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
votes
1answer
25 views

How do I find this basis given matrix representations?

Here is the question: Consider the multiplication operator $L_A:{\mathbb R}^2\to {\mathbb R}^2$ defined by $L_A(x)=Ax$ where $A=\left[\begin{array}{cc}2 &0\cr1 &-1\end{array}\right]$. Find an ...
0
votes
2answers
25 views

Decomposition of a diagonal matrix into a product of particular matrices.

Could you tell me please, if it's possible to find a decomposition of a diagonal matrix $ 3 \times 3 $ : $ D ( \lambda_1 , \lambda_2 , \lambda_3 ) = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 ...
0
votes
1answer
30 views

Is this function of matrix F convex?

A scalar function $f(F)=vec(F)^{H}\Big(\Omega^{H}\big(M\otimes F(I+F^{H}F)^{-1}F^{H}\big)\Omega\Big)^{-1}vec(F)$ convex? $\Omega$ is arbitrary matrix and $M$ is positive definite. F can be reviewed ...
0
votes
0answers
7 views

Is it possible to obtain a Hermitian, positive semidefinite matrix as some sum of non-commuting matrices?

I am working with generalized Pauli matrices given by $X \vert j \rangle = \vert (j+1)mod~p \rangle$, where $p$ is a prime number. $Z = \vert j \rangle = \omega \vert j \rangle$, where $\omega = ...
2
votes
3answers
34 views

How to explain the calculation of the determinant of a $4\times4$ matrix

In my linear algebra lecture notes, I am studying an example which concerns the calculation of the determinant of a $4 \times 4$ matrix, by first reducing the matrix to upper triangular form. (See ...
2
votes
0answers
31 views

Does this matrix normal form have a name and has it been used?

In a research paper in Theoretical Computer Science, we are using a certain matrix normal form, which I was not able to find in the literature (I have to admit that my Linear Algebra got a bit rusty, ...
0
votes
2answers
40 views

If $A$ is idempotent and $B=(I-A)$, then $BA'=I$ [on hold]

Given that $A$ is idempotent and $B=(I-A)$, then prove that $BA'=I$. I try this by taking two idempotent matrices..but i am confused
1
vote
1answer
44 views

The number of linearly independent solution of the homogeneous system of linear equations $AX=0$

I came across the following multiple choice question: The number of linearly independent solution of the homogeneous system of linear equations $AX=0$, where $X$ consists of $n$ unknowns and $A$ ...
0
votes
0answers
40 views

Writing an abelian group as a direct sum of cyclic groups

I have this problem Determine an isomorphic direct sum of cyclic groups, where $V$ is an abelian group generated by $x,y,z$ and subject to:: $x+y=0, 2x=0, 4x+2z=0, 4x+2y+2z=0$ So I wrote ...
0
votes
1answer
22 views

How to use the crank-nicolson method

I'm going over my study questions for an exam I have tomorrow in Applied Numerical Methods and I know everything except for one thing. There's a sample question about using the Crank-Nicolson method, ...
-3
votes
1answer
13 views
4
votes
1answer
27 views

Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in ...
1
vote
1answer
21 views

Possibilities of minimal polynomial for a matrix

I came across a problem recently in my linear algebra studies that went something like this: Let $A$ be a linear transformation on a finite-dimensional space $V$ with characteristic polynomial $(x ...
0
votes
0answers
11 views

Magnitude of linear transformation map

In a 2D situation: given a 2x2 matrix A and a vector $\vec u$ , does the magnitude $|A \vec u|=|\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \begin{pmatrix} ...
0
votes
0answers
17 views

Parallelogram with vertices 0, Xa, Xb, Xa+Xb (X is matrix, a and b are vectors)

There is a paralellogram with vertices 0, a, b, and a+b, whose area is $34$. What is the area of the parallelogram which has vertices 0, Xa, Xb, and Xa+ Xb, where X = \begin{pmatrix} 3 & -5 \\ -1 ...
0
votes
1answer
25 views

Find transformation matrix $T$ relative to new bases

T is a linear transformation represented as $\left(\begin{array}{ccc}1 & 1 & 0 \\0 & 2 & 0 \\3 & 1 & 0 \\0 & 1 & 1\end{array}\right)$ w.r.t the standard basis. Now ...
0
votes
2answers
55 views

What is the simplest way to find an inverse matrix?

let $A = \left( \begin{array}{cccc} 1 & -1 & 2 & -1\\ -1 & 2 & -3 & -2 \\ 2 & -3 & 7 & 5 \\ 3& -2 & 6 & -3\end{array} \right)$ I want to find the ...
0
votes
0answers
18 views

Inverse of a rigid transformation

I would be grateful for any help with the steps required to complete this calculation. You may assume that I have some experience with matrices from before, but I am obviously no master! So we have ...
1
vote
4answers
64 views

Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
-4
votes
0answers
29 views

Relation between Eigenvectors and the commutivity between square matrices?

I'm stuck on this in a proof, any help is greatly appreciated! Thanks so much in advance! [EDIT] Okay, so basically I'm stuck on a proof, I need to proof that all eigenvectors of B are unique, ...
0
votes
1answer
32 views

About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
-2
votes
0answers
87 views

If $AB$ has zero trace for every matrix $B$ of zero trace, then $A$ is a scalar matrix

Let $A$ be an $n \times n$ real matrix such that $\operatorname{Trace}(AB)=0$ whenever $\operatorname{Trace}(B)=0$. Show that $A=cI$ for some $c \in \mathbb{R}$. My attempt Let $U$ be the subspace ...
2
votes
2answers
29 views

Relation Between Eigenvalues of Block Matrices

Is there any relation between eigenvalues, or spectral radii, of $M$, $M_1$, and $M_2$ block matrices? \begin{equation} M= \begin{bmatrix} A&B\\B^T&C \end{bmatrix} \end{equation} ...
-1
votes
0answers
11 views

How do you calculate the kernel of a matrix

How to find a kernel of a matrix. The teaching slides wasn’t very helpful. Can someone show me how you would find a kernel. 3...0... 0...-3 2...0...-1...1 3...-3...3...13 1...-1...1...4 If anyone ...
1
vote
0answers
44 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
3
votes
0answers
16 views

How to use a Matrix Pencil to decompose Exponentials

I'm an Engineering wanting to do some analysis on decomposing exponentials. My problem is that in order to get the highest resolution results I need to decompose a signal that exponentially decays in ...
3
votes
2answers
106 views

Is this matrix positive semidefinite for all $n$?

This is an extension of my previous question (see here). In this follow-up problem extra ones have been added in the non-diagonal matrix elements. We want to prove the positive semi-definiteness of ...
1
vote
0answers
35 views

Eigenvector and eigenvalue of an infinte, symmetrical matrix

How to get eigenvectors and eigenvalue of an infinite matrix like $$ A= \begin{pmatrix} 1&0&1&0&\dots\\ 0&1&0&1&\dots\\ 1&0&1&0&\dots\\ ...
0
votes
0answers
14 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequality of the Schatten-p (quasi-)norm, ...
1
vote
1answer
33 views

How to calculate Cartesian coordinates for an element after rotation has been applied?

I have a square on a Cartesian coordinate system with origin (0,0) on top left (yellow arrow from the picture). The initial coordinate of the square from the ...
0
votes
0answers
49 views

Matrix norm optimization problem : $\min_{\textit{ }x} \| A x B \|_4$, $x$ in the “unit” circle

Bonjour, Let $A$, $B$, $C$ and $D$ complex matrices. Is there a way to find a matrix $x$ (edit: non trivial) as: $\min_{\textit{ }x} \| A x B \|_4^4$ Or, more complicated, $x$ as $\min_{\textit{ ...
4
votes
1answer
53 views

Positive semidefinite matrix proof

Let $e_i$ the $i$-th column of the identity matrix. Is there an easy way to prove that the matrix $$\left[\matrix{\mathbb{I}_n & e_1e_2^T & \cdots & e_1e_n^T\\ e_2e_1^T & ...
1
vote
1answer
58 views

On matrices with trace value zero

I would like to ask you something regarding the trace of a matrix (the value of the diagonal after adding all its members, a value which is said to remain constant independently from base changes): ...
0
votes
2answers
59 views

What are some interesting properties of $A - 2I$ when the matrix $A= \tiny \begin{pmatrix} 1&*&* \\ 0&2&* \\ 0&0&3\\ \end{pmatrix}$

Without using the Jordan forms, What are some interesting properties of $A - 2I$ when the matrix $A= \begin{pmatrix} 1&*&* \\ 0&2&* \\ 0&0&3\\ \end{pmatrix}$. Attempt: $A - ...
-2
votes
0answers
31 views

Does new matrix also have integral eigen values?

$K_n$ is complete graph on n vertices. Laplacian matrix of $K_n$ has integer eigenvalues. If we are taking compliment of a $K_m$ (alongwith n-m isolated vertices) ; $m<n$ in $K_n$, Does Laplacian ...
1
vote
1answer
18 views

Which way to write the transition matrix arrow?

On a study guide my professor writes the problem: Let $B\:=\:\left\{\left(1,-1\right),\left(-2,1\right)\right\}$ and $B'=\left\{\left(-1,1\right),\left(1,2\right)\right\}$ be bases for $\mathbb{R}^2$ ...
-5
votes
0answers
29 views

How to solve this determinant problem? [closed]

If $A$ is a $3\times3$ matrix and $|A|=2$, find $$\lvert A^{-1}+4\mbox{adj}(A)\rvert$$ Thank you very much!
0
votes
3answers
30 views

Find determinant value

\begin{vmatrix} 3 & 2 & 0 & 0 & . &. & . & . &0 &0 \\ 1 & 3 & 2 & 0 & . &. & . & . &0 &0 \\ 0 & 1 & 3 & 2 & . ...
3
votes
1answer
31 views

How can I project a matrix on the set of symmetric positive definite matrices with trace 1?

Given a square matrix $A \in \mathbb{R}^{n \times n}$, I need to compute $$ \min_{X \in \Omega} \lVert A - X\rVert^2$$ where $\Omega = \{X \in \mathbb{R}^{n \times n} |\, tr(X) = 1, X \text{ is ...
0
votes
2answers
17 views

A matrix $Q$ has orthonormal columns, but $QQ^T \neq I$

I have to find an example of a matrix $Q$ that has orthonormal columns, but $QQ^T \neq I$. If a matrix has orthonormal columns, it does not imply that the matrix is orthogonal, so that it is a ...
2
votes
0answers
27 views

How to solve the equation $Au+Bv=C$

How do I solve $Au+Bv=C$ Where $A$ and $B$ are constant known matrices that are nxn, $C$ is a constant known nx1 vector while $u$ and $v$ are unknown nx1 vectors with the condition given that $u_i = ...
3
votes
0answers
88 views

Does this have a name?

While messing around, I seem to have stumbled upon an interesting family of matrices: $$\mathbb{S} = \bigg\lbrace A\in\mathbb{M}_{n\times n}(\mathbb{R}) : A^{T}A=AA^{T}=\frac{1}{2} (A + ...
2
votes
0answers
42 views

algebraic equation with trace

I have a problem which I don't know how to attack. Actually I am not even sure there is a way to do it. Is it possible to solve an equation of this form $$A²-\frac{tr(A)²}{4}Id_{4\times 4}=B$$ where ...
0
votes
1answer
22 views

Find basis for kernel and matrix representation

Problem 4 from https://math.berkeley.edu/~ogus/Math_54-07/Exams/midsol1.pdf $\beta$ is a basis of $P_3$, the set of all polynomials of at most degree 3.$\beta = (x^0,x^1,x^2,x^3)$. Let $T$ be a ...
0
votes
0answers
17 views

Jacobian of an error function containing state space vector (for linearization)

1) I have a vector of robot poses, that indicates the state space --> $X = \{(x_1,y_1),(x_2,y_2)....(x_N,y_N)\} \equiv \{a_1,a_2,...a_n\}$. 2) I have odometry measurement values at each time step ...
0
votes
1answer
44 views

Find the matrix $A$

Let $A$ be a matrix such that $A\vec{x}=\begin{bmatrix}2 \\ 4 \\6 \end{bmatrix}$, where $\vec{x}=\begin{bmatrix}2 \\ 0 \\0 \end{bmatrix}+c\begin{bmatrix}1 \\ 1 \\0 \end{bmatrix}+d\begin{bmatrix}2 ...
-7
votes
0answers
50 views

can anyone help me in my homework in algebra [closed]

Consider the linear system: $$\begin{cases}x+y-z=1\\ 2x+3y+az=3\\ x+ay+3z=2\end{cases}$$ For what values of $a$ does the system have: a) No solution; b) More than one solution; c) A unique ...
0
votes
0answers
22 views

Self-adjoint linear operator $f_A$ of a scalar product $g_A$

Let $A$ be a real symmetric matrix of order $n$. Let $x,y \in \mathbb{R^n}$. Then, we have that $$g_A(u,v)=X^{\top}AY,$$ where X and Y are column vectors with at each entry one component of $x$ ...
2
votes
1answer
30 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...
0
votes
0answers
25 views

Adjoint of a 3x4 matrix

How do I find the adjoint of this matrix? I am familiar with finding the adjoint of an $n x n$ matrix, but this has thrown me. $$A= \left( \begin{array}{ccc} 1&-1&0\\ 0&0&1\\ ...