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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6
votes
1answer
59 views

Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
0
votes
2answers
20 views

what is the parametric function of the new Bezier curve?

The cubic Bezier curve can be given in matrix form as If a cubic Bezier curve is rotated by an angle 30 around x-axis what is the parametric function of the new Bezier curve?
0
votes
0answers
13 views

Lower bound on eigenvalues of class of matrices

I'm trying to get a lower bound on the smallest eigenvalue of these matrices. Let $\{\phi_i\}_{i=1}^{N+1}\subset R^N$ be unit norm vectors such that $\langle \phi_i, \phi_j\rangle=-\frac{1}{N}$ for ...
1
vote
0answers
30 views

Generator Matrix

I have a C in $F_2^6$ $(x_1,x_2,x_3,x_4) \to (x_1,x_2,x_3,x_4,x_1+x_2,x_3+x_4)$ for $x = (1,0,1,1)$ i get $c = (1,0,1,1,1,0)$ we know that $$c = G . x$$ G is the Generator Matrix in the solution ...
2
votes
1answer
37 views

Multiplication for matrices

Can I prove following theorem $$(\lambda A)*B =A(\lambda B)=\lambda (A*B)$$ like this? My proof of the first equal sign We know that an easier notation for a generell matris, e.g A, is $$A= \ ...
0
votes
2answers
51 views

Determinant of complex matrix

How is the determinant of a complex matrix calculated? Is it the same algorithm as for real matrices, but the determinant itself is complex instead of real? (I was unable to find any hints with ...
0
votes
1answer
32 views

How to apply Vandemonde determinant to show nilpotent here?

This is question 5.5 from Erdmann & Wildon (paraphrased): Consider 2 square matrices over complex number $A,B$. Let $C=AB-BA$. Assume that $AC=CA$ and $BC=CB$. (i) Show that $trace(C^{m})=0$ for ...
0
votes
2answers
62 views

How do you find a non zero vector in Linear Algebra?

The question is; The vectors $a_1 = (1, 1, 0)$ and $a_2 = (1, 1, 1)$ span a plane in $\Bbb R^3$. Find the projection matrix P onto the plane, and find a nonzero vector $b$ that is projected to zero. ...
0
votes
0answers
40 views

Determining whether a function $\delta: M_{3\times 3}(\Bbb F)\rightarrow \Bbb F$ is $3$-linear

I've been working some suggested problems for a class, and I can't seem to understand the proper way how approach these problems. I tried to follow the following example and apply it to another ...
2
votes
1answer
24 views

Relating the Error in Matrix Inversion to the Determinant

If $M$ and $\tilde M$ are invertible square matrices which are almost the same (you get to pick the norm) $$\tilde M-M<\epsilon$$ Then can we say that their inverses are almost the same (of ...
1
vote
1answer
60 views

Linear maps, matrix transformation

Suppose $T$ is an element of $L(P_3(\Bbb R), P_2(\Bbb R))$ is the differentiation map defined by $Tp = p'$. Find a basis of $P_3(\Bbb R)$ and a basis of $P_2(\Bbb R)$ such that the matrix of T with ...
0
votes
2answers
28 views

Matrix chain product

I’m Reading a book , and I’m stuck at a property of a product of matrix chain , it says that given $$A_{i..k }=A_{i}\times A_{i+1}\times A_{i+2} \times ...A_{k}$$ where every matrix is a ...
0
votes
1answer
22 views

Matrix Multiplication By Rows

Lets y be a row vector $(y_1 y_2... y_n)$ and A to be $nxm$ matrix \begin{array}{ccc} a_{11} & a_{12} & ... &a_{1m} \\ a_{21} & a_{11} & ... &a_{2m} \\ ... & ... & ...
2
votes
1answer
71 views

What causes commutativity of matrices?

My understanding is that the multiplication of two matrices is NOT commutative most of the time. One exception is two matrices, A and B, that are inverses of the other. This condition leads in turn, ...
1
vote
0answers
34 views

solution of infinite dimension linear sysmtem

Let $\{a_n\}_{n\ge0}$ and $\{b_n\}_{n\ge0}$ be decreasing sequences such that $a_0=A$, $\lim_{n\to\infty}a_n=0$ and $b_0=B$, $\lim_{n\to\infty}b_n=0$. For fix $n$, one can construct a $n$-dimensional ...
0
votes
0answers
25 views

Finding the least square fit for 3 parameters in Linear Algebra

I know how to find least square for $y = mx+b$ when we have two parameters. But this question has $3$ parameters, am trying to think of how to approach it but so far no success, I can't find any ...
1
vote
1answer
81 views

Given $A$, find invertible $B$ such that $B^{-1}AB$ is positive

Given $A \in Mat(n,n,\mathbb R)$, is there always an invertible matrix B, such that $B^{-1}AB$ is positive, assuming all eigenvalues of A are positive and simple ? If yes, is it possible to classify ...
1
vote
1answer
37 views

Finding a matrix projecting vectors onto column space

I can't find $P$, for vectors you can do $P = A(A^{T}A)^{-1}A^T$. But here its not working because matrices have dimensions that can't multiply or divide. help
0
votes
1answer
44 views

Faithful Representations of C*-algebras

Can anyone give me an example of a represetation of the algebra $M_n(\mathbb{C})$ that is not faithul? If it's not possible, could you explain me why it is not?
-1
votes
0answers
9 views

Is this matrix singular and of certain class of matrix?

2n*2n Matrix is given by $$ \pmatrix{ a_0&b_0&a_1&b_1\cdots & a_{n-1} & b_{n-1} &a_{n}&b_n\\ c_0&d_0&c_1&d_1\cdots & c_{n-1} & d_{n-1} ...
1
vote
1answer
13 views

Matrix translation by (1x2) vector

I'm having trouble figuring out how to approach this matrix translation question: Find the equation of the image line produced by translating all of the points on the line $y = 3x -1$ by the ...
0
votes
0answers
15 views

All permutation matrices that convert one Hadamard matrix into another Hadamard matrix.

Given a Hadamard matrix $H$, I know that applying row and column permutations, along with multiplying a row or a column with a -1 results in another Hadamard matrix $H^{'}$ equivalent to the first. ...
1
vote
1answer
26 views

Irreducible matrix equivalent connectedness of matrix graph?

If a matrix is irreducible, based on the following definition A matrix is reducible if there are two disjoint sets of indexes $I,J$ with $|I|=\mu$, $|J|=\nu$, $\mu+\nu=n$ such that for every ...
0
votes
1answer
21 views

How to construct orthogonal basis from a missing vectors?

I have $m$ vectors with a missing element each. $v_i=(*, a_{2i},\cdots,a_{ni})^\mathrm{T}\,\forall\, i\in\{1, \cdots, m\}.$ I would like to add the missing element $*$ to all $v_i$'s such that all ...
2
votes
1answer
38 views

Checking connectivity of adjacency matrix

What do you think is the most efficient algorithm for checking whether a graph represented by an adjacency matrix is connected? In my case I'm also given the weights of each edge. There is another ...
2
votes
2answers
24 views

Decomposing a square matrix into two non-square matrices

I have a matrix $A$ with dimensions $(mxm)$ and it is positive definite. I want to find the matrix $B$ with dimensions $(nxm), (n << m)$, which follows the following expression: $$A = B'B$$ Here ...
2
votes
2answers
147 views

How to prove these 2 matrix problems?

I'm reading a book and it gives that $\frac{\partial}{\partial A}Tr(AB)=B^T$, then it shows we can obtain $\frac{\partial}{\partial A}Tr(ABA^T)=A(B+B^T)$. But it seems we should have ...
0
votes
1answer
24 views

Getting linear combinations in linear algebra?

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;
0
votes
0answers
12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
2
votes
1answer
36 views

Effective way of checking if all eigenvalues of a matrix are integers

Given A matrix with integer entries, it should be checked if all its eigenvalues are integers. Of course, the characteristic polynomial could be calculated, but is there any faster (or easier) ...
2
votes
0answers
32 views

Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
0
votes
1answer
42 views

Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.
2
votes
2answers
30 views

2x2 inverse of a complex matrix with complex determinant

Firstly, my question may be related to a similar question here: Are complex determinants for matrices possible and if so, how can they be interpreted? I am using: $$ \left(\begin{array}{cc} a&b\\ ...
0
votes
0answers
29 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
3
votes
4answers
97 views

Can you use row and column operations interchangeably?

Is it possible to use row and column operations "at the same time" on a matrix $A$? So, for example, first subtracting $row_1$ from $row_2$, and then choosing to multiply $column_3$ by a constant $c$? ...
1
vote
1answer
41 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
1
vote
0answers
27 views

When is the LU decomposition unique?

I want to find out when a matrix decomposition $A = LU $ (L lower and U upper matrix) is unique? Clearly, if $A$ is not invertible, there is no chance that this decomposition is unique. Hence, ...
0
votes
1answer
35 views

Symmetric matrix problem

$A$ is a symmetric matrix and has a eigenvalue $\lambda$ of order m why $\lambda$ has m independent eigenvector
3
votes
1answer
57 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
0
votes
0answers
27 views

Error in Matlab code, help please

I have a Unitary matrix MM1 and a theta function , I am trying to create another analytic function f by by summing up the products MM1(i, j)*thet(j) i.e say when j =1 f(1) = (MM1(1,d)*thet(0)) + ...
1
vote
4answers
33 views

Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$

I came across the fact that the rank of a nxn-matrix A with $A^2=0$ is at most $\frac{n}{2}$. The easiest way to proof this is using the inequality $rank(A) + rank(B) -n \le rank(AB)$. With $A=B$ and ...
0
votes
0answers
19 views

Is it a Wishart matrix? [duplicate]

Now I am having some problems about the Wishart matrix. Please help me, thank you! We know that $m \times m$ random matrix $\boldsymbol{A} = \boldsymbol{H} \boldsymbol{H}^H$ is a (central) ...
2
votes
0answers
25 views

Is it a Wishart matrix or not?

Now I am having some problems about the Wishart matrix. Please help me, thank you! We know that $m \times m$ random matrix $\boldsymbol{A} = \boldsymbol{H} \boldsymbol{H}^H$ is a (central) ...
1
vote
1answer
16 views

Cross-product is a left singular vector?

Assume A is a 3x2 matrix with rank(A)=2. u1 and u2 are already left singular vectors... How would I go about proving that the cross-product of the two is also a left singular vector? Hints would be ...
0
votes
1answer
28 views

Cholesky factorization and non-positive definite matrices

When Cholesky factorization fails, is there an alternative method to obtain the $\mathbf{L}$ matrix in: $\mathbf{A}=\mathbf{L}\mathbf{L}^{*}$ I'm dealing with a matrix not guaranteed to be ...
1
vote
2answers
76 views

On the nilpotence of the matrix $AB-BA$ [closed]

Given $n\times n$ matrices $A,B$ satisfy: $rank(AB-BA)=1$ Prove that $(AB-BA)^{2}=0$ Generalize the problem if possible. Any solution not mention Jordan canonical form would be appreciated!
-1
votes
2answers
27 views

When does the system of equations have initly solution, no solutions, and only one solution [closed]

I am trying to solve this math problem. So far I am bad at it. I nedd to determine for what given x does the system of linear equations has no solutions, has one solution, or infinetly many solutions. ...
0
votes
1answer
29 views

Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
5
votes
3answers
156 views

$A+A^2B+B=0$ implies $A^2+I$ invertible?

Let $A$ and $B$ be two square matrices over a field such that $A+A^2B+B=0$. Is it true that $A^2+I$ is always invertible ?
3
votes
3answers
51 views

Homotopy of Involutory Matrices?

I want to construct a homotopy from the matrix $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ ...