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
0answers
48 views

Does this family of special matrices have a name?

These are the bisymmetric matrices that are "pyramid" shaped as follows: $$f(14) =\begin{bmatrix}1&1&1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\1& ...
0
votes
2answers
75 views

Can we find the inverse for a vector

Can we inverse a vector like we do with matrices, and why ? I didn't see in any linear algebra course such a concept of vector inverse and I was wondering if there is any such thing and if not, why.
0
votes
2answers
37 views

Generating a random binary matrix with fixed number of nonzeros

I want an algorithm (just the idea, not the actual code) to generate a random $n$ by $n$ matrix with binary entries, but with the condition that the number of nonzeros must be a fixed number $c$. Any ...
1
vote
1answer
30 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
2
votes
2answers
42 views

Properties of the matrix square root

In a paper I am reading, it is claimed that if $A, B \in \mathbb{R}^{n \times n}$ are positive definite, then $$ A^{1/2} (A^{−1/2} B A^{−1/2})^{1/2} A^{1/2} = A (A^{-1}B)^{1/2} $$ because of the ...
4
votes
3answers
218 views

All Two by Two Matrices Satisfy a Certain Property Problem

Show that if $A$, $B$ are $2 \times 2$ matrices over $\mathbb{R}$ then there exists a real number $\lambda$ so that $$ (AB-BA)^2 = \lambda I $$ I can do this problem using brute force (i.e. looking ...
0
votes
0answers
45 views

$\text{Ker}A=\text{span}(u) \implies A=mat_C\left( u\wedge . \right)$

i found this equality and i wonder how can i find the right term $$\dfrac{1}{2}\left(\begin{matrix}0&1&1 \\ -1&0&1\\ -1&-1&0 ...
2
votes
1answer
25 views

Commutativity of the square root of matrices

Let $A, B \in \mathbb{R}^{n \times n}$ two positive definite matrices such that $AB = BA$, that is $A$ commutes with $B$. It is easy to prove that $A^{1/2}$ commutes with $A$, indeed $AA^{1/2} = ...
1
vote
0answers
26 views

GMRES and Preconditioning

I am using GMRES to approximate the solution of a system of equations $Ax=b$, I am using a preconditioner $P$ to make GMRES converge faster. My question is how do I know if the preconditioner I am ...
2
votes
1answer
60 views

determinant of matrix $X$

Please hint me. ‎How ‎can I ‎calculate ‎determinant ‎of ‎matrix ‎‎$‎X‎$‎?‎ \begin{equation*}‎ ‎\mathbf{X}=\left(‎ \begin{array}{ccc}‎ A&B&‎\cdots&B\\‎ B&A&‎\cdots& B\\‎ \vdots ...
1
vote
1answer
27 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
2
votes
1answer
39 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
3
votes
0answers
25 views

Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
1
vote
1answer
24 views

Commutativity of matrix square root

Let $A, B \in \mathbb{R}^{n \times n}$ and let us assume that $A^{1/2}$ exists. I have often seen people write something like $$ AB = A^{1/2}\, B\; A^{1/2} $$ when both $A$ and $B$ are symmetric, in ...
2
votes
2answers
82 views

Find smallest $n \in \mathbb{N}$ s.t $A^n=I$

Let $A$ be $2 \times 2$ matrix: $$ \left( \begin{matrix} \sin\frac{\pi}{18} \\ \sin\frac{4\pi}{9} \end{matrix} \begin{matrix} -\sin\frac{4\pi}{9} \\ \sin\frac{\pi}{18} \end{matrix} \right) $$ ...
0
votes
1answer
11 views

Show that rodriques formula is a linear transformation?

Can someone help me out on how to find the the matrix representation and show proof that it is a linear transformation? It is the rodrigues roation formula and the matrix representation should ...
0
votes
0answers
19 views

Where does this matrix rotation formula come from?

Im in a book and they use this rotation matrix formula in the picture. Where does it come from. I know that the c in the matrix is for Cos and the s is for Sin. Is there a proof?
2
votes
1answer
30 views

How to find the inverse of the matrix over $\mathbb Z_5$

How to find the inverse of the matrix over $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
0
votes
1answer
25 views

Next step to show that these matrice expressions are equal?

This is a problem from Discrete Mathematics and its Applications I know invertible means it is possible to take the inverse of this matrix. This is definition of a power of a square matrix from my ...
0
votes
0answers
22 views

Linear Algebra - verification of my answer, basis for $ImT$

I'd like to verify this answer, because I think that the answer in my book is incorrect. I'll be very glad if someone could tell me, if the basis I found for $ImT$ is correct. Let : $T:R^3 ...
0
votes
0answers
28 views

Determinant of matrix n x n [duplicate]

How to calculate $det\begin{bmatrix}1 & x_1 & x_1^2 \dots x_1^{n-1} \\ 1 & x_2 & x_2^2 \dots x_2^{n-1} \\ \\ 1 & x_n & x_n^2 \dots x_n^{n-1}\end{bmatrix}$?
0
votes
0answers
19 views

Positive/Negative Definite Bordered Hessian?

I understand how to check a function for concavity and convexity using the Hessian matrix and the rules for the determinants of the leading principal minors. I understand if these rules are violated, ...
2
votes
1answer
18 views

$B - A \in S^n_{++}$ and $I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}$ equivalent?

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Now suppose that $A,B \in S^n_{++}$ are two ...
2
votes
1answer
41 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
2
votes
1answer
57 views

Finding orthogonal matrix that maps one vector to another

Let $w, v \in \mathbb{R}^k$ be two known vectors such that $||w|| = ||v||$ ($|| . ||$ is the usual Euclidean norm). My questions are related with the problem of finding $Q$ orthogonal such that $v = Q ...
0
votes
1answer
20 views

Rewrite an expression in terms of basis vectors

Given any vector k $\epsilon$ $R^{3}$ consider k= $\sum_{j=1}^{3}$ $c_{j}u_{j}$ where $u_{1}$,$u_{2}$,$u_{3}$ are the orthonormal basis vectors (I don't know how to make them bold sorry about that, ...
4
votes
1answer
165 views

How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
0
votes
1answer
40 views

Matrix Differentiation using Kronecker operator issue

Let X an $n\times n$ variable matrix and given vectors and matrices $p_1$ ($1\times n$), $p_2$ ($n\times 1$), $\Omega$ ($n\times n$). What is the derivative of the function $f(X)=p_{1}X^{-1}\Omega ...
24
votes
3answers
2k views

Checkboard matrix, brand new or old?

Ok so what I found was a square matrix of order $n×n$ where $n$ follows $2m+1$ and $m$ is a natural number the pattern these matrices follow is as follows: for a $3×3$ matrix: $$ A = \left( ...
1
vote
2answers
26 views

limit of a function with a matrix exponential

I spent too many time trying to solve this problem...and finals are coming. Please help me! I just can't see a method to do this demonstration: "For an $A_{n \times n}$ matrix, demonstrate that a ...
0
votes
1answer
25 views

One eigenvalue and eigensystem

Matrix $A \in \mathbb{K}^{n,n}$ has one engenvalue $\lambda \in \mathbb{K}$ and its engensystem $V_{\lambda}$ has dimension that equals to $n$. How to show that $A = \lambda I_{n}$?
2
votes
1answer
44 views

proof on similarity of matrices

Could you please help me with the following problem? Let $A$ be an $n$$\times$$n$ complex matrix. Prove that $A$ is similar to $B$, which is an $n$ $\times$ $n$ real matrix, if and only if $A$ is ...
0
votes
1answer
31 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
1
vote
0answers
15 views

Applications of Matrix in simplifying algebra [closed]

Inversions (and the Mobius Transformation, though it belongs to complex numbers) are pretty good tools in simplifying an algebric mess. What other tools exist apart from this and how may we use them? ...
2
votes
2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
0
votes
1answer
22 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
2
votes
1answer
54 views

Example of a non singular square matrix such that $A+A^{-1} = 0$

Is there any example of a non singular square matrix $A$ such that $A+A^{-1} = 0$? Are they any specific type of matrices or can these be found under any category of matrices (such as symmetric, ...
2
votes
0answers
23 views

Problems in metric space including matrices. [closed]

Let $M(n, \Bbb R)$ denote the set of a real $n \times n$ matrices. We can always define a linear isomorphism between $M(n, \Bbb R)$ and $\Bbb R^{n^2}$....where the isomorphism is defined as for any ...
1
vote
1answer
31 views

Matrix multiplication computation

Any tips how to solve this? $$ \left[ \begin{matrix}1 & 2 & 0 \\ -2 & -5 & 1 \\ 11&15&5 \end{matrix}\right] \times \mathbf{X} \times \left[ \begin{matrix} -4&5&1\\ ...
1
vote
1answer
18 views

Vector notation question

Just a short question regarding notation: If this matrix represents a vector and I want to solve it for $t=2$, may I write it as follows: $ \left( \begin{array}{ccc} vt\\ vt-gt\\ \end{array} ...
0
votes
0answers
7 views

Composition of a rotation and a homothetic transformation of different centers?

Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$. Furthermore let $h_{\lambda,S}$ be the homothetic transformation of center $S\neq \Omega$ and ratio $\lambda$. What ...
1
vote
1answer
35 views

the relation between positive definite matrix and its ellipsoid

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Then, we associate with each $A \in S^n_{++}$ ...
2
votes
1answer
52 views

Do linear operators $A$, $B$ satisfying $A = B+BAB$ commute?

I have two linear continuous operators $A$, $B$ on Banach space $X$ (for example, square matrices), satisfying the equation $$ A = B + BAB, $$ and such that the continuous inverses $(\mathrm{Id} ...
2
votes
0answers
19 views

Transformations invariant wrt. $L_1$ norm.

$A$ is a real matrix of size $n \times k$, where $k \leq n$. $A$ has independent columns. Characterize the class of matrices $M \in \mathbb{R}^{k \times k}$ such that: $\forall x \in \mathbb{R}^k.\; ...
1
vote
0answers
28 views

Good resource to learn geometric interpretations of matrices [duplicate]

I need a good resource to learn matrices and all its properties through geometry. I feel geometry gives an insight into many matrix operations and a good resource will be useful for many students who ...
0
votes
1answer
20 views

Find all solutions to $Bx =[7, -10, 7, 0]^T$

$$ B=\left[ \begin{array} k1 & 0 & 2 & 1\\ -3 & 2 &-1 & 5\\ 2 & -1 & 1 & 4\\ 0 & 3 & 2 & 4\\ \end{array} \right] ...
0
votes
2answers
50 views

Commutative matrices

Knowing that $AB=BA$, find the matrices that commute with the matrix \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} I have assumed that multiplying matrix $\begin{pmatrix} a & b ...
5
votes
1answer
28 views

How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...
0
votes
1answer
22 views

conjugate of matrix multiplication [closed]

I have been trying hard to find what the conjugate is of the product of the matrices $A$ and $B$, but I did not have luck. Could you please help me? Thank you very much in advance.
0
votes
0answers
16 views

The conditions on marginals that guarantees a certain class of measures

$x$ and $y$ are $m\times n$ matrices. $a, B,C$ are $m\times 1$ matrices. $b, A$ are $n\times 1$ matrices. $$\sum a_i=1, 0\leq a_i\leq 1, \forall i$$ $$\sum b_j=1, 0\leq b_j\leq 1, \forall j$$ ...