1
vote
1answer
13 views

Similar Matrices Conditions:

Sorry this question was already asked but my english is not good. For matrix to be similar, does it have to have all of these properties or SOME of them? Same determinant Same Trace Same ...
0
votes
1answer
27 views

Prove a statement for the infinite matrix

We are given infinite two dimensional matrix $\{a_{i,j}\}_{i,j=1}^\infty$. And we know that matrix contain only natural values and each number appears in the matrix exactly 8 times. Task is to prove ...
1
vote
1answer
39 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
2
votes
1answer
36 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
0
votes
0answers
31 views

What does my teacher mean by 'choosing' from a vector?

I'm revising some lecture notes from a class I missed, I'm just struggling to figure out what she means at this point. What is choosing x1=0, x2=1... etc mean? Could someone explain?
1
vote
2answers
39 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
0
votes
0answers
35 views

Prove that B is bounded ?? [on hold]

Let G be a Banach space and let B be a subset of G. Suppose that f∈G* we have f(B) = {f(x); x∈B} is bounded in R. Prove that B is bounded. Such that G* is the dual space.
2
votes
1answer
36 views

One step Gauss Seidel method

Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with A = $\begin{bmatrix} 4 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 4 \end{bmatrix}$, b = $\begin{bmatrix} 4\\ ...
0
votes
0answers
23 views

Linear algebra.Proof proportinal between minors and cofactors

$B$ is square matrix. Order of matrix $B$ is $n$. First $m$ lines form the matrix $C$, $rank (C)=m$.Last $n-m$ lines form fundamental system solutions of homogeneous linear equation with matrix $C$ ...
0
votes
1answer
30 views

Question about Symmetric matrix

Ok my book says this matrix $A = \left ( \array{ -2 & 1 \\ 1 & -3 } \right )$is symmetric. But, I don't understand b/c if it were a symmetric matrix, wouldn't it be ...
0
votes
1answer
26 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
2
votes
1answer
46 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
0
votes
3answers
41 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
0
votes
1answer
15 views

Duality and Optimality Conditions

I have seen the solution and it involves adding a $x_5$ and $x_6$ to the inequalities. I really do not understand why this happens? I have not seen any questions like this yet. Any pointers would ...
2
votes
2answers
48 views

Is it possible to triangularize a matrix only by adding scalar multiples of rows to each other?

I am working on showing if $B$ is a $s \times s$ matrix, $D$ is a $t \times t$ matrix, $C$ is a $s \times t$ matrix, and $0$ is a $t \times s$ zero matrix, then $\det(A)=\det(B)\det(D)$, where $$A = ...
2
votes
3answers
44 views

Evaluate determinant of an $n \times n$-Matrix

I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix: $$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots ...
0
votes
3answers
38 views

How to find rank of a matrices?

Here is the question given in my text book IF the rank of the matrix $\begin{bmatrix}-1 & 2 & 5\\2 & -4&a-4\\1&-2&a+1\end{bmatrix}$ is 1, then the value of a is: a) ...
0
votes
3answers
39 views

Calculate Matrix A from eigenvalues, but no given eigenvectors

Here is my question: Write down a nontriangular 3 by 3 matrix whose eigenvalues are 6, 9, 2. I understand that you can calulate Matrix A using the formula A=V$\Lambda$$V^-1$, but is there a way to ...
0
votes
1answer
15 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
0
votes
3answers
54 views

For which values of $k$, we have $A = A^{-1}$?

I got this question in hw. Can anyone help me solve it? Let $ A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right) $ For which values of ...
0
votes
1answer
30 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
2
votes
1answer
42 views

Is there a quicker way to show that a set of vectors is a spanning set?

Let's say set $S = \{(2,1,4), (1,-1,1), (3,2,5)\}$ and the vector space V is $R^3$. In this case, the number of independent vectors is equal to the dimension of the vector space. I know that one way ...
2
votes
1answer
47 views

Inverse of a matrix is expressable as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
0
votes
2answers
31 views

Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.
7
votes
1answer
99 views

Prove that determinant of matrix equal $\pm1$ or $0$

We are given square binary matrix $A_n$. Data contained by A comply the following rule: if row has any 1's then they would appear there only successively (row $(1\space 1\space0\space1 )$ is ...
1
vote
2answers
63 views

If $ \det A$ is nonzero then $A$ is invertible

The problem is prove if $A$ is an $n\times n$ matrix with $\det A\neq 0$, $$A^{-1} = \frac{1}{\det A} ...
1
vote
1answer
29 views

Linear Algebra Eigenvalues question

This question doesn't look too hard but I just can't seem to figure it out. Let $A$ and $B$ be n x n matrices. Show that if none of the eigenvalues of A are equal to 1, then the matrix equation $XA ...
1
vote
1answer
19 views

Prove that adjacency matrix has negative eigenvalue

We are given non-oriented graph without loops. Task is to prove that adjacency matrix of that graph has negative eigenvalue. I put some effort into drawing a proof here , but it seems that I'm ...
0
votes
1answer
32 views

Exploring Determinants of Matrices. [closed]

I have a homework and i have to explore different patterns of determinant. I have find a unique pattern with determinants and make a conjecture. Your ideas about different patterns will be welcomed. ...
1
vote
1answer
25 views

Does this question make any sense - eigenvalues and norms

Im having difficulties understanding this question: show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an ...
1
vote
2answers
46 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
2
votes
2answers
36 views

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5. I will just add that this task is slightly ahead of my knowledge of field theory. So any pointers would be ...
0
votes
1answer
28 views

Show that matrix under addition is isomorphic with the group of complex numbers under addition

Q: Let $M = \{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} | a, b \in \mathbb R \}$. Show that $(M,+)$ and $(\mathbb C,+)$ are isomorphic. Show that $(M^{*},*)$ and $(\mathbb C^{*},*)$ are ...
1
vote
2answers
37 views

Example: Sum of non-commuting matrices is not normal

I'm trying to find an example of two non-commuting normal matrices, such that their sum is not normal. I know that unitary, orthogonal, hermitian and symmetric matrices are all normal. I figure it ...
0
votes
3answers
67 views

How to compute the nth power of a matrix

How would I compute $ \big ($ $\begin {matrix} -5 & 8 \\ -4 & 7 \\ \end {matrix}$ $\big )$$^5$ Using the relationship between the diagonal matrices and the nth power of a matrix? My ...
0
votes
1answer
42 views

Symmetric Positive Definite Matrix Proof

Suppose that $H^+ = H - (\mathbf y^TH \mathbf y)^{-1} H\mathbf y \mathbf y^T H + (\mathbf y ^T \mathbf s )^{-1}\mathbf s \mathbf s^T $ where H is symmetric and positive definite. Supposing that ...
3
votes
1answer
24 views

$a^{(m+n)}_{ij} \geq a^{(m)}_{ik}a^{(n)}_{kl}$ for non-negative Matrix $A$

Let $A$ be a non-negative infinite Matrix (all entries $\geq 0$). $a_{ij}^{(n)}$ denotes the $ij$-th entry of $A^n$. Does the following inequality hold: $a^{(m+n)}_{ij} \geq ...
0
votes
2answers
40 views

Eigenvalues of negative companion matrix

Here's a homework question I've been stuck on for a while. Given $A = \left[ \begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a_0 \\ -1 & 0 & 0 & \cdots & 0 & ...
1
vote
1answer
18 views

Diagonalisable matrices question

Hello, I am puzzling through this for homework. I understand most of the theory here, but I'm struggling to put together a coherent proof. I can easily see that (b) follows from what is shown in (a) ...
2
votes
2answers
65 views

Question about Implicit function theorem

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y ...
1
vote
1answer
65 views

Find high powers of a matrix with the Cayley Hamilton theorem

Let A = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ -1 & -1 &-1\\ \end{bmatrix} Compute $A^{10000} + A^{9998}$ I know this should be done by the Cayley-Hamilton theorem. I get ...
0
votes
1answer
41 views

Finding a matrix such that $AX=I_3$

How would I go about answering this question? I thought of multiplying both sides by $A^{-1}$ but that will not get me a $4 \times 3$ matrix, any hints/suggestions would be appreciated. ...
3
votes
2answers
77 views

Tricky Question on Induction and Characteristic Polynomials

I am to prove via induction that for any $n \times n$ matrix $A$, the characteristic polynomial of $A$ has degree $n$; $(-1)^n$ as the coefficient of the $\lambda ^n$ terms; $(-1)^{n-1}\cdot ...
0
votes
1answer
28 views

Linearization with Jacobian Matrix

Assuming I have $\frac{dx}{dt}=5x^2+2xy+x$ $\frac{dy}{dt}=xy-y$ which leads to a jacobian matrix $$\begin{pmatrix} 10x+2y & 2y \\ y & x-1 \end{pmatrix}$$ one of the fixed points is ...
2
votes
1answer
31 views

Nilpotent operator of index $n$

Let $T: \mathbb R^n \to \mathbb R^n$ be a linear operator such that $T^{n-1} \neq 0$ but $T^n = 0$. Prove that $\text{rank}(T)=n-1$ and give an example of such operator. PS. This was on a homework, I ...
3
votes
1answer
44 views

matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now ...
0
votes
1answer
43 views

Gauss Elimination - Diagonal dominant matrices don't need row changes

I was asked to prove the following statement: let $A$ be an $n$ by $n$ matrix with real entries such that $\forall k \in \mathbb N, k\leq n$: $$\sum_{i \neq k} |A_{i,k}| < |A_{kk}|$$ Show that if ...
0
votes
3answers
25 views

Composite linear map Rank and Image

I have been pondering on this question, I did part $(a)$ wherein you had to prove that $\operatorname{Im}(T)= \operatorname{Im}(T^{2})$ , but I am struggling to get the concept of part $(b)$, any help ...
1
vote
3answers
39 views

Which of the following does not form a basis for $\mathbb{R}^{3}$

u = $\begin{bmatrix}1\\0\\1\end{bmatrix}$ v = $\begin{bmatrix}0\\1\\1\end{bmatrix}$ Which of the following does not form a basis for $\mathbb{R}^3$ when taken together with u & v? (a) = ...
0
votes
1answer
49 views

Proof: $ker(g) \subset ker(f)$ ..

Let $V = \mathbb{R}_{\le 3} [x]$ with basis $ B = (1, x, x^2, x^3)$. And $f: V \to \mathbb{R}, p \to \int_{-1}^1 p(x) dx$ and $g: V \to \mathbb{R}^3, p \to ^t( p(-1), p(0), p(1) )$. (1) I had to ...