer
66 views

solve linear system using gaussian elimination

I want to solve a linear system of the form Ax=b. First of all I create the augmented matrix (A|b). I apply some elementary row operations and i obtain the REF form of A. After than, I do not know ...
0
votes
0answers
32 views

What is max norm of a matrix?

I am going to implement a function in C to calculate the max norm of a vector. But I cannot understand the definition of max norm. The following the is the link to the definition of max norm in ...
0
votes
1answer
43 views

Bilinear transformation and eigenvalues

I have a proof to do and I am stuck on proving that if there exist a matrix $A$ with eigenvalue $\lambda_i$ and $B$ with eigenvalues $\mu$ such that $A = (B+I)(B-I)^{-1}$ then we have ...
1
vote
1answer
172 views

Find determinant of the matrix NxN

We are given matrix $M_{n,n}$, where $m_{ij} = \begin{cases} a_i \cdot a_j,\ \mbox{if}\ i \ne j \\a_i^2+k,\ \mbox{if}\ i=j \end{cases}$ Hence, M gotta look like that: $ \left( \begin{array}_ ...
2
votes
1answer
54 views

Matrix Groups in Abstract Algebra

QUESTION: Let $h= \begin{pmatrix} -1 & 1\\-1&0 \end{pmatrix} \in GL_2(\Bbb R)$. Find $\langle h\rangle$. I'm stuck on the solution, but here is what I have: Let $h=\begin{pmatrix} ...
0
votes
0answers
36 views

Representative value of non-square matrix

First of all, I apologise if this question is inappropriate, I wish I could be more specific - but due to the nature of it, as I am actually asking for a suggestion of some technique, that's hard to ...
2
votes
1answer
66 views

Establishing an inequality between the $2^{nd}$ largest eigenvalue of $A$ and a related matrix.

Let $A$ be an irreducible, aperiodic matrix with non-negative entries, with $1 \in \ker(A - I)$, $w \in \ker(A^\top - I)$, $w_i > 0$ $\forall i$. Define $W = \text{diag}(w)$. I am studying the ...
1
vote
2answers
73 views

How to prove “rank is not less than the number of non-zero eigenvalues”?

I know to prove this using core-nilpotent decomposition. But if it feels like using a big tool for a small problem, is there any other better, simple proofs?
1
vote
1answer
53 views

Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. [Lay P160 Ch 2 Sup Q4]

Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To verify this, compute $ (I \color{orangered}{-A} ...
2
votes
2answers
121 views

If $AX=XA$ for all $X$, then $A = \alpha I$ for some $\alpha$

Let $A$ be a $2 \times 2$ real matrix such that $AX=XA$ for all $2 \times 2$ real matrices $X$. Show that $A= \alpha I$ for some $\alpha ∈R.$ I am absolutely stuck, i thought $X$ and $A$ are ...
1
vote
0answers
241 views

random nonsingular matrices using matlab

Does anybody know how to generate a random nonsingular matrices using matlab? I use sprand (m, n , dens, 1)function to specify the condition number to be 1 right now.But it is too slow.Is there any ...
2
votes
1answer
48 views

skew-diagonalization of a matrix

I think about the skew-diagonalization of a matrix, for example, let $A=\begin{pmatrix}a & b \\ c& d \end{pmatrix}\in SL(2,\mathbb{R})$ , if $trace(A)=0$, is it conjugate to $\begin{pmatrix}0 ...
1
vote
2answers
149 views

Odd-dimensional skew-symmetric matrix is singular, even in a field of characteristic 2

I'm familiar with the usual proof $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A)$ which only works in fields of characteristic not equal to 2. To get a proof that works in characteristic 2 I can ...
1
vote
1answer
31 views

Determinants of matrices with constrained entries

Let $A \in GL(n,\mathbb{Z})$, written $A = (a_{i,j})$. Define the Height of $A$ to be $\max_{i,j} |a_{i,j}|$. The Laplace expansion of $\det(A)$ clearly implies that if $A$ has height $N \in ...
2
votes
1answer
45 views

Skew-Symmetric after base change symmetric?

Are there invertible matrices $A,B \in \textrm{GL}(\mathbb{C}^3)$ such that for every skew-symmetric matrix $S \in \textrm{Mat}_{3 \times 3} (\mathbb{C})$ the matrix $A \cdot S \cdot B$ is symmetric? ...
1
vote
0answers
25 views

Vector*Matrix multiplication through Fast Transforms

I have recently read a paper in which the authors indicated they used a Fast Cosine Transform to implement a Vector*Matrix multiplication. The idea is to decrease complexity when implementing such ...
0
votes
0answers
22 views

Change of basis - free submodule

Let $R$ be a commutative ring and $(e_1, \dotsc, e_n)$ be a basis for $R^n$, the free $R$-module of rank n. Let $A$ be an $n \times n$ matrix with entries in $R$. Let $f_i = \sum \limits_{j=1}^n ...
3
votes
1answer
97 views

Is there a way in matrix math notation to show the 'flip up-down', and 'flip left-right' of a matrix?

Title says it all - is there an accepted mathematical way in matrix notation to show those operations on a matrix? Thanks.
2
votes
1answer
35 views

Proof the Similarity of Matrices

Suppose $A$ is a $3\times 3$ matrix with entries in a field $F$ of characteristic $0$, and assume $\operatorname{Tr}A = 6$, $\operatorname{Tr}(A^2)=14$, and $\det A = 6$. ($\operatorname{Tr}$ denotes ...
0
votes
1answer
74 views

Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
1
vote
0answers
108 views

Map points between 3D Coordinate systems

I am trying to find a way to relate two 3D coordinate systems. I have 24 points for each system and found this, but it only works for 2D coordinate systems: ...
2
votes
2answers
90 views

Tridiagonal Matrix's Rank and Eigenvalues

Please proof that a tridiagonal matrix with positive entries on minor diagonals has rank n-1 or n and the eigenvalues of this matrix are real. ...
1
vote
1answer
69 views

Finding invertible 3x3 matrix A such that Stab(M)A=AStab(N) for given matrices M,N

I'm reading that it is possible to find an invertible $3 \times 3$ matrix $A$ such that for the matrices $M = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 ...
4
votes
2answers
76 views

Why does $A^2=I$ imply $nullity(A)=0$?

$A$ is a square matrix, why does $A^2=I$ imply $nullity(A)=0$? This is the key step in the solution, which I can't get it. Please help