Skip to main content

Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

Filter by
Sorted by
Tagged with
1 vote
0 answers
25 views

What problems will arise if we define matrix multiplication this way?

$a\in \mathbb{R}^n$ $a=(a_1,a_2,\dots,a_n)$ lets define this to be equivalent to $(a_1,a_2,\dots,a_n,0,0,0 \dots,0)$ (finite many zeros) by this I think we can make an $n \times m$ matrix $A$ ...
0 votes
0 answers
4 views

Convergence of power method for dominant complex eigenvalue

Suppose $A\in\mathbb{R}^{n\times n}$ is a diagonalisable matrix. Let the eigenvalues of $A$ are given by, $$|\lambda_1 | = |\lambda_2| > |\lambda_3| \geqslant |\lambda_4|\geqslant \cdots\geqslant |\...
1 vote
1 answer
28 views

How to find the basis of $U$?

Let $U=$span{$u_1,u_2,u_3,u_4$} $\subseteq$ $\mathbb{R}^4$ where $u_1=\begin{pmatrix} 2 \\ 4 \\ 4 \\ 2 \end{pmatrix}$ $u_2=\begin{pmatrix} 3 \\ 5 \\ 3 \\ 1 \end{pmatrix}$ $u_3=\begin{pmatrix} 3 \\ 3 \\...
1 vote
0 answers
19 views

Smallest $k$ s.t. $\operatorname{rank} A^m=\operatorname{rank} A^k$ for all $m>k$.

Problem: Let $A\in \mathbb C^{n\times n}$ be a matrix. Determine the smallest number $k\in \mathbb N$ (dependent on $A$), such that $\operatorname{rank} A^m=\operatorname{rank} A^k$ for all $m>k$. ...
2 votes
1 answer
75 views

Arithmetic Sequence in 2×2 matrices

Let $$ S = \left\{ \begin{pmatrix} x & y \\ z & w \end{pmatrix} \in \mathbb{R}^{2\times2}: \; x,y,z,w \; \text{ is an arithmetic sequence}\right\}. $$ Find all matrices $C \in S$ that ...
2 votes
2 answers
594 views

Left and Right Polar Decomposition

Let $A \in \mathbb{C}^{n \times n}$ and suppose we have $A = PU = VQ$ are left and right polar factorisations, respectively. That is, $P^*=P$, $Q^*=Q$, and $U^*U = V^*V= I$. I want to prove the ...
0 votes
0 answers
11 views

Find appropriate rotation matrix

Consider the following elements in linear algebra: $\pmb{G}: L \times K$ matrix $\pmb{F}: T \times K$ matrix $\pmb{H} = \pmb{G} \pmb{F}^{\intercal}: T \times L$ matrix. The apex $\intercal$ denotes ...
1 vote
1 answer
39 views

Forming a matrix (i.e., linear operator) based on implicit description of coefficients

I'm implementing an algebraic multigrid solver based on "A Multigrid Tutorial, 2ed", and in chapter 3, the formation of the prolongation operator (a matrix denoted $P = I_{2h}^{h}$) is ...
0 votes
1 answer
26 views

Proving A*Α=E for chosen coefficients.

I have this problem : Let $ζ = e^{2πi} $ and $A \in Mat(n;\mathbb{C})$ so that $[A]_{jk} = \frac{ζ^{jk}}{\sqrt{n}}$ for every $j, k$. I have to prove $A^{*}A=E$. (Where $A^{*}=\bar{A}^t$.) This is ...
1 vote
1 answer
23 views

A case problem about rank-1-perturbation of diagonal matrices

I have the following prediction for rank-1 perturbations of diagonal matrices, but I don't know how to prove (or disprove it). Given $v:= [v_1,...,v_K] \in (0,1]^K$, we define $a:= \sum_{k=1}^K v_k = \...
2 votes
2 answers
3k views

Jordan form of a specific 2x2 matrix

I'm trying to follow an algorithm for finding the Jordan form for the matrix: $$ \begin{pmatrix} 0 & -1 \\ 4 & 4 \\ \end{pmatrix} $$ Its eigenvalue is 2 with ...
4 votes
0 answers
82 views
+200

Matrix involving reciprocal factorials

Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 2/(2n+2i)!$. Is there a formula for the ...
3 votes
1 answer
127 views

Normal Form of Linear Maps Between Matrices

I am looking for a reference which features the following result -- for lack a better term I will call this "normal form" (of a linear map between matrices) -- and which explores its ...
0 votes
1 answer
31 views

Matrix Transformation of a Polynomial

I'm having trouble understanding matrix representations of polynomials. The question in particular is: $$ f → (x^{2} − 1)f'' + 2xf'. $$ I know for this question we have to use the standard bases of n ...
0 votes
2 answers
715 views

Matrix equations, reversible matrix

A reversible matrix is ​​given $A \in R_{3\times 3}$ and matrix $B$ formed from $A$ as a result of the operations $r_1-2r_3$, $r_2+r_3$, $2 \cdot r_2$, where $r_i$ is the $i$-th row of the matrix. ...
3 votes
0 answers
96 views

Leading term of $\det (A + \lambda B)$

Let $A$ and $B$ be $n \times n$ matrices. If $B$ is nonsingular, then the polynomial $p(\lambda) = \det (A - \lambda B)$ is clearly the characteristic polynomial of $B^{-1}A$ times $\det B$. What is ...
0 votes
1 answer
35 views

An elementary inequality problem with 6 variables

I am trying to prove/disprove the following statement. Given $x,y,z \in \mathbb{R}$ and $0\leq a,b,c <1$ such that $x+y+z =0$ and $a^2x+ b^2y+c^2z<0$. Then $ax+by+cz<0$. In my context, $[x,...
0 votes
1 answer
25 views

Property of the matrix operation to get reduced row echelon form?

If you have a matrix $A$, you can perform row operations to turn it into the reduced row echelon form $\hat{A}$. I believe you can encode these row operations into a single matrix $R$ such that $RA = \...
1 vote
1 answer
37 views

On invertibility of a real matrix

Given that $A$ is a real 'symmetric' matrix. I need to prove or find a counter example for the statement that $I+iA$ is always invertible, where $I$ is the identity matrix of the order same as that of ...
-1 votes
0 answers
30 views

Solve for variable matrice in matrice equation (linear algebra)

Say I have three matrices $A (3\times3)$, $B (2 \times 2)$ and $C (3 \times2)$. The question is to find $X$ in the matrice equation $A^2XB = C$. But I don't quite understand how to solve this? My ...
1 vote
0 answers
38 views

If A,B,C are three matrices of order $2$ , such that $|A+B|−|C|=−10 , |B+C|−|A|=25 , |C+A|−|B|=15$ then value of $|A+B+C|$ is equal to

If $A, B, C$ are three matrices of order $2$, such that $|A+B|-|C|=-10$, $|B+C|-|A|=25$, $|C+A|-|B|=15$ then value of $|A+B+C|$ is equal to ($|A|$ denotes the determinant of matrix $A$) My Approach: ...
0 votes
1 answer
3k views

Difficulty with the derivative of $L_2$ norm

So I was under the impression that the L2 norm squared of a vector x is just 2x, but the example in the screenshot I have linked to says otherwise. What gives? I can't figure out why there's an extra ...
2 votes
2 answers
124 views

Theorem on non-diagonalisable matrix

My professor gives me the theorem on non- diagonalisable matrices: Let a matrix the $A \in M_{n\times n}(\mathbb{R}).$ $A$ has $k$ independent eigen vectors $\Leftrightarrow$ A is similar to $$ \begin{...
0 votes
3 answers
1k views

Simultaneous diagonalisation of quadratic forms

Is there a linear transformation that simultaneously reduces the pair of real quadratic forms $$x^2-y^2$$ and $$2xy$$ to diagonal forms? My attempt I know that neither of these forms are positive ...
0 votes
0 answers
52 views

Find th determinant of a matrix with x under diag, 1 on diag and numbers to n above diag

$$ \begin{pmatrix} 1&2&3&\cdots&n \\ x&1& 2 &\cdots&n-1\\ x&x&1&\cdots&n-2\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x&x&x&\...
1 vote
0 answers
18 views

Proof that if $I, J \trianglelefteq R$ then $M_n (I) \cdot M_n (J) = M_n (IJ)$

I want to proof the following proposition. If $I, J \trianglelefteq R$ then the product of matrix ideals $M_n (I) \cdot M_n (J) = M_n (IJ)$. Let's consider the matrices $A = (a_{ij}) \in M_n (I)$ and $...
1 vote
2 answers
70 views

Jacobian matrix from mass action kinetic ODE network

I have an ODE system consisting of >1000 equations for which I would like to obtain the Jacobian matrix. For simplicity, let's consider the following example: ...
1 vote
1 answer
145 views

Relationship between matrix sum eigenvalues and matrix summand eigenvalues

In the blue bolded line, the author claims that because $A = 5 eye(4) - ones(4)$, \begin{gather*} eigenvalues(A) = eigenvalues(5 eye(4)) - eigenvalues(ones(4))\\ Ax = \lambda_A x\\ (5eye(4) - ones(4)) ...
3 votes
1 answer
40 views

Proving Similarity of 2 Matrices

I am given two matrices $U$ and $V$. I'm also given that $U=JK$ and $V=KJ$, where $J$ is an invertible matrix. I am supposed to prove similarity but I'm unsure if my proof is sufficient. $$V=KJ$$ $$VJ^...
3 votes
2 answers
82 views

Diagonalization/Eigenvalues of block matrices

Suppose I have two square $2^n \times 2^n$ matrices $A$ and $B$. Suppose furthermore that $A$ and $B$ are both symmetric matrices with real coefficients. Therefore their eigenvectors can be chosen to ...
28 votes
5 answers
45k views

Gradient of squared Frobenius norm of a matrix

In linear regression, the loss function is expressed as $$ W \mapsto \frac1N \left\| X W - Y \right\|_{\text{F}}^2 $$ where the matrices $X$ and $Y$ are given. Taking the gradient yields $$ W \mapsto \...
0 votes
0 answers
29 views

How to find the general form of a vector acted upon by linear and non linear operations

I stumbled upon this problem, originally I considered it easy to solve, I tried, and failed, and now I am beginning to believe it is quite a formidable adversary; so I am posting here, in hope of ...
7 votes
2 answers
3k views

How to prove the range of $AA^T$ is the same as range of $A$?

I have seen quite a number of questions regarding similar issues, like this and this. However, all the answers were trying to approach the topic via a non-straightforward way, that is to prove the ...
1 vote
1 answer
1k views

Proof of Cauchy's determinant identity

I am reading Matrix Analysis 2nd. Page 66 gives the proof of Caucuh's determinant identity as follow: where $(1.3.23)$ is as follow: I am very confused about these questions: How does second ...
1 vote
1 answer
1k views

Determinant of tridiagonal (banded) matrix

Struggling with homework. I know that this is a banded matrix ( bandwidth = 3) but I don't know how to approach computing the determinant. I tried to compute it with matlab for n= 3,..,7 but I didn't ...
1 vote
0 answers
23 views

Perron Eigenvalue versus Determinant for a class of circulant-like matrices.

Let $\mu_0(dx), \ldots, \mu_{k-1}(dx)$ be $k$ non-null (strict) subprobability measures on $[0,1]$, with max of their support $\eta_0, \ldots, \eta_{k-1}$. Let also $[n] \in \{0, \ldots, k-1\}$ be $n$...
1 vote
1 answer
1k views

Prove that $k(u,v) = f(k_1(u,v))$ is a valid kernel, where $f$ is a polynomial with positive coefficients

I want to show that $k(u, v) = f(k1(u, v))$, is a valid kernel where $f$ is a polynomial with positive coefficients. Proof. Since each polynomial term is a product of kernels with a positive ...
5 votes
0 answers
91 views

If $B^3=B$, is $B$ diagonalizable?

Let $B\in M_n(\mathbb{F})$ such that $B^3=B$. Is $B$ diagonalizable? If $B^3=B$, then $B^3-B=0$. Consider the polynomial $p(x)=x^3-x$. If $p(B)= B^3-B=0$. Since we know that the minimal polynomial of ...
-3 votes
0 answers
55 views

$lim_{ n \to \infty} [(1/2,1/2,0); (0,1/2,1.2);(1/2,0,1/2)]^n*[1,2,3]$ [closed]

this matrix is not diagonalizable over Real number, how should I approach now?
4 votes
2 answers
379 views

What is canonical spectral theorem?

My teacher has given me the following definition of the canonical spectrum theorem: Let a matrix the $A \in M_{n\times n}(\mathbb{R})$, and set of eigenvalues, $\sigma(A)$={$\lambda_1$,$\lambda_2$........
0 votes
0 answers
22 views

Doubling the numbers within an image matrix

I have an image Matrix tag that looks like this ...
0 votes
2 answers
8k views

matrix multiplication, change in order

I need to change the order of the matrix multiplication in the form $A \cdot B C$ where $A$ is $2 \times 2$, $B$ is $2 \times 3$ and $C$ is $3 \times 3$ matrix. Is it true $A \cdot B C=C^T A \cdot B$ ...
3 votes
1 answer
615 views

Let $A_1,A_2,\dots,A_d$ be $d\times d$ matrices that are strictly upper triangular. Then, the product of $A_1,A_2,\dots,A_d$ is the zero matrix.

SOLUTION: Let $B_i$ be given by $A_1A_2\dots A_i$. It can be shown inductively that $B_i$ is also strictly triangular, but with at least $i$ zero rows and columns. Therefore, $B_d$ will be the zero ...
1 vote
3 answers
87 views

Let $A\in M_2(\mathbb{Z})$ s.t. there is a positive integer $n$ satisfying $A^n=I_2$. Show that $A^{12}=I_2$.

Let $A\in M_2(\mathbb{Z})$ s.t. there is a positive integer $n$ satisfying $A^n=I_2$. Show that $A^{12}=I_2$. I have no idea where to start. Suggestions?
2 votes
1 answer
122 views

If $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{rank}(AB-BA)=\text{rank}(A^*B-BA^*)$?

I already asked this question for general $B$ and it was answered negatively here, so if the statement is true, one has to use the assumption on $B$ as well. In the original question I already ...
3 votes
2 answers
284 views

Find a "canonical form" for the linear operators that are both self-adjoint and unitary in a finite-dimensional complex inner product space.

Find a "canonical form" for the linear operators that are both self-adjoint and unitary in a finite-dimensional complex inner product space. What's the meaning of "find a canonical form" in this ...
0 votes
0 answers
50 views

Under what kind of conditions is the following matrix non-singular?

Given $m$ complex numbers $z_1,\ldots,z_m \in \mathbb{C}^m$, I am interested in the singularity of the following $2m \times 2m$ real matrix \begin{equation} M := \begin{bmatrix} 1 & \cdots & 1 ...
1 vote
1 answer
48 views

Diagonalizability of a special matrix involving a parameter k

I'm taking a linear algebra course this semester and we just got our second assignment. I've been working on one particular problem and I think I've made some progress, but I want to make sure my ...
5 votes
2 answers
2k views

The proof of positive semi-definite for a kernel

How to prove the following kernel $K$ over $\mathbb R \times \mathbb R$ is positive semi-definite: $$K(x_i, x_j) = e^{-\lambda[\sin(x_i - x_j)]^2},$$ where $\lambda > 0$. It looks like the gaussian ...
1 vote
2 answers
71 views
+50

Gram schmidt swapping two vectors

The question has background here but it's really just a linear algebra question. Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process (with no normalization of vector) ...

1
2 3 4 5
1121