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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0
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1answer
36 views

Is this “truncating” matrix function well known?

I'm working with a kind of "truncating" matrix function $\tau_r:M_{n\times n}\to M_{n\times r}$, where $r\leq n$, defined by $\tau_r(A)=B$, where $b_{ij}=a_{ij}$ for $j\leq r$. Is this a well known ...
1
vote
2answers
46 views

Eigenvectors and Kronecker product

Let us define $$ v:=v_A\otimes v_B\quad (*) $$ where $v_A$ is a fixed vector in $\mathbb{R}^{d_A}$, $v_B$ is any vector in $\mathbb{R}^{d_B}$ and $\otimes$ denotes the Kronecker product. To rule out ...
3
votes
0answers
49 views
+50

What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
0
votes
1answer
46 views

Matrix derivative of a special function

I need some help for calculating the matrix derivative of a special function. I have checked Wikipedia and Matrix Cookbook, but could not get the answer or idea. Let us define $f(X)$ as ...
1
vote
2answers
41 views

How can you expand the adjoint of a matrix into a polynomial with matrix coefficients?

This book contains an algorithm which claims that a matrix $sI - A$, where $A$ is some $n \times n $ square matrix and $s$ a variable can be expanded into $$adj(sI - A) = K_0 s^{n-1} + K_1 s^{n-2} ...
2
votes
2answers
76 views

How to derive this matrix equation

$$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r} \right) \right\| }_F^2 $$ I think $ ...
-1
votes
2answers
47 views

If $A,H\in GL_{n}(\mathbb{R})$, What happen for $H A H^{-1}$, if $h(i,j)\longrightarrow‎ 0$?

If $A$ and $H$ is a two $n\times n$ matrixs, such that $\det(A)\neq 0$ and $ \det(H)\neq 0$, what happen for $H A H^{-1}$, if $h(i,j)‎\longrightarrow‎ 0 $? Is $H A H^{-1}‎\cong A$?$ \ \ \ (1\leq ...
0
votes
1answer
26 views

General Element of U(4)

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$. Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ ...
1
vote
1answer
28 views

Row equivalence implies independent columns?

I need to prove that "given" two matrices are row equivalent, a set of columns of the first matrix are linearly independent iff the corresponding columns of the second matrix are linearly independent. ...
1
vote
1answer
14 views

Linear transformation with matrices in base

Consider the vector space of real $2 x 2$ matrices and take as base $\{{E_{11},E_{12},E_{21},E_{22}}\}$. Where $E_{ij}$ represents the matrix with a $1$ in the $i$-th row and $j$-th column and the ...
3
votes
2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
1
vote
3answers
20 views

Finding the rank of an non-invertible matrix

I have a $3\times3$ matrix with three different eigenvalues $0,1, 2$. The question is: what is the rank of this matrix? If the matrix was invertible, I could say that the rank was equal to $n=3$. ...
0
votes
0answers
17 views

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? [duplicate]

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? Please provide an example (with numbers if possible).
9
votes
1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
1
vote
2answers
50 views

What are the Eigenvectors in the following matrix?

I have the matrix A: \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4\\ \end{bmatrix} I found $\lambda I_n - A$ to be: \begin{bmatrix} (\lambda -4) & -2 & -2\\ -2 ...
0
votes
2answers
25 views

Jordan Normal Form matrix decomposition into the sum of parts that commute.

I'm learning about Jordan Normal Form matrices, and I've read that we can decompose a Jordan Normal Form matrix into the sum of two parts $$J=N_J+D_J$$where $D_J$ is the diagonal part (i.e. the ...
0
votes
1answer
69 views

Find a matrix of the linear map in the given basis

Let $Y = \{y_1, y_2, y_3\}$ be a basis for $R^3$ where $y_1 = (1, 1, 1)$, $y_2 = (4, 1, 1)$ and $y_3 = (1, 1, 2)$. Let $W = \{w_1,w_2\}$, $w_1 = (1, 1)$ and $w_2 = (2, 4)$ be a basis in $R^2$. Need ...
0
votes
1answer
12 views

If the Rref(A) of a 3x3 matrix is I(A), is this a valid eigenvector?

For the vector A: EDIT: I had originally multiplied the matrix by -1. Apologies. $$ \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \\ ...
1
vote
1answer
61 views

Exponential of 4x4 matrix

It is asked to calculate $e^{tA}$, where $$A=\begin{pmatrix} 0&1 & 0&0 \\ 3\omega ^2&0 &0 &2 \omega \\ 0& 0 & 0 &1 \\ 0& -2 \omega &0 ...
0
votes
0answers
22 views

What does it mean to find the principal directions and radii, given a matrix?

Do I compute the SVD, getting $$U \Sigma V^*$$ and then read off the column vectors in U that correspond to the positive singular values? These column vectors span the range of the matrix A, and I ...
1
vote
0answers
36 views

minimize smallest eigenvalue

Assume $P_A,P_B$ are probability transition matrices (each element is nonnegative and row sum is 1) and $v$ is probability row vector (each element is nonnegative and sum of elements is 1). How to ...
0
votes
0answers
32 views

Help explain “3d algebra”

The following is part of my lecture note, but I get lost after the first paragraph. I know what is "even" permutation and "odd permutation" which I learned from my abstract algebra course, and figured ...
-1
votes
2answers
60 views

A square matrix is called skew-symmetric if $A^T=-A$. Prove that if $A$ and $B$ are skew-symmetric matrices, then $A+B$ is skew symmetric.

A square matrix is called skew-symmetric if $A^T=-A$. Prove that if $A$ and $B$ are skew-symmetric matrices, then $A+B$ is skew symmetric.
2
votes
1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
1
vote
1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
1
vote
1answer
27 views

Condition for guaranteed minimum-rank solution

Consider the following rank minimization problem of a positive semi-definite matrix $X$: \begin{equation*} \begin{aligned} & \underset{X}{\text{minimize }} & & rank(X) \\ ...
1
vote
0answers
35 views

Inverse of $A^\top BA+C^\top DC$?

I'm numerically solving a system of equations of the form: $$Mx = b$$ where: $$M = A^\top BA+C^\top DC,$$ $B$ and $D$ are block-diagonal, $A$ and $C$ are $n\times m$ matrices with $m \leq n+3$. ...
0
votes
0answers
33 views

What kind of transformation does each matrix determine?

I have been given 5 matrices, been asked 'What kind of transformation does each matrix determine?', to multiply them and to 'Explain how the transformations determined by each matrix in the pair are ...
0
votes
1answer
27 views

A complex matrix with real eigenvalues

Let $A$ be a $10\times 10$ matrix with complex entries and all eigenvalues non-negative real numbers and at least one eigenvalue strictly positive . Then there exist a matrix $B$ ...
1
vote
1answer
43 views

If $V$ is a vector space and $U$ & $W$ are subspaces of $V$, such that $U \oplus W = V$! Need help with proofs!

Consider the map $\rho : V \to V$, defined by $\rho(v) = u − w$, where $v = u + w$, $u \in U$, $w \in W$. Show that: i. $\rho$ is well defined and it is linear; ii. $\rho(u) = u$, $\forall u ∈ U$; ...
0
votes
0answers
12 views

Conversion of network-like matrix

I have given a network in the following form (Example): x1 + x2 - x3 = 0 x3 + x4 - x5 = 0 x5 + x6 - x7 = 0 where = is something like a node, where flow needs to ...
2
votes
1answer
30 views

Prove that this matrix is total unimodular

Is there an easy way to prove that this matrix is total unimodular ? $$ \begin{bmatrix} 1 & F_1 & 0\\ 1 & 0 & F^T_1 \\ 0 & F_2 \end{bmatrix} $$ $1$ is the identity matrix, ...
0
votes
0answers
26 views

For any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?

Is it true that for any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And is it true that for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ? ( One problem I'm ...
2
votes
3answers
39 views

Is $A(x, y, z) = (xy + z, yz - x)$ a linear function?

Is this a Linear function? If yes, what is its matrix? $$A:\mathbb{R}^3\to\mathbb{R}^2, A(x, y, z) = (xy + z, yz - x);$$ Since they are all first degree variables, i think that this is a linear ...
1
vote
0answers
34 views

Generalized inverse of matrix product involving a positive semi-definite matrix

I have the following: A real square positive definite matrix $A$, and a real square conformable positive semi-definite matrix $B$. I form the product $$C = A^{-1}BA^{-1}$$ and I wonder, is it true ...
0
votes
1answer
23 views

Showing that $HTH = e^{-i \frac{\pi}{8} \sigma_x}$ (quantum gates)

I'm trying to prove that an arbitrary single qubit unitary (read: unitary two by two matrix, and thus rotation up to a phase) can be composed from Hadamard and T gates, given by $ H = ...
2
votes
0answers
74 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
3
votes
2answers
40 views

Is the entrywise nonnegative part of a real positive semidefinite matrix still positive semidefinite?

Let M be a real positive semidefinte matrix and consider the entrywise nonnegative matrix M' obtained from from ...
0
votes
0answers
20 views

Scaling Matrix?

I have two matrix problems which I have no idea of how to start solving. If possible could someone guide me through this? Links to videos would be great so I can solve future problems myself 1) Find ...
8
votes
1answer
70 views

Find all $n\times n$ matrices $A$ satisfying $\det(I+A^n)=(1+\det(A))^n$

Problem: Find all $n\times n$ matrices $A$ satisfying $$\det(I+A^n)=(1+\det(A))^n.$$ Clearly, the identity matrix $I$ works because $$\det(I+I^n)=\det(2I)=2^n=(1+\det(I))^n.$$ Are there any ...
-1
votes
1answer
20 views

SVD of matrix whose columns are orthogonal vectors [closed]

Write the SVD of a matrix $A$ whose columns are orthogonal vectors ($w_1, ...,w_n$). Can please someone give me a tip how to start writing the SVD ? Thanks
3
votes
1answer
52 views

$S = \left\{ x^* Ax\mid x \in C^n ,\ x^*x = 1 \right\} \implies S\;$ is compact and convex

Let $\,A \in {\mathbb{C}^{n \times n}}\,$ and $\,S = \left\{ {{x^*}Ax \mid x \in \mathbb C^n,\ {x^*}x = 1} \right\}.\,$ Why is $A$ compact and convex?
0
votes
1answer
30 views

Proof that transpose of Hadamard Matrix is also a Hadamard matrix

The question is self explaining from the title, but let me elaborate it. In most of the articles/books I've read, fact that the transpose of Hadamard matrix is also a Hadamard matrix is used, but I ...
-1
votes
1answer
30 views

matrix multiplication questions [duplicate]

$A$ and $B$ are two matrices, when is $(A-B)(A+B)=A^2 - B^2$
0
votes
1answer
34 views

Help Finding Elementary Matrix.

I apologise in advance if this is simple, but I'm losing my brain over this question. I'm unsure how to make the matrix format work either. I'm trying to find elementary matrices so that ...
1
vote
0answers
8 views

Request reflection matrix about these types

Supposed there's $(a,b)$ point and going to be reflected and find the mapping. The baseline formula will I use is $\begin{pmatrix} x' \\ y' \end{pmatrix}=M_{R} \begin{pmatrix} x \\ y \end{pmatrix}$. ...
0
votes
2answers
33 views
+50

Single transformation matrix of $A \circ B$ and $B \circ A$ with certain conditions

Let $A$ is 2x1 translation matrix and $B$ is 2x2 matrix of reflection or rotation matrix (reflection, rotation, etc.). Suppose I want to find the mapping of a $y=mx+c$ line and the mapping is done by ...
1
vote
1answer
36 views

Intuition about an orthogonal projection operator for matrices

Let $A \in \mathbb{R}^{m \times n}$ be of rank $r$, and $A = U\Sigma V^T$ be its SVD with $\Sigma \in \mathbb{R}^{r \times r}$. Let $P_U = UU^T$ and $P_V = VV^T$ be orthogonal projectors onto the ...
2
votes
1answer
23 views

Cayley transform a matrix that is invertible when added to the identity

Let A be an nxn matrix such that (I+A) is invertible. I need to prove that the Cayley Transform of A, denoted by $A^c$, is such that $(I+A^c)$ is invertible. The Cayley Transform is defined as ...