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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1answer
33 views

Matrix factorization and its rank

Ask a fundamentla problem: Suppose a matrix $A \in R^{n \times m}$ can be factored into $A = UV'$, with $U \in R^{n \times k}$ and $V \in R^{m \times k}$ If $m,n \geq k$, what the rank of matrix $A$ ...
0
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1answer
43 views

Understanding matrix with greater than or equal

I'm reading a book and it states the following: $$ \begin{bmatrix} v\\ u\\ \end{bmatrix} \ge k $$ Does this mean that BOTH v and u are greater than or equal to k? If ...
0
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2answers
105 views

Coefficient Matrix of $T:\mathbb{R}^{3}\rightarrow\mathbb{R}$

Consider a vector $\vec{u}=\begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}$, such that $\left|\vec{u}\right| =1$. We define a linear transformation $T:\mathbb{R}^{3}\rightarrow\mathbb{R}$ given by $\vec{x}\...
1
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1answer
59 views

Inverse of Matrix of Standard Vectors

Having a bit of trouble with the reasoning behind this question: "Consider a $n\times n$ matrix $A$. Assume that for each standard vector $\vec{e}_i$, there exists another vector $\vec{v_i}$ such ...
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0answers
36 views

Solutions to linear equations

Am I right in thinking that the following augmented matrix equation only has one solution: $\begin{bmatrix} 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 10 \end{bmatrix} $ i.e., if the ...
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2answers
63 views

If $A^n$ is normal, is $A$ normal?

My question is : Given an invertible matrix $A$ ( with complex entries ) , if $A^n$ is normal,is $A$ normal? This is related to the question : If $A$ is an invertible $n\times n$ complex matrix and ...
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0answers
47 views

Comparing Bases in $\mathbb{R}^{n}$

A bit of trouble with the following question: Let $\mathcal{B}$ be the basis of $\mathbb{R}^{n}$ consisting of the vectors $\vec{v_1},\vec{v_2},\cdots,\vec{v_n}$, and let $\mathcal{E}$ be some other ...
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0answers
79 views

Equivalence proofs concerning matrices and systems of linear equations

Let $\mathbf A$ be the augmented $m × (n + 1)$ matrix of a system of $m$ linear equations with $n$ unknowns. Let $\mathbf B$ be the $m × n$ matrix obtained from $\mathbf A$ by removing the last ...
0
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1answer
33 views

Is the observed widest-width of an oblate sphere constant under all rotations?

This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically. Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, c<...
6
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3answers
278 views

Square root-related calculations with matrices

If $\mathbf A$ is an $n \times n$ matrix such that $\mathbf A^6 = \mathbf I_n$ (the identity matrix), is it true that either $\mathbf A^3 = \mathbf I_n$ or $\mathbf A^3 = \mathbf -I_n$? I'm ...
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1answer
511 views

Constructing the Matrix B “column by column”

I'm going through the various ways to construct a B-matrix of a linear transformation and I'm hitting a snag with one of the methods. We have $A = \begin{pmatrix} -3 & 4 \\ 4 & 3 \\ \end{...
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0answers
162 views

Linear Algebra Proof for matrices

Could someone possibly help me in proving this: Let $A$ be the augmented $m \times (n + 1)$ matrix of a system of m linear equations with $n$ unknowns. Let $B$ be the $m \times n$ matrix obtained ...
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1answer
53 views

FP3 Vectors question

$$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$$ $$\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$$ $$\mathbf{c}=c_1\mathbf{i}+c_2\mathbf{j}+c_3\mathbf{k}$$ Use appropriate determinants ...
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2answers
69 views

Show an integral of continuous $\gamma:\mathbb{R}\to GL(n,\mathbb{R})$ is invertible.

Let $$\gamma:\mathbb{R}\to GL(n,\mathbb{R})$$ be a continuous map such that $$\gamma(s+t)=\gamma(s)\gamma(t),\quad\gamma(0)=I,$$ for all $s,t\in\mathbb{R}$, and let $\psi:\mathbb{R}\to[0,\...
0
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1answer
145 views

Diagonally Dominant Tridiagonal Matrix always has an inverse

We are given the task to go and find a specific theorem from a Mathematics textbook at our university library and look up the proof. The problem, however, is that I do not know what the theorem is ...
0
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1answer
30 views

How is this norm $\|A\|_*$ called and what is it?

In my lecture notes I have found the notation $\|A\|_*$ for a matrix norm. Do you know the name of this norm (such that I can read the definition of it), or do you even know the definition of it? ...
0
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1answer
18 views

How can we use the symmetry of this complex matrix?

Find the Jordan normal form of $A\in \mathbb C^{4,4}$ if A is symmetric, $A^2=A$ and $\operatorname{rank} A=3$. So $A^2=A$ implies that the only eigenvalues are $0$ and $1$. From $\operatorname{rank}...
0
votes
3answers
253 views

No two 2x2 matrices in Jordan form are similar?

Let $S$ be the set of 2x2 matrices in Jordan Normal form $\begin{pmatrix}x&a\\ 0&y\end{pmatrix}$ with $a=0$ or $1$ and $x \leq y$. How do I show that no two matrices in $S$ are similar? Thank ...
4
votes
1answer
822 views

Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix.

I would like to prove that every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix. How is this possible? I'm not sure where to begin really- I know that a nilpotent ...
0
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1answer
50 views

Finding rotate matrix which solves equation

I try to solve the following problem: given a unit vector v, find rotate matrix R such that R*v = (0,0,..0,1) (vector that it's (n-1) components are 0 and the n'th component is 1). I know that if I ...
0
votes
1answer
44 views

Finding an eigenvectors and eigenvalues to a matrix

I got a question : Given A a matrix which the sum of all elements in each row equals to a constant $\alpha$, find eigenvector and eigenvalue it is belong to. I have no clue from where to start, ...
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2answers
136 views

Orthogonal Matrix question

$$A= \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} $$ is an orthogonal matrix. a) Prove that $A^{-1}=A^T$ b) show further that $a^2=d^2$ and that $b^2=c^2$. ...
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0answers
222 views

How to find the number of connected components of a graph by using its 16x16 adjacency matrix?

Good day, I have this exercice that provides me with the 16x16 matrix of adjacency of a graph and it asks me to find the number of connected components of the graph and I need to give a spanning tree ...
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1answer
51 views

Homogenous System

This question caught me off guard. I believe I did it the right way, but it was a bit confusing and I wanted a bit more explanation of the process and to see if what I got was accurate. This is the ...
4
votes
1answer
70 views

Is $\left(\sum_{n=0}^\infty\frac{1}{n!}A^n\right)v=\sum_{n=0}^\infty\frac{1}{n!}(A^nv)$?

Suppose we have a convergent power series of matrices $$A=\sum_{n=0}^\infty a_nX^n,$$ for $X\in M_n(\mathbb{C})$. Is it true that if $v\in\mathbb{R}^n$ then $$Av=\sum_{n=0}^\infty a_n(X^nv)?$$ ...
4
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3answers
86 views

Are there real solutions to $\exp(X)=-I$?

As we know, the equation $$e^x=-1,\quad x\in\mathbb{C}$$ has no real solution (in fact $x=i\pi+2ki\pi$, $k\in\mathbb{Z}$). I am now considering the generalization of this question to $2\times 2$ ...
0
votes
4answers
45 views

If two invertible matrices agree on a vector, does this imply their determinant agrees as well?

As stated, if we let $A, B \in M_n(\mathbb{R})$ be invertible and there is some $v\in R^n$ such that $$Av = Bv$$ does it follow that $\det(A) = \det(B)$? Additionally, does this hold if we let $A, B ...
0
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3answers
139 views

Square root of determinant equals determinant of square root?

Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$? I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer ...
0
votes
1answer
49 views

Find the inverse of defined operation $\Delta$

We defined the operation $ \Delta $ as $ (a,b) \Delta (c,d) = (ac + \delta bd, ad + bc) $ where $ a, b, c ,d \in \mathbb{Q} $ I have already proven that this operation is both commutative and ...
0
votes
1answer
89 views

Limit of regular symmetric matrix

I've got this statement about the topic. I'm trying to figure it out as it is given without proof. I know a symmetric matrix is a square matrix $A$ such that $A = A^T$ and a regular matrix is one ...
0
votes
1answer
62 views

Symmetric matrix-Spectral theorem

Assume we have a matrix $A$ let's say $100 \times 4$. We determine the product $B=A^{T}A$ Then by the spectral theorem \begin{equation} B =U^{T} \lambda U \end{equation} $B$ is a symmetric matrix $...
0
votes
2answers
46 views

Prove that Det(A-E)=0 if and only if AC=C

We have some $n \times n$ matrix $A$ and $n \times 1$ vector C. Let $E$ be the identity matrix. $$Det(A-E)=0 \iff AC=C.$$ Me and a few friends have been trying to prove it, but none of us could. ...
2
votes
1answer
99 views

If $A$ is a square matrix that is linearly independent, is $AA$?

I'm just not sure how to start this problem from Linear Algebra Done Wrong. The problem is to prove that if the columns of $A$, square matrix, are linearly independent, then the columns of $A^2$ = $AA$...
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2answers
15 views

A question on Matrix limits

Consider $M(n, \mathbb R)$ , the set of all real square matrices of size $n$ , as an NLS with norm $||A||:=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=Trace(AA^t)$ , then is it true that a sequence of ...
2
votes
2answers
152 views

Why use homogeneous coordinates?

I am having trouble understand the use of homogeneous coordinates for when describing transformations in 3D space. From what I have seen, the only difference between a transformation matrix in ...
2
votes
2answers
88 views

Minimal polynomial of an $n\times n$ matrix $A$ is $x^3+2x+2$; then $3$ divides $n$

Let $A$ be an $n × n$ matrix with rational entries such that the minimal polynomial of $A$ is $x^3 + 2x+2$. Prove that $3$ divides $n$. I think there is no rational root of this polynomial but how ...
0
votes
1answer
71 views

Transformations between coordinate systems

I have three three-dimensional orthogonal coordinate systems, O, A and B. A and B are the result of two different transformations from O. I now want to calculate the transformation matrix R, which ...
1
vote
1answer
86 views

How to obtain a convergent solution iteratively for a linear system of equations?

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} &...
0
votes
1answer
20 views

eigenvalue of a specific matrix

I am looking for a way to calculate the eigenvalues of this matrix. the last row contains complex numbers in general.
0
votes
1answer
38 views

Finding the value for the reproduction rate that will cause population stabilization

So essentially I have a matrix equation AB A is a 4x4 matrix containing reproduction rate, survival rate and maturity rate. B is a 4x1 matrix containing the populations for each age group. How would ...
0
votes
2answers
37 views

Suppose A is $n$ x $n$ and the equation A $\vec{x} = \vec{b}$ has a solution for each $\vec{b}$ in $\mathbb{R}^n$

Explain why A must be invertible. Can someone explain why? I am a little confused here.
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2answers
151 views

A real and normal matrix with all eigenvalues complex but some not purely imaginary?

I'm trying to construct a normal matrix $A\in \mathbb R^{n\times n}$ such that all it's eigenvalues are complex but at least one of then also has a positive real part (well, at least two then, since ...
0
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0answers
64 views

Decomposition of a matrix into a product of two-level unitaries

I am working through example 4.12 and 4.13 from 'Quantum Computing: From Linear Algebra to Physical Realizations' (Pages 83-84 and 405-406) and I can't seem to figure out some what the '*' in figure (...
0
votes
1answer
49 views

Let $x = (11, 2)^T$ . Find both reflection matrices $M$ such that $Mx$ is a multiple of $e_1$.

How would I go about solving this? I believe my professor said that it deals with householder matrices. I feel like I should calculate $v = x + ||x|| e_1$ and then calculate $u = \frac{v}{||v||}$ ...
0
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1answer
49 views

Use given identity to computer exponent of 4x4 matrix

I've been given an identity (that I don't know how to prove unfortunately), and been asked to use it to compute exp$(xM)$, where $$ M = \begin{bmatrix} 1 & 1 & 1 & 1 \\ ...
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0answers
53 views

Given a Positive Definite Matrix, find conditions of elements inside the matrix

I have a question that asks me to use the following symmetric positive definite matrix of order $n + 1$ $$B = \begin{bmatrix} \alpha & a^T \\ a & A \end{bmatrix} $$ With this matrix, I ...
2
votes
2answers
105 views

System of Equations

How would I solve a $4 \times 3$ matrix? I've tried making it into an augmented matrix but I ended up with all zeros at the bottom. Please help! $$\begin{align}\begin{cases}x_1+x_2+x_3+x_4&=1 \\...
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2answers
66 views

Calculate $e^{xA} $

$$ A = \begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix} $$ I have the answer, but I don't know the ...
0
votes
0answers
57 views

Given justification for induced norm for nonsingular matrix

Help!! Determine whether the following statement is true or false and give a justifcation: If A is nonsingular, then for any induced norm, $$ ||A^{-1}|| = ||A||^{-1}$$ Thanks!
1
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1answer
32 views

About a determinant identity.

If $A$ is any matrix and $B$ is a rank $2$ matrix of the same dimension then it follows that for any real $t$, $det(A -B) = [1-\partial_p + \frac{1}{2}\partial_p^2 ]det(A + pB) \vert _{p=0}$ I ...