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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3answers
42 views

Find a matrix $A \in \mathbb{R}^{2 \times 2}$ such that $ \|Ax\|_{2}=\|x\|_{2}$ for every $ x\in \mathbb{R}^2 $

How to find a matrix $A \in \mathbb{R}^{2\times 2}$, $A\neq I_{2}$ such that for every $ x\in \mathbb{R}^2$ we have $\|Ax\|_{2}=\|x\|_{2}$. Is that even possible?
2
votes
1answer
41 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in ...
2
votes
1answer
27 views

Under what conditions does $AEA^{-1}=E $

In matrix multiplication when does $AEA^{-1}=E $? if any are the identity then it's trivial. Extra question, are there non diagnol matrices solutions or a proof one can't exist?
5
votes
3answers
45 views

Find a matrix $E$ such that $EA= B$

I am asked to find a matrix $E$ such that $EA= B$. I am given matrix $A$ which is $4\times 4$ and matrix $B$ $4\times4$. Would I find $E$ the following way or is incorrect? $$EA=B$$ $A^{-1} [EA = ...
0
votes
1answer
43 views

Kernel of a polynomial

Let's say we have a 2nd degree polynomial $a+bx+cx^2$ and it is given that $T:P2\rightarrow R$ given by $T(p)=\int_{0}^{1}p(x)dx$ We are asked to find the kernel of $T$. Now, I know that depending ...
1
vote
2answers
55 views

Properties of a $3 × 3$ matrix $A$ that contains two equal rows.

A $3 × 3$ matrix $A$ contains two equal rows. State whether each of the following is true or false. (a) $A$ has an inverse. (b) The rows of $A$ are linearly independent vectors. (c) The determinant ...
2
votes
2answers
51 views

For which $\lambda$ do we have solutions

I'm trying to find for what values of $\lambda$ the following matrix has either no solutions, infinitely many or unique solutions. $$A=\begin{pmatrix} 1 & 1 & \lambda & 1 \\ 4 & ...
1
vote
1answer
15 views

Which one is the correct definition of natural norm?

In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ...
1
vote
1answer
18 views

What is the “default” direction of feature / parameter vector?

This seems like a pretty obvious thing, so it's never really explained, but I can't understand it. Many book chapters use the expression: $\boldsymbol w^{T}\boldsymbol x$ as a form of denoting ...
1
vote
1answer
54 views

Can any linear transformation be represented by a matrix?

Use $\cal L$ to denote a linear transformation on some vector space. We know any matrix $\bf{A}$ can be viewed as a linear transformation by defining $\cal L:= \cal L(\bf{v})= Av$ where $\bf{v}$ is a ...
2
votes
0answers
21 views

Show that matrix is totally unimodular

I want to show that this matrix is totally unimodular: \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & ...
1
vote
1answer
32 views

What is a particular use of Gram-Schmidt orthogonalization?

We have a linear space V of m x n matrices. I know that we can use Gram-Schmidt to construct an orthonormal basis but the natural basis for this space (where every ij-th element is 1 and the rest 0) ...
-1
votes
0answers
10 views

Property or feature of a symmetric or skew symmetric matrix for comparison

I have a symmetric matrix of size $n\times n$, where $n$ may range from $10$ to $50$. Let us call this matrix $M_1$. If from $M_1$, I delete an $m$th row and an $m$th column, all other elements of ...
0
votes
0answers
14 views

Property / Feature of a symmetric or skew symmetric matrix for comparison

I have a symmetric matrix of size $n \times n$, where $n$ may range from $10$ to $50$. Let us call this matrix as $M_1$. If from $M_1$, I delete an $m$th row and an $m$th column, all other elements ...
0
votes
0answers
12 views

Cholesky of a special block matrix using cholesky of sub matrix

Let $A$ be a real $N\times N$ positive semi-definite matrix. Let $r$ be a real $N\times 1$ vector. Then consider the matrix \begin{align} B = \begin{bmatrix}A & r \\ r^T & 0\end{bmatrix} ...
0
votes
1answer
36 views

Linear - Algebra - Matrices

Determine the value of b that would guarantee that the below linear system is consistent. $$\eqalign{x_1 − 2x_2 − 6x_3 &= -4\cr 5x_1 − 4x_2 − 2x_3 &= -7\cr −11x_1 + 4x_2 − 18x_3 &= ...
0
votes
0answers
23 views

Minimum sum of matrix coefficients

Let be $A$ a finite matrix with positives coefficients of order $m\times n$. Lets consider the sum of $k=min(m,n)$ coefficients of matrix $A$, such that at most one element by row and column are ...
6
votes
5answers
87 views

In mathematics, what is an $N \times N \times N$ matrix?

In mathematics, what is an $N \times N \times N$ matrix? I think this is a tensor but definitions of tensors that I have read are so overly complicated and verbose that I have trouble understanding ...
1
vote
1answer
32 views

Symmetric block matrices with zero trace

In my research in graph theory i am getting symmetric matrices with trace zero of this kind $$ \begin{bmatrix} 0 & 1 & 2 & 3 & 3 \\ 1 & 0 & 1 & 2 & 2 \\2 & 1 ...
1
vote
1answer
68 views

Matrix equation that doesn't change its form under a change of basis.

I would like to continue the thread with preserved properties under a change of basis (I'm developing my reasearch), however the problem is so distinct that I've decided to ask a new question (I hope ...
-1
votes
0answers
16 views

Condition number plane fit

I fit a 3D plane to 3D points. I setup the corresponding linear system $Ax=0$ by removing the mean of all the points and stacking them as rows into $A$, and solve for a non-trivial ($x\neq0$) using ...
0
votes
0answers
26 views

What's the “dimensionality” of a matrix (not dimension!)?

I know that a $m\times n$ matrix has dimension... well, $m\times n$. However, in a paper (not a mathematics paper!) I encountered the term dimensionality of a matrix, whose definition is nowhere to ...
0
votes
2answers
32 views

Transition Matrix and Invariant Probability

Given the transition matrix for a 2 state Markov Chain, how do I find the n-step transition matrix P^n? I also need to take n--> inf and find the invariant probability pi?
-2
votes
1answer
90 views

Show that $det \left( {A+B} \right) \geq det\left( {A} \right)$ [closed]

Let $A$ is a positive define symmetric matrix,and $B$ is a semi-positive definite symmetric matrix.Does $$det \left( {A+B} \right) \geq det\left( {A} \right)$$ hold?Could someone help me with ...
1
vote
0answers
20 views

Minimum Column/Row Matrix “Covering”

I'm not sure if Column/Row "Covering" is the correct terminology. I have a square matrix. I would like to know how to determine the minimum number of lines (rows and/or columns) needed to "cover" the ...
1
vote
1answer
26 views

Necessary condition for matrix multiplication commutative (and if for permutation matrix)

This is a very old and popular question: the following link is the similar question When is matrix multiplication commutative? And there is a very famous theorem: If $A,B$ are simultaneously ...
4
votes
1answer
42 views

Characteristic Polynomial and Minimal Polynomial

(True or false): Suppose $A$ is an $n \times n$ matrix and $A^{k} = 0$ for some k. Then the characteristic polynomial is $x^n$ I am inclined to believe this is true since if $x^{k} = 0$ then the ...
2
votes
0answers
20 views

Find series for spiraling over matrix of size $nxn$ filled with numbers from 1 up till and including $n^2$

The following question exists: When starting from the number 1 and adding four numbers on each row a $4x4$ matrix is formed as follows: ...
1
vote
1answer
42 views

Why $Z_n$ is normally distributed?

We know $\epsilon_n \sim N(0,1)$, and $$Z_n = \frac {\mu_n^T(I-M_n)\epsilon_n} {\sqrt {\mu_n^T(I-M_n)\mu_n}},$$ where $M_n=X_n(X_n^TX_n)^{-1} X_n^T$, $\mu_n=X_n\beta_n$. Why $Z_n \sim N(0,1)$ ?? ...
0
votes
2answers
39 views

A matrix of a single 1 in each row and 0 elsewhere

Is there a particular name given to a matrix of m rows and n columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance: ...
0
votes
1answer
19 views

Show that a linear transformation on $\Bbb R^{2n}$ preserves the symplectic form $\Omega$ if and only if $A^T \Bbb J A = \Bbb J$

Hope everyone is well. I'm really needing some help with this question I've been doing for the matrix groups course I'm taking. Consider the skew-symmetric billinear form (on the vector space $\Bbb ...
0
votes
0answers
20 views

Demonstration involving inequality of traces of product of psd matrix

Let, $ \forall i \in [1, N]: P_i \in \mathbb{R}^{n \times n}, P_i \succ 0, w_i \in \mathbb{R}, \bar{P} = \sum_{i=1}^N w_i P_i$. Then, I want to demonstrate that $ \sum_{i=1}^N w_i ...
0
votes
1answer
24 views

Need Help with determining Eigenvector of this Particular Singular Matrix ($\lambda = 0$)

It is my understanding that a singular matrix will always have an eigenvector of some values for eigenvalue = 0. Let $$A =\begin{pmatrix} 2.25 && -2.25 \\ -2.25 && 2.25 ...
4
votes
2answers
49 views

When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...
0
votes
2answers
41 views

Find eigenvalues given A and eigenvectors

I have the following problem: I know how to compute the eigenvectors given the matrix and then finding eigenvalues. I could turn A into a triangular matrix and then compute for lambdas, but I wanted ...
0
votes
1answer
25 views

Find the Jordan Canonical Form of a nilpotent matrix

A is a $20x20$ nilpotent matrix. $$rank(A)=11, rank(A^2)=5, rank(A^3)=2, rank(A^4)=0$$ I know that the minimal polynomial is $m_{\lambda}=\lambda^4$. There's one eigenvalue which is $0$ (because A is ...
0
votes
2answers
55 views

Trouble computing eigenvectors of 3x3 matrix

I have the following matrix: $$\left[\begin{matrix}-4 & 20 & -33 \\ 0 & 1 & 12 \\ 0 & 0 & 5\end{matrix}\right]$$ Since it is a triangular matrix, we have the eigenvalues: ...
0
votes
2answers
38 views

Kernel and image of a linear map

Let $$A= \begin{pmatrix}3 & 1 \\ 1 & -2 \\ 2 & 2 \end{pmatrix}.$$ and $L$ be the transformation defined by : $$L : \mathbb{R}^2 \to \mathbb{R}^3: X\mapsto AX.$$ 1) ...
1
vote
0answers
13 views

Faster way to find inverse for a given matrix with a particular structure

Let $x_1,\dots,x_N$ be a set of $d\times 1$ vectors where $d$ is typically within $50$ and $N$ is in the range of $50000$'s. I construct the $N\times d$ matrix $X$ by stacking up $x_i$ as its rows, ...
1
vote
1answer
17 views

When does $A\mathbf{v} = \lambda B\mathbf{v}$ admit a basis of solutions?

Let $A, B \in \mathbb{C}^{n \times n}$ be Hermitian matrices, and consider the so-called generalized eigenvalue problem $$A\mathbf{v} = \lambda B\mathbf{v}$$ where $\lambda \in \mathbb{C}$ is called a ...
0
votes
1answer
81 views

Eigenvalues of matrices of order $p+q$

How to find eigenvalues of following symmetric matrices? $$A=\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \;(p \; \text{times}) & 1 & 1 & 1 & \cdots & 1 \;(q \; ...
5
votes
1answer
63 views

An intuitive way to understand the Jacobi's formula.

Suppose that $\mathbf A=\mathbf A(t)$ is a matrix whose entries are parametrized by a variable $t$. The Jacobi's formular states that $$ \frac d{dt}\left( \det \mathbf A\right)= \text{Tr}\left( ...
0
votes
1answer
28 views

Proof of nilpotent matrix in Complex Numbers [closed]

n $\in \mathbb N$ V is a n-dimensional Vectorspace of $ \mathbb C$ and $\phi$ is a endomorphism of V with $$ im(\phi) \subseteq ker(\phi)$$ Prove that $\phi$ is nilpotent. Additionally find ...
3
votes
0answers
38 views

Matrix with roots of unity entries

For a given prime p, i am interested in the norms of matrices which have root of unity entries, i.e., $M_{k,l} \in \{1, \zeta, \dots, \zeta^{p-1}\}$ where $\zeta = \exp{(2\pi I/p)}$. Are there any ...
0
votes
1answer
53 views

Why are the real eigenvalues of an orthogonal matrix always 1 or -1? [closed]

I heard that all real eigenvalues of an orthogonal matrix are either $1$ or $-1$. Why is that?
2
votes
1answer
67 views

Question regarding matrices with same image

Let $A$ and $B$ be $m\times n$ matrices over $\mathbb{Z}$ such that image($A$) = image($B$), where $A$ and $B$ are considered as maps $\mathbb{Z}^n \rightarrow \mathbb{Z}^m$. Does there exist an ...
1
vote
2answers
50 views

Is the following matrix diagonalizable?

Determine if this matrix is diagonalizable. $$ C= \begin{bmatrix} \frac{\sqrt2}2 & 0 & -\frac{\sqrt2}2 \\ 0 & 1 & 0 \\ \frac{\sqrt2}2 & 0 & \frac{\sqrt2}2 \end{bmatrix} $$ I ...
3
votes
1answer
53 views

How does the sum of the absolute values of the diagonal entries of a matrix change when the matrix is written in a random basis?

The set-up is as follows: I have a complex, Hermitian matrix $H$ with $\mbox{Tr }H=0$, and such that the trace norm $\|H\|_1=1$ (i.e. the sum of the singular values $=1$). Let me define the functiona ...
0
votes
0answers
15 views

Total support in matlab

I need help writing an algorithm in Matlab telling me if a radom matrix has total support or not. I'm trying to use the Linprog formula, but I don't understand it.
1
vote
2answers
47 views

Help me to Improve my method of creating a diagonal matrix?

Based on a past exam question: Q:Consider the matrix $$A = \pmatrix{5&-6&0\\4&-5&0\\-1&1&2}$$ with entries from $\mathbb{C}$. Find a diagonal matrix $D$ and an invertible ...