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
0answers
7 views

Spectral norm of random matrices times a diagonal matrix

Consider some scalars $d_1\leq \dots\leq d_k$ and let $D = \text{diag}(d_i)\in\mathbb{R}^{k\times k}$. Assume that $X\in \mathbb{R}^{k \times k}$ is a random matrix of i.i.d. normal entries (i.e. $X_{...
0
votes
0answers
10 views

What is the determinant of a multivector?

I am looking for a nice and short definition of the determinant for geometric algebra? For example, in $Cl_1(\mathbb{R})$ with basis $\mathbf{e}_0$: $$ \mathbf{e}_0=\pmatrix{0&1\\-1&0} $$ I ...
0
votes
0answers
6 views

Is this function invariant to $U(n)$? $f(\mathbf{u})=((\mathbf{u}^T\mathbf{u})^H)(\mathbf{u}^T\mathbf{u})$

Is this function invariant to $U(n)$? $$ f(\mathbf{u})=((\mathbf{u}^T\mathbf{u})^H)(\mathbf{u}^T\mathbf{u}) $$ where $\mathbf{u}\in\mathbb{C}^n$. $$ f(M\mathbf{u})=((\mathbf{u}^TM^TM\mathbf{u})^H)(\...
1
vote
1answer
25 views

Eigenvalues of a block matrix with all diagonal blocks but one

Let us consider a block matrix of the form $$A= \begin{bmatrix} -(k+\mu)I & B \\ kI & -(\gamma + \mu)I \end{bmatrix},$$ where $I$ is the $n\times n$ identity matrix, $\gamma, k$ and $\mu$ ...
1
vote
0answers
13 views

Pseudoinverse of a matrix with columns of exponential decays

I want to calculate the pseudoinverse $A^+$ of a matrix $A$ whose columns are exponential decays: $$ \begin{pmatrix} e^{-\alpha_{0}t_{0}} & e^{-\alpha_{1}t_{0}} & e^{-\alpha_{2}t_{0}} ...
0
votes
0answers
5 views

Balance equation of stationary distribution (Markov chains)

Let $P$ be a transition matrix on the finite state space $S$. Show that every stationary distribution $\pi$ satisfies the following equation for all $A \subset S$: $\sum_{i \in A} \sum_{j \in S \...
0
votes
3answers
40 views

Does there exist a non-symmetric involutory matrix?

Let $A$ be a real involutory matrix i.e. $$A^2 = I.$$ Is it necessarily symmetric? Any help will be highly appreciated. Thank you very much.
0
votes
0answers
12 views

Equivalence of semidefiniteness and quadratic inequality

My question is about a logical equivalence in Robert Freund's lecture notes "An Introduction to Semidefinite Programming (SDP)" from MIT's OCW. The claim is that for $Q$ an $n\times n$ real ...
0
votes
0answers
14 views

Bounding the norm of the inverse matrix

Very strangely, I do struggle with answering the following, seemingly elementary, question. For a square matrix $A$, is there an upper bound for the norm of its inverse? In terms of e.g. the norm of ...
0
votes
0answers
15 views

I have a quadratic minimization problem such that, $e=x^TAx+x^Tb+k$, I wish to find its minima using recursive least square.

I have arrived at a generic quadratic minimization problem, such that, \begin{equation} e=x^TAx+x^Tb+k \end{equation} where,$x=[x_1,x_2,x_3]^T \in \mathbb{R}^3$, $b=[\gamma,0,0]$ and $k$ is a constant....
0
votes
0answers
16 views

Commutator of 2 Rotational Matrix in x and z axis

How do i find the commutator of rotational matrix in x and z axis? Dx = \begin{pmatrix}1&0&0\\ 0&cos\gamma &-sin\gamma \\ 0&sin\gamma &cos\gamma \end{pmatrix} Dz = \begin{...
3
votes
1answer
53 views

Determinant of $2\times 2$ block matrices

I am trying to solve the problem here: Let $A,B,C,D,$ be commuting $n\times n$ matrices over the field $F$. Show that the determinant of the $2n\times 2n$ matrix $$\begin{bmatrix} A&B\\C&D \...
1
vote
1answer
41 views

Compute an $n\times n$ determinant with factorial and powers of $x$

Compute $$ D_{n}= \begin{vmatrix} 1 & 0 & 0 & 0 & 0 & \ldots & 1\\ 1 & 1! & 0 & 0 & 0 & \ldots & x\\ 1 & 2 & 2! & 0 & 0 & \ldots &...
3
votes
1answer
39 views

Identifying quotient groups of matrices

Let $\newcommand{\GL}{\mathrm{GL}}H_n = \{A \in \GL(n,\Bbb C) : \lvert\det(A)\rvert = 1\}$. Prove that $H_n$ is a normal subgroup of $\GL(n,\Bbb C)$ and that $\newcommand{SL}{\mathrm{SL}} \SL(n,\Bbb ...
0
votes
0answers
13 views

Proof that the difference of any two semi-simple matrices is also semi-simple

Let $A$ and $B$ be semi-simple matrices. For $A-B$ to make sense, $A$, $B$ and $A-B$ need to be matrices of the same size $n$ over the same field $F$. If $F$ is algebraically closed, like when $F=\...
0
votes
1answer
26 views

Are all the conditions of the Moore-Penrose inverse definitions necessary?

The Moore-Penrose inverse of a real or complex matrix $M$ is the unique matrix defined by four conditions. Can any of these four conditions be relaxed with no loss of uniqueness? I noticed there was a ...
2
votes
1answer
60 views

How to find the determinant A from an equation having A as variable?

I'm currently struggling because I can't find the answer do this. If anyone can help me, it would be great. A is a $5\times 5$ non scalar matrix, $(A+2)(A+A^3+1)^2 (A^2+A^3+1)^3 =0 $ a) ...
0
votes
0answers
16 views

Matrix Operations: Name of function to rotate each column such that the first element is on the main diagonal

Suppose I have a matrix, A = [ 1 1 1 2 2 2 3 3 3 ] And I would like to create or find some function g such ...
-1
votes
0answers
28 views

Matrice Algebra

I am reading a paper and stumbled upon this piece: $$(2u-1)^TQ(2u-1) = 4u^TQu - 4(1^TQ)u + 1^TQ1$$ I have two conflicting results but both results are different $4u^TQu-2u1^TQ-2u^TQ1+1^TQ1$ another ...
1
vote
2answers
24 views

When is a series of matrices divergent. How to define divergence in this case?

In Quantum Mechanics we deal with series of operators represented as matrices like $$e^A = 1+ A + \frac{A^2}{2} + \dots$$ and similarly for $\sin(A) $, etc., where $A$ is a matrix. Now my question ...
1
vote
1answer
35 views

Approximation of a matrix multiplication

Let $A_{n \times n}$ be a positive-definite matrix. There is a way to approximate the following product? $$\left(I-\frac{1}{n}11^{T}\right) A \left(I-\frac{1}{n}11^{T}\right)$$ where $1$ is a column ...
1
vote
1answer
25 views

If $B$ is positive definite, then $(g^TBg) (g^TB^{-1}g) \ge (g^Tg)^2$

I am trying to prove something in Optimization, and it comes down to proving the inequality $$(g^TBg) (g^TB^{-1}g) \ge (g^Tg)^2$$ where $B$ is positive definite and $g$ is any vector. I am able to ...
0
votes
1answer
38 views

Expected value of $1$'s in a matrix product defined over $\mathbb{Z}_2$

Let $\mathbf{A}$, $\mathbf{B}$ be random boolean matrices of $n \times n$ size, such that the matrix entry is $1$ with probability $p$ and $0$ otherwise. All entries are independent. How many $1$'s on ...
0
votes
1answer
21 views

Applying Isometries to Matrix Inequalities [closed]

If $A$, $B$ are two matrices such that $A-B\geq 0$, i.e., the difference is positive semidefinite. Let $V$ be an isometry. Is it true that $V (A-B) V^\dagger \geq 0$? Or under what conditions does ...
0
votes
1answer
17 views

Calculating the characteristic polynomial of a 3x3 matrix

I had to calculate the eigenvalues of the following matrix. $$H=h\begin{pmatrix}A+\frac{1}{2}(B+C) & = & \frac{1}{2}(B-C) \\ 0 & B+C & 0 \\ \frac{1}{2}(B-C) & 0 & A+\frac{1}{2}...
2
votes
1answer
30 views

How to find a solution for a matrix with 1 equation and 3 unknown variables?

The task is to find all solutions for $A_1 x = 0$ with $x\in \mathbb R^3$ $$A_1 = \begin{pmatrix} 6 & 3 & -9 \\ 2 & 1 & -3 \\ -4 & -2 & 6 \\ \end{pmatrix} $$ The given ...
0
votes
1answer
23 views

Find out whether linearity for the functions $f$ and $g$ persists

Given $$f: \mathbb{C}^3 \rightarrow \mathbb{C}^2, \begin{bmatrix} a\\ b\\ c \end{bmatrix} \mapsto \begin{bmatrix} ia+b\\ c \end{bmatrix}, \,\,\,\,\,\,\,g: \mathbb{C}^3 \rightarrow \mathbb{C}^2, \...
0
votes
1answer
23 views

Demonstrate this matrix derivative expression with the formulas of this table.

I'm doing this problem: Calculate $\frac{∂||Ax-b||^2}{∂x} = 2A^T (A-b) $. Knowing that $\frac{∂||x||^2}{∂x} = 2x $ so far I have this: $$\frac{∂||Ax-b||^2}{∂(Ax-b)} \frac{∂(Ax-b)}{∂x} = 2(Ax-b)A$$ ...
1
vote
1answer
46 views

Doubt regarding the proof of row rank = column rank

Wikipedia provides two methods to prove row rank of a matrix is equal to its column rank. My doubt is regarding the second method. But the wikipedia page mentions that this proof is valid only for ...
-3
votes
1answer
38 views

$A^{-1}XB = I$ Solve for X matrix equation

$A^{-1}XB = I$, $A$ and $B$ are given and they are square matrixes. If I want to solve this matrix equation for $X$, I need to change it to the form like this $X = A×B×I$?
0
votes
1answer
16 views

Eigenvector of a complete graph Laplacian

Can somebody help me prove why $v=\begin{bmatrix} 1 \dots 1\end{bmatrix}^T$ is the eigenvector of every complete graph Laplacian matrix? Thanks!
0
votes
1answer
20 views

product of matrices, and its norm

I have a product of matrices $\prod\limits_{i=1}^{n} a_i$, If $b$ is an eigenvalue of $a_i$ for any $i$, then $|b|<1$. (1) Under what norm or condition, $\|\prod\limits_{i=1}^{n} a_i\|<r<1$ ...
0
votes
0answers
71 views

to reduce an equation into the simplest form using translation and rotation [closed]

Given the equation $3x^2+10xy+3y^2-2x-14y-5=0$, I want to reduce it into the simplest form using translation and rotation of the coordinate axes by $2 \times 2$ orthogonal matrix. How can I proceed ...
0
votes
1answer
20 views

Doubt regarding groups formation under matrix multiplication

When considering the set of matrices: $$Sp(n) = \{ S \in \text{GL} (2n, \mathbb{R}) \hspace{2mm} \text{s.t.} \hspace{2mm} S^T \Omega S= \Omega\} \tag{1}$$ where $$\tag{2} \Omega = \begin{pmatrix} 0 ...
-1
votes
0answers
26 views

How to solve the following determinant?

I don't understand the proof given in this book. Please help me.
0
votes
0answers
8 views

the row vector has the same sum of products with two matrix

Assume there are matrices $A=[a_{ij}]\in\mathbb{R}^{N\times N}$ and $B=[b_{ij}]\in\mathbb{R}^{N\times N}$, and $A$ is symmetric. I want to get a new matrix $C = [c_{ij}]\in\mathbb{R}^{N\times N}$ with ...
0
votes
0answers
26 views

Product of a matrix and its transpose is doubly stochastic?

Let $A$ be an $m \times n$ matrix with $\mathrm{rank}(A) \leq \min(m,n)$. What are the conditions $A$ needs to fulfill such that $A^T A$ is a doubly stochastic matrix?
0
votes
2answers
23 views

Block matrix determinant property(sum)

In $2\times 2$ matrix, $\det \begin{bmatrix} a&b \\ c+e&d \end{bmatrix} =\det\begin{bmatrix} a&b \\ c&d \end{bmatrix}+ \det \begin{bmatrix} a&b \\ e&0 \end{...
-3
votes
0answers
20 views

reduced row echelon form with zeros column [closed]

The rref is like rref image And a website I saw shows that the parametric form of the rref is parametric form image I'm confusing that why the parametric form isn't parametric form image 2 because ...
0
votes
0answers
25 views

Name for a matrix with sum of elements along every row/column is equal?

Is there a specific name for a symmetric matrix where sum of elements along every individual row is equal to the sum of elements along every individual column? For example, \begin{equation*} A = \...
0
votes
0answers
25 views

Prove that an elementary row operation on the product of two matrices .

In my math textbook there is a method for finding inverse of a given matrix using row operation but the book does not give any proof for that. I searched for the proof of the on the internet and had ...
0
votes
1answer
46 views

$A$ is positive definite iff $\det(A_k) > 0$

Let $A$ be a symmetric $n \times n$ matrix, and $V = \mathbb{R}^n$. Define $\langle v,w \rangle = v^t A w$ with $v,w \in V$. Show that $A$ is positive definite iff $\det(A_k) > 0$ for $1 \leq k \...
0
votes
0answers
14 views

How can I express a matrix construction with zero columns in index notation?

Suppose we have three matrices $W_\alpha$, with components $W_{\alpha_{ij}}$ for $\alpha = 1,2,3$ and $i,j$ taking also the values $1,2,3$. Each matrix contains vectors $\mathbf{z}_\alpha$ as columns, ...
2
votes
2answers
55 views

Compute an $n \times n$ determinant

Compute the following determinant $$ \begin{vmatrix} a_{1}-b_{1}+x & a_{1}-b_{2} & \ldots & a_{1}-b_{n} \\ a_{2}-b_{1} & a_{2}-b_{2}+x & \ldots & a_{2}-b_{n} \\ ...
0
votes
0answers
17 views

Determine the orthogonal complement of x in span [p1, p2, p3]. [duplicate]

Let vectors P1, P2, and P3 be defined as follows: P1 = [1 2 3 4] $^{T}$, P2 = [4 -2 -6 -7]$^{T}$, and P3 = [3 4 -2 1]$^{T}$ Let $x = [1 \, 2\, 3\, 7]^{T}$ This question is compromised into two ...
0
votes
1answer
19 views

Inverting symmetric matrix

Is it possible to find an analytical expression for the innverse of the matrix $A$ defined by : $$i\neq j \;:\; A_{ij}=x_ix_j$$ $$A_{ii}=\alpha x_i$$ With $\alpha$ a constant and $x$ a vector. If ...
0
votes
0answers
7 views

Condition number of normalized matrices

I would like to study the condition number of a normalized matrix $\hat X$. Let $X \in \mathbb{R}^{n \times n}$. We obtain $\hat X$ by taking all the rows of $X$ and normalizing them such that the $\...
0
votes
1answer
21 views

Accuracy of low rank approximation

I am currently studying about randomized low-Rank Approximation of a matrix. In the problem's statement, given $m$ x $n$ $A$,it is referred that we want to minimize $\|A-Q_{k}Q^{T}_{k}A\|$ and for ...
0
votes
2answers
28 views

Eigenvectors for eigenvalue with multiplicity $\mu = 2$

I'm looking for a way to determine linearly independent eigenvectors if an eigenvalue has a multiplicity of e.g. $2$. I've looked for this online but cannot really seem to find a satisfying answer to ...
1
vote
1answer
23 views

transform a 2 dimensional ode 1 system to 2nd order one dimension system [closed]

Given a matrix $M$ of $2 \times 2$, and an ode: $$y'=My$$ let $y=(v_1,v_2)$. find a second order ode such $v_1,v_2$ are solutions.

1
2 3 4 5
854