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
39 views

Clarification for a linear algebra problem stated: Find All Solutions to $AX = B$

I am working on a Linear Algebra HW problem which goes like: Find all solutions $X = \left[\begin{matrix} x & y \\ z & w \end{matrix}\right]$ to the matrix equations $AX = B$ when $A = ...
0
votes
1answer
16 views

Finding base for a set of vectors

Given these sets of vectors: $$ T=\{(2,1,-1),(1,0,-1),(5,1,-4)\} $$ $$ S=\{(1,2,1),(1,1,2),(3,4,5)\} $$ 1) Find a base for the subspaces: $Sp(S)$, $Sp(T)$, $Sp(S\cup T)$ 2) Describe the vectors ...
2
votes
2answers
39 views

linear algebra problem in matrices

I have no idea how to approch this, any help will be greatly appreciated: Given: Matrix A of order $(k\times n)$ Matrix B of order $(n\times k)$ with $k\neq n$, prove that its not possible for ...
-1
votes
0answers
24 views

Show A is not similar to a Diagonal Matrix

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $A = \begin{bmatrix} 4 & 0 & 0 &0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & -2 & -3 \\ 0 & -1 ...
1
vote
1answer
22 views

The $5\times 5$ matrix whose $(i,j)$th entry is $1$ if $j$ is a multiple of $i$, and $0$ otherwise

The $5\times 5$ matrix whose $(i,j)$th entry is $1$ if $j$ is a multiple of $i$, and $0$ otherwise. I know that I’m supposed to show the work I’ve done, but I just have no idea what to do with this. ...
0
votes
2answers
18 views

Find the dim of the solutions for $Ax=0$

Let $A$ be a matrix: $$ A=\begin{pmatrix} 1 & 1 & -5 & -6 & 1 \\ 2 & 1 & -7 & -7 & 1 \\ 1 & 2 & -8 & -11 & 5 \\ ...
1
vote
1answer
20 views

Finding eigenvalues for a vectorspace such that the matrixrepresentation is a diagonal matrix

Problem: Let $T$ be a linear operator on the vectorspace $V = M_{2 \times 2}(\mathbb{R})$ and let $T\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & b \\ c & a ...
0
votes
2answers
22 views

Convergence rate of the power method for finding eigenvectors

Let $M$ be a real-valued square matrix with an eigenvector $w$ strictly larger (in absolute value of the corresponding eigenvalue $\lambda$) than all others, and let $v$ be any vector not orthogonal ...
3
votes
3answers
42 views

Derivative of $tr((AX)^tAX)$

I'm trying to calculate the derivative (with respect to the matrix $X$) of the function $f(X) = tr((AX)^t(AX))$, Chain's rule gives that $\nabla_X(f(X))=\nabla_X(tr(AX))\nabla_x(AX)$ However I'm ...
7
votes
1answer
61 views
+50

$L^2$ Bounds for Markov Chains.

Consider a non-negative, square stochastic $n \times n$ matrix $P$ (rows sum to one, $P$ is ergodic). We are interested in characterizing the set of $n \times n$ invertible matrices $A$ such that we ...
0
votes
3answers
85 views

Find all matrices where the matrix is its own inverse and the determinant is 1

I need to find all the matrices: $$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ such that $$ad-bc=1$$ and $$A^{-1}=A$$ How would I go about doing this? I know that $AA=I^2$, ...
5
votes
1answer
149 views
+100

Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
0
votes
0answers
10 views

how to visualize this statement: Matrix M falls in a Ball-set.

So the question is simple: Assume you are told that a matrix M has the following property: $\|M\|_2<1$, i.e. it falls in unitary ball. When we say it is inside a ball set, if you imagine a ...
3
votes
2answers
65 views

Street Fighter: is the game balanced?

Suppose that $A$ is a matrix that describes the matchup information of any pair of Street Fighter characters e.g., considering $3$ characters, assume that the first row/collumn is associated with a ...
0
votes
1answer
27 views

What method is used to find the determinant of this $4 \times 4$ matrix?

This is a pre-solved example in my book, I don't understand how they solved it. What method is used? Find the determinant of $A = \begin{bmatrix} 0 & 1 & 0 & 2\\[0.3em] -1 ...
2
votes
0answers
47 views

Determinant of a sum

We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is ...
2
votes
1answer
26 views

What is the difference between coordinates transformation and change of coordinates?

In the context on 3D computer graphics, what is the difference between coordinates transformation and change of coordinates? It can just be a matter of notation, but my book makes a clear distinction ...
1
vote
1answer
35 views

Matrices Further Mathematics A-Level

I can solve the first part but after that, I'm not sure what to do. How do I relate what I need to show with the original matrix of A and subsequently prove the rest of the question?
1
vote
1answer
18 views

Differential Equations Hermitian Matrix Proof

I would like some help with the following proof below. Thanks for any help in advance. Prove that if $u(t) ∈ \mathbb C^N$ is a solution to the initial value problem $iu’ =Au$, $u(0)=u_0$, where $A$ ...
1
vote
1answer
162 views

Differentiation with respect to a matrix (residual sum of squares)?

I've never heard of differentiating with respect to a matrix. Let $\mathbf{y}$ be a $N \times 1$ vector, $\mathbf{X}$ be a $N \times p$ matrix, and $\beta$ be a $p \times 1$ vector. Then the residual ...
2
votes
1answer
43 views

Holder type inequality

If $A$ is a symmetric and positive semidefinite matrix is it true that $$\sum_{i,j=1}^n A^{i,j}x^iy^j \leq \sqrt{\left(\sum_{i,j=1}^n A^{i,j}x^ix^j\right)\left(\sum_{i,j=1}^n A^{i,j}y^iy^j\right)},$$ ...
1
vote
3answers
55 views

Invert a $2\times 2$ Matrix containing trig functions [duplicate]

Invert the $2\times 2$ matrix: \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} My thought was to append the $2\times 2$ identity matrix to the right ...
3
votes
1answer
42 views

Calculate a matrix to the power of “n” given an eigenvector

I have a question that I simply cannot solve. I do not want a direct answer to the question but simply an explanation as to the steps one would take to go about solving it, that way I can try it ...
2
votes
1answer
28 views

Why is it that while taking the inverse matrix a Wronskian pops up in this solution?

I was working on an ordinary differential equation solution when I saw another way that could be used to solve using matrices such that \begin{align*} \left(\begin{matrix} y_1\left(x\right) && ...
0
votes
3answers
42 views

Eigenvalues of skew-symmetric matrix

Prove that all of the eigenvalues of skew-symmetric matrix are complex numbers with the real part equal to 0. Has anyone got a clue how to do it?
0
votes
0answers
22 views

Finding the generalized eigenvector for the following matrix

For the given matrix A = $\begin{pmatrix} 1 & 0 & 2\\ 0 & 1 & 3\\ 0 & 0 & 1 \end{pmatrix} $ I've found that the eigenvalues are given by $\lambda = 1$ where $am(\lambda)=3$ ...
1
vote
1answer
32 views

Eigenvalues of the matrix $AA^*$

Suppose $A \in M_{n\times n}(\mathbb C)$ and let $B=A A^*$. Show that all the eigenvalues of $B$ are non-negative real. Can you please give me an hint how to start the proof? All I know is that ...
1
vote
0answers
24 views

In the SVD of $A = U \Sigma V^T$, how does one know that V actually spans the row space $C(A^T)$ of A and U the column space $C(A)$?

In the SVD of $A = U \Sigma V^T$, how does one know that V and U actually span the column and row space of A (respectively for each one)? I do know how to find such a U and V and $\Sigma$ by just ...
3
votes
1answer
30 views

Find the value of the expression-

Consider a matrix $A=\begin{bmatrix}3 & 1\\-6 & -2\end{bmatrix}$, then $(I+A)^{99}$ equals ? So how can I expand this ? The solution paper gives the answer as $I+(2^{99}-1)A$
0
votes
1answer
48 views

Prove that linear system has no solution

$A$ is a matrix $4\times3$, $rank(A)=3$ Also known that all elements of $A$ are nonzero values, $a_{ij}\neq0$ $c_i$ columns of $A$ ,$A=[c_1;c_2;c_3]$ $F$ is a diagonal matrix $4\times4$, all the ...
1
vote
2answers
16 views

If A is positive semidefinite, then $A+\alpha I$ with $\alpha\neq 0$ is positive definite? [duplicate]

If $A$ is a symmetric positive semidefinite matrix, then $A+\alpha I$ with $\alpha> 0$ is positive definite? Or are there some conditions to $\alpha$ so that it verifies? $I$ is the identity ...
2
votes
1answer
35 views

How to solve $C = X^\top C X$?

All matrices are $n \times n$. $C$ is real symmetric positive definite. How to solve $C = X^\top C X$ for $X$? I am interested in characterizing both the set of real matrices satisfying the equation ...
1
vote
2answers
25 views

Given a general 3D Matrix operation … who can I apply “1/2” of the effect of it ?

Given a general 3D Matrix operation ... who can I apply "1/2" of the effect of it ? I have an object with a given orientation in space and a given position ... and another version of same object ...
0
votes
1answer
18 views

Invertible? rewriteable

The question is as follows: Let $A$ be an $n$-by-$n$ matrix with the characteristic polynomial $p$. Prove that $A$ is invertible if and only if $p(0) \ne 0$. Let $A$ be invertible. Show that ...
1
vote
1answer
74 views

“Prove that for any $n \times n$ matrix $A$ and $B$, $AB-BA \neq I$”

So this is a exercise from the course compendium for a matrix course I'm currently taking. "Prove that for any $n \times n$ matrix $A$ and $B$, $AB-BA \neq I$" Is the proof that I have constructed a ...
0
votes
0answers
15 views

Converting points in a right hand z vertical coordinate system to left hand y vertical [migrated]

I have a series of points in space from Fanuc (robot manufacturer). The points are in a right hand system with positive z up. I need to convert this system to Direct X, which is left handed and has ...
0
votes
0answers
32 views

Positive semidefiniteness of a matrix -to examine convexity- What mistake have I made?

$f(x) = \begin{cases} e^{x_1x_2} & x_1, x_2 \geq 0 \\ +\infty & otherwise \\ \end{cases}$ The Hessian of this matrix is: $H = \begin{pmatrix} x_2^2e^{x_1x_2} & ...
-3
votes
1answer
36 views

Let A be an $n \times n$ square matrix. Is $A^t$ $A$ diagonalisable? [closed]

Let $A$ be an $n \times n$ square matrix. Is $A^t$$A$ diagonalisable? True or false
0
votes
1answer
15 views

inversion of a symmetric matrix after that a column has been changed

Suppose $Z\in \mathbb R^{n\times k}$ and $S=Z^TZ$. Let now $Z(i, x)$ be the matrix $Z$ where the $i-th$ column has been replaced with $x$. Given $S^{-1}$ is there a quick way to compute ...
1
vote
0answers
15 views

Summation of elements of a matrix in matrix notation

I have come across the following proof in a research paper. I feel the given end formula is wrong. Any help to correct it is greatly appreciated. where $V = \{v_{ij}\} \hspace{0.3cm} \text{with} ...
2
votes
1answer
45 views

How to prove these two statements?

Let A,B,C,D be real matrices (not necessarily square) such that $$A^T=BCD$$$$B^T=CDA$$$$C^T=DAB$$$$D^T=ABC$$ For the matrix S=ABCD, prove that $$S^3=S$$ and $$S^2=S^4$$ My little brother got this in ...
0
votes
0answers
28 views

Determining if an asymmetric matrix is positive. [duplicate]

Can we use leading principal minors to determine positive definiteness of an asymmetric matrix? I've been searching around the net but all sources I found apply this method on symmetric matrices but ...
0
votes
1answer
23 views

Proving or demonstrating that an adjacency matrix of a directed graph represents a cycle(s)

I'm currently struggling with this concept for my master's thesis in a computing discipline. If we have an adjacency matrix for a directed graph, $G$, where $A[i, j] = 1$ indicates a directed edge ...
1
vote
1answer
17 views

An alternative proof to: $P_{[X,Z]}=P_X+M_XZ(Z'M_XZ)^{-1}Z'M_X$

With $P_A=A(A'A)^{-1}A'$ and $M_A=I-P_A$ denoting the usual orthogonal projection matrices, I'm trying to find an alternative proof to $$ P_{[X,Z]}=P_X+M_XZ(Z'M_XZ)^{-1}Z'M_X.\tag{i} $$ I already ...
2
votes
1answer
37 views

Maximum value of $f(x) = \log_{(\tan x + \cot x)}(\det A)$ for a diagonal matrix $A$

If $$A =\begin{pmatrix} d_1 & 0 & 0 & 0 \\ 0 & d_2 & 0 & 0\\ 0 & 0 & d_3 & 0\\ 0 & 0 & 0 & d_4\\ \end{pmatrix}$$ ...
1
vote
1answer
29 views

$a_{ij}=i $ if $i+j=n+1$ and $0$ otherwise; compute det $A$

The entries of the matrix is specified by this rule, $A=(a_{ij})\in M_n(\mathbb R)$, $a_{ij}=i$ if $i+j=n+1$ and $0$ otherwise. Compute det $A$ > I have seen ...
0
votes
2answers
38 views

What is the relationship between matrix position to array index of corresponding matrix?

I want to know the exact relationship between matrix position and array index, where array contains the matrix data in each row appended format. For example: I had a matrix of $3 \times4$ as follows: ...
1
vote
2answers
208 views

Difference between dimension and rank of matrix

Let $A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 3 & -1 & 1 \\1 & 5 & 3\\ \end{array} } \right]$ and $V$ be the vector space of all $X\in \mathbb{R^3}$ such that $AX = 0$. Then ...
4
votes
2answers
85 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
0
votes
1answer
47 views

How to prove 2x2 rotation matrix is a manifold [duplicate]

How can I prove that this matrix is a manifold? $\begin{pmatrix} \cos(a) & -\sin(a) \\ \sin(a) & \cos(a) \end{pmatrix}$ Thanks!