2
votes
3answers
26 views

Spanning Matrix

Consider the following matrix: $A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \end{pmatrix}$ The columns of $A$ span $\mathbb{R}^2$. The columns of $A$ span $\mathbb{R}^3$ ...
0
votes
2answers
41 views

$A^{T}A$ is diagonal. What can I say about $A$?

Is there any special property about the elements of $A$ if $A^{T}A$ is diagonal? I imagine you need some sort of symmetry but I can't see what it should be. Edit: Sorry, maybe it's better phrased ...
0
votes
1answer
37 views
1
vote
2answers
41 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
1
vote
1answer
33 views

Some simple matrix identities

I've recently been learning some linear algebra and I've isolated what seem to be some important matrix relations (often used tacitly). I would be most grateful if someone could just check that I have ...
0
votes
0answers
9 views

How is this a substitution? Linear algebra transformation matrix misunderstanding

I found the following matrix equation in '3D Surveillance System Using Multiple Cameras', (authors: Ajay Kumar Mishra, Bingbing Ni, Stefan Winkler, Ashraf Kassima) (link here): I don't follow the ...
2
votes
1answer
20 views

Column/Row Space check

I have the following matrix: \begin{bmatrix} 1 & 2 & 0 & 1 & 0\\ 3 & 6& 1 & 6 & 1\\ 2 & 4 & -1 & -1 & -1\\ 4 & 8 & 0 & 4 ...
0
votes
2answers
28 views

How does dot product work in matrix algebra?

I am working on a weighted minimization problem. Without the weights, the error function can be expressed as $e^T e$. With weights, $e$ first need to element-wise multiple by $w$, then the same ...
0
votes
0answers
25 views

Find solutions to magic puzzle with sums

I need help to solve the folowing puzzle using linear algebra (matrix and Gauss-Jordan Method): (for example the second horinzontal line: w + w + w + z = 45 or the ...
1
vote
1answer
25 views

How to avoid complex value for square root of a symmetric matrix?

I want to find square root of a matrix $Z$ which is a symmetric matrix using eigen values. So I find the eigenvalues($A$) and eigenvectors($B$) of $Z$ and find $B A^{1/2} B$. But because of small ...
2
votes
1answer
4 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
0
votes
0answers
25 views

Updating the LU Factorization

I am looking for a way to update the $LU$ factorization of a general $m \times n$ matrix after adding a column to the matrix. I have to iterate this procedure so I will begin with a matrix that is $m ...
9
votes
2answers
96 views

Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
0
votes
0answers
18 views

A linear-algebraic property of stochastic matrices.

All matrices are real, $n \times n$. By a stochastic matrix, I mean any non-negative real matrix with rows summing to one. Denote the set of all stochastic matrices by $\mathcal{S}$. By $I_k$ I mean ...
1
vote
2answers
43 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
0
votes
0answers
25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
0
votes
0answers
39 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
0
votes
2answers
31 views

Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
0
votes
1answer
39 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
2
votes
3answers
49 views

A linear map that is multiplication by a matrix

The problem statement, all given variables and data Let $T$ be multiplication by matrix $A$: $$A= \begin{bmatrix} 1 & -1 & 3 \\ 5 & 6 & -4 \\ 7 & ...
0
votes
3answers
55 views

Proof concerning matrix composition.

I have a statement which I don't know how to prove. All matrices are real, $n \times n$. For all $0 < k < n$ the following has to hold. It is impossible do define a matrix $A$ of rank ...
1
vote
0answers
17 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
1
vote
0answers
16 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
0
votes
1answer
45 views

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
2
votes
1answer
33 views

Perron–Frobenius theorem

What is exactly the Perron–Frobenius theorem? In different books papers I read different statments, and I don't know what is the truth. In wikipedia there are also a lot of statements under this ...
8
votes
1answer
86 views
+50

Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
0
votes
0answers
18 views

How do you find the change of coordinates matrix from B to C (defined below)?

B = {b1, b2} C = {c1, c2} b1 = 5c1 - 3c2 b2 = 8c1 - 5c2 How do you find the change of coordinates matrix from B to C? Also, how would you find [x]c for x = -4b1 + 3b2? I assumed the change of ...
0
votes
0answers
11 views

Constructing an oblique projection via formula.

Assume $\Phi$ is an arbitrary given $n \times k$ real matrix with $k < n$ and with independent columns. Consider the family of oblique projections on the column space of $\Phi$. All members of ...
1
vote
1answer
52 views

Why must $b=0$ for this linear system to have infinitely many solutions for all $a$?

Consider the parameterized linear system of equations represented by the augmented matrix: $$ \left[ \begin{array}{ccc|c} 1 & 0 & a & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & ...
-2
votes
1answer
29 views

Proving result on matrix rank

Is it true that, if $A=QR$ with $Q$ unitary matrix and $R$ an upper triangular matrix, and $A\in\mathbb{C}^{n\times n}$, then the rank of $A$ is the same as that of $R$? And if so, how do I prove it?
3
votes
3answers
86 views

Minimum linear subspaces cover problem

Given a set of vectors $V=\{v_1,v_2,...,v_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$ ($V_i$ may not be a subset of $V$), I want to find minimum number of sets from $\{V_1,V_2,...,V_m\}$, denoted as ...
0
votes
1answer
67 views

Self-adjoint on dot product

Let be $V = M_3(\mathbb{R})$ the vector space of the real antisymmetric matrix and let be $\phi$ the scalar product defined by $\phi(X,Y) = tr(^tXY)~ \forall X, Y \in V$. Let be $A$ a symmetric ...
1
vote
2answers
27 views

How do you find the change of coordinates matrix from a given matrix to the standard basis?

I'm not sure how to approach this problem. The examples I've come across on the internet show how to find the change of coordinates matrix from a matrix to another matrix, such as B to C (for ...
0
votes
0answers
33 views

Solving system of equations in rationals

Do there exist solutions to solve system of $n-2$ equations with $n-2$ variables where $n$ is fixed even integer and $a_i,b,c\in\mathbb{Q},i\in\{0,1,2,\cdots,n-5\}$ $$\left\{ ...
0
votes
1answer
26 views

Transpose of higher dimension matrices

We all know transpose of 2D matrix A Old $A_{ij}$ will be replaced by $A_{ji}$ in the transpose matrix and vice versa Question If A is 3D matrices of $3\times 3 \times 8$ then what is old ...
2
votes
0answers
17 views

Counting the operations of a problem

I have a square matrix $A\in\mathbb{R}^{n\times n}$, it has a LU decomposition. $L$ and $U$ are triangular and $L$ has ones on the main diagonal. I'm counting the number of operations for ...
1
vote
1answer
64 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
0
votes
1answer
46 views

Tensor product of Frobenius algebras

In proving the fact that the tensor product of any two finite-dimensional Frobenius algebras $R$ and $S$ over the same field $k$, it is usually defined a $k$-bilinear pairing $E: W×W→k$ where ...
0
votes
1answer
22 views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 ...
0
votes
1answer
24 views

Describe the solution set of the system

Consider the linear system below: $$\begin{array}{ccccccc} x_1&-&2x_2&+&&&x_4&=&1\\ 2x_1& -& 5x_2& -& 2x_3& +& k^2x_4 &= &-2\\ ...
4
votes
2answers
184 views
+50

Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal

Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, ...
0
votes
1answer
31 views

Matrix Multiplication - When do you only multiply by one number and add vs. multiplying all numbers?

*I wasn't sure where to put this. Just let me know if I should delete it or if there is another category/website where this question would fit better. Thanks! Or if you know the answer & don't ...
3
votes
2answers
104 views

Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [on hold]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
3
votes
1answer
48 views

Rank of the product of 3 matrices

Suppose I have 3 n by n matrices $A,B,C$ with $ABC=0$, what could be the maximal rank of $CBA$? I guess the answer would be n but I failed to prove it( tried to use Rank-Nuillity Theorem but I don't ...
0
votes
1answer
22 views

Inverse Matrix Multiplication

Let $A \in F^{n*n}$ a inverse matrix and $B\in F^{n*n}$ a none inverse matrix We can say that because A is row equivilate to $I_n$$ \implies $ $AB$ is none inverse matrix?
0
votes
0answers
39 views

Sum/diff of matrix units

I understand what the product of matrix units means, but I don't understand what the sum/difference of two different matrix units represents. For example, what does ${e_{2,2}}-{e_{5,5}} $ equal? ...
3
votes
2answers
61 views

Special solutions to Ax = 0

I solved most of it, just not sure about one point. The problem statement, all given variables and data Suppose A is the matrix shown below: $$ \begin{pmatrix} 0 & 1 & 2 ...
2
votes
1answer
72 views

Proof $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular

I need help proving $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular, as part of the proof of the Cayley-Hamilton theorem The result makes a lot of sense but I can't prove it ...
1
vote
1answer
48 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
1
vote
0answers
47 views

Matrices over field with characteristic $p$

For $A,B$ $n\times n $ matrices over a field $F$ with characteristic $p$ if $AB-BA=cI$ for $c\in F$ does this imply that $c=0$? Intuitively I would say that it doesn't but I cannot think of a ...