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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6
votes
3answers
131 views

Invertibility of a Kronecker Product

Prove that A Kronecker Product B is invertible if and only if B Kronecker Product A is invertible. I dont have a clue where to start to be honest. I am not very familiar yet to the Kronecker Product ...
1
vote
1answer
892 views

Derive a rotation from a 2D rotation matrix

I have a rotation 2D rotation matrix. I know that this matrix will always ever only be a rotation matrix. $$\left[ \begin{array}{@{}cc} \cos a & -\sin a \\ \sin a & \cos a \\ \end{array} ...
6
votes
2answers
262 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
0
votes
2answers
64 views

Simple question of matrix algebra

Is the vector space $\mathbb R^{mn}$ the same as $\mathbb R^{nm}$? I need to know this to answer a question for Calculus, couldn't find it anywhere else.
1
vote
1answer
30 views

Matrix equations question?

I have to solve $B(Y+C)+3A=5(A+Y)$ for given matrices of $B$, $C$ and $A$. so what I do is $BY+BC+3A=5A+5Y$, and then $BY-5Y=2A-BC$ and then $Y(B-5)=2A-BC$. here I subtitute $B-5=M$ and $2A-BC=N$ and ...
2
votes
2answers
1k views

Finding the Solutions of the two systems by using the inverse.

I am having a difficult time understanding where I went wrong with the following: $$\begin{matrix}4x-y = 1 \\ 2x+3y = 3 \end{matrix} $$ $$\begin{matrix}4x-y = -3 \\ 2x+3y = 3 \end{matrix} $$ I found ...
-1
votes
3answers
179 views

matrix representation of a trigonometric rotation

Hey guys!I have a couple of doubts regarding this exercise, for a) I think that the Matrix rotation of P is [(cos t, -sen t) , (-sen t, cos t)] and for Q [(-cos t, -sen t), ( sen t, cos t)] , is ...
1
vote
1answer
308 views

Explicit conjugacy on 2 linear systems involving flow.

We need to find an explicit conjugacy between the flows of these 2 systems 1st system $X'$ = $AX$ and second system $Y'$ = $BY$ A = $$\begin{bmatrix} -1 & 1 \\ 0 &2\end{bmatrix}$$ B= ...
1
vote
0answers
62 views

Can someone explain the Kronecker Product?

I am in the final two weeks before the Calculus 2 exam and we just started with the topic 'vector differentiation'. We use a reader which is written by a PHD student but it is everything expect ...
0
votes
2answers
116 views

Diagonalized matrix question?

So I have the matrix \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 3 \end{pmatrix} I have to find a diagonalized matrix for this matrix.So I have found the ...
0
votes
1answer
77 views

Prove that a matrix is the permutation matrix of a permutation

Prove that a matrix is the permutation matrix of some permutation just when ...
1
vote
1answer
169 views

A question about Elementary Row Operation: Add a Multiple of a Row to Another Row

The task is that I have to prove the following statement, using Linear Algebra arguments: Given a matrix A, then: To perform an ERO (Elementary Row Operation) type 3 : (c * R_i) + R_k --> ...
0
votes
1answer
775 views

Matlab question: Converting a permutation matrix into a vector showing row exchanges

Let me preface that I am an absolute beginner with Matlab. I am trying to perform $PA=LU$ factorization on a matrix, however I am having difficulty with the permutation matrix. When I execute ...
0
votes
1answer
78 views

Finding all invertible matrices $A$ where $A = A^{-1}$ and $A^{-1} = A^T$ [duplicate]

Finding all invertible $A$, a $2\times 2$ matrix that satisfies $A = A^{-1}$ and $A^{-1} = A^T$. Hint: The identity $\cos^2t + \sin^2t = 1$ may be useful. I have no idea how to start this. Any help ...
3
votes
0answers
116 views

How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?

The question is pretty straight-forward: how do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$ ?
1
vote
3answers
133 views

Matrix with $1$s in the diagonal and off diagonal entries with absolute value less than $1$ invertible?

I ran into this problem recently. If $A$ is a $n\times n$ matrix with $1$s in the diagonal and all off diagonal entries have absolute value less than $1$, is $A$ invertible? It it definitely true ...
0
votes
2answers
112 views

Method for finding eigenplanes of a linear transformation.

Every introductory linear algebra course teaches methods for finding eigenvalues and associated eigenvectors of linear transformations $T$ acting on $\mathbb{R}^n$ and describes the geometric ...
0
votes
2answers
344 views

Find the general form of the solution to the system of equations below

Find the general form of the solution to the system of equations below. \begin{align} 2x_1-x_2+8x_3-x_4&=0\\ -4x_1+3x_2-18x_3+x_4&=0\\ 2x_1+x_2+4x_3-3x_4&=0. \end{align} My attempt: $$ ...
0
votes
1answer
163 views

Are these vectors in the span of $\mathbb R^3$?

If I have vectors $\vec v_1=\left[\begin{matrix}0\\0\\-3\end{matrix}\right]$,$\vec v_2=\left[\begin{matrix}0\\-3\\9\end{matrix}\right]$, and $\vec ...
7
votes
2answers
670 views

$AB-BA$ is a nilpotent matrix if it commutes with $A$

I saw this in a MathOverflow post and am putting it here for posterity. Problem: Let $A$ and $B$ by square matrices and set $C=AB-BA$. If $AC=CA$, prove $C$ is nilpotent.
1
vote
1answer
233 views

Change of basis matrices

Let $E=(e_1, e_2,e_3,e_4)$ be the standard basis in $\mathbb R^4$ and let another basis be given by $$ B = (\begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix} ,\begin{pmatrix} 2\\ 1\\ 0\\ 0 \end{pmatrix} ...
4
votes
4answers
123 views

is matrix invertible?

The characteristic polynomial of a $3\times3$ matrix $A$ is given by $$ \chi_A(x) = x^3 + ax^2 +bx +c $$ and takes the values $\chi_A(-1) = 4$, $\chi_A(2)=4$ and $\chi_A(-3) = -16$. Is $A$ ...
1
vote
2answers
218 views

$\det(\exp X)=e^{\mathrm{Tr}\, X}$ for 2 dimensional matrices

I want to prove that for $X\in M_2(\mathbb{R})$ the formula $\det(\exp X)=e^{\mathrm{Tr}\, X}$ holds, writing $X$ in normal form gives $X=PJP^{-1}$, where $J$ is the Jordan matrix, now $\exp ...
2
votes
3answers
46 views

Image and vectors question?

So I have the image $\displaystyle \operatorname{Im}f=\{\lambda_1(1, 2 ,0)+\lambda_2(2 ,1, 3) \}$ and I have to find the values of $\lambda$ so that the vector $\displaystyle (1,\lambda,\lambda^{2}) ...
0
votes
1answer
176 views

2D Eculidian matrix to 2D cartesian graph/plan

Can anyone help ? I am trying to convert a 2D matrix of distances to a 2D graph. For instance, I would like to go from this : ...
0
votes
0answers
121 views

What are the eigenvectors of this normal matrix?

Suppose $A$ is a real $n\times n$ normal matrix, and consider the matrix $$B=(I_n-\theta A)^{-1}(A+A^T)(I_n-\theta A^T)^{-1},$$ where $\theta$ is a real scalar such that $(I_n-\theta A)$ is ...
2
votes
0answers
55 views

An matrix inequality: true or false

My purpose with this post is to know if the following inequality is true or false: $$||S_n^{-1} x_1 x_1^T a_1 + ... + S_n^{-1} x_n x_n^T a_n|| \le x_1^T S_n^{-1} x_1 ||a_1|| + ... + x_n^T S_n^{-1} ...
1
vote
1answer
94 views

Good source for self study of matrix decompositions

What is a good source for study of various types of matrix decomposition, which is both comprehensive and also includes applications? It should at least cover LU, RQ, SVD, spectral, Schur, and ...
2
votes
1answer
287 views

Lower bound for matrix sorting?

Consider the problem of sorting a $n$ by $n$ matrix i.e. the rows and columns are in ascending order. I want to find the lower and upper bound of this problem. I found that it is $O(n^2logn)$ by just ...
0
votes
0answers
21 views

Weighting things relative to a number

I have a collection of data that has the following structure: [Customer: 1, OrderCount:11 ], [Customer: 2, OrderCount:22 ], [Customer: 3, OrderCount:1 ] ... I ...
3
votes
2answers
1k views

What is inverse of $I+A$?

Assume $A$ is a square invertible matrix and we have $A^{-1}$. If we know that $I+A$ is also invertible, do we have a close form for $(I+A)^{-1}$ in terms of $A^{-1}$ and $A$? Does it make it any ...
1
vote
3answers
62 views

Help with Matrices operations and groups

I have just finished a chapter concerning matrices and I am trying to solve a problem with Matrices and * operations. Specifically the question is as follows: Denote by $A$ the set of $3 \times ...
1
vote
0answers
169 views

Square root of a squared block matrix

I’m trying to compute the square root of the following squared block matrix: \begin{equation} M=\begin{bmatrix} A &B\\ C &D\\ \end{bmatrix} \end{equation} (that is $M^{1/2}$) as function of ...
1
vote
1answer
157 views

Can we always recover a matrix from its eigenvalues and eigenvectors?

If we're given all the eigenvalues of a square matrix $A$ and the corresponding eigenvectors of each eigenvalue, then in what case(s) is it possible theoretically to recover $A$ from this much ...
3
votes
1answer
468 views

Matrix Pseudo-Inverse using LU Decomposition?

What is the step by step numerical approach to calculate the pseudo-inverse of a matrix with M rows and N columns, using LU decomposition? So far, I have found this, but it uses singular value ...
11
votes
1answer
942 views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...
2
votes
1answer
5k views

Finding values such that a matrix has a unique solution

I have a matrix where I'm supposed to find the values of a and b so that the matrix has a unique solution. I have looked through my textbook and there aren't any examples of how to go about this. $$ ...
1
vote
1answer
74 views

On the convexity of the element-wise norm 1 of a pseudoinverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
3
votes
3answers
778 views

When does a system of equations have no solution?

I have performed Gaussian elimination on this matrix to reduce it to $$ \left[ \begin{array}{@{}ccc|c@{}} -3&-1&2 & 1 \\ 0& \frac{-5}{3}& \frac{10}{3} & \frac{8}{3} \\ ...
4
votes
1answer
162 views

Eigenvector of a sparse structured matrix corresponding to the eigenvalue 1

I have a matrix with the following sparsity pattern: $M = \begin{bmatrix} \ast &\ast &0 &0 &0 &0 &0 &0\\ 0 & 0 &\ast &\ast &0 &0 &0 &0 \\ 0 ...
2
votes
1answer
526 views

How to solve for matrix D in $ABC^{T}DBA^{T}C=AB^{T}$?

So the problem I'm having is trying to solve for matrix $D$ in the following equation, assuming the matrices are all $n\times n$ size and invertible. $$ D ~~\text{in}~~ ABC^{T}DBA^{T}C=AB^{T} $$
2
votes
2answers
202 views

Gaussian-Jordan Elimination question?

I have the linear system $$ \begin{align*} 2x-y-z+v&=0 \\ x-2y-z+5u-v&=1 \\ 2x-z+v&=1 \end{align*}$$ Very well. I form the matrix $$ \left[ \begin{array}{@{}ccccc|c@{}} ...
1
vote
3answers
77 views

Existence of Matrix inverses depending on the existence of the inverse of the others..

Let $A_{m\times n}$ and $B_{n\times m}$ be two matrices with real entries. Prove that $I-AB$ is invertible iff $I-BA$ is invertible.
7
votes
2answers
146 views

Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$

Given $A\in F^{n \times n}$ prove: $$\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$$ $\operatorname{rank}(A^{n+1}) \leq \operatorname{rank}(A^n)$ is easy, just from: How to prove ...
0
votes
2answers
110 views

Given a diagonalizable matrix A, must $A^2$ and $A$ be row equivalent?

Given a diagonalizable matrix A prove or give a counter example: $A^2$ and $A$ are row equivalent
4
votes
4answers
4k views

Properties of zero-diagonal symmetric matrices

I'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices. The only thing I can come up with is that it cannot be ...
0
votes
1answer
762 views

An example homography matrix than shows after transformation, the point inside the triangle could be outside

assume we have 3 vertices and a point inside the triangle. with a homography matrix [a,b,c] [d,e,f] [g,h,1] Also the final location of the 3 vertices and the point that is not inside the ...
7
votes
2answers
2k views

Are one-by-one matrices equivalent to scalars?

I am a programmer, so to me [x] != x—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for 1 x 1 ...
1
vote
0answers
52 views

calculate Variance?

This is related to the following question I posted earlier Optimize matrix multiplications If the values in $E$ are the values of some random variable $X$ and $$x= \frac{v_1^T E v_2}{v_1 ^T M v_2},$$ ...