0
votes
1answer
28 views

Diagonalization of Skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. and we have the relation $C=UDU^{-1} ...
0
votes
0answers
37 views

proof that T^k is a positive operator

so the book (Axler linear algebra done right) asks me to prove that if $T$ is a positive operator then $T^k$ is also positive , now the book defines a positive operator as an operator which is self ...
0
votes
1answer
21 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
5
votes
1answer
50 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
1
vote
3answers
47 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
0
votes
0answers
5 views

Transform gradient to reference element

Minimal example of the problem My attempt I think this is not a linear solution like \begin{equation} \nabla u = \nabla A_K x + \nabla b_K \end{equation} which must be wrong because $A_K$ is a ...
1
vote
1answer
27 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
-1
votes
2answers
48 views

Matrix with eigen values given find [closed]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
0
votes
2answers
34 views

Matrices Problem

I am doing the Cambridge O/L 2012 M/J P1 4024/12 Paper, Question number 12 (b). $$m = \begin{pmatrix} 3 \\ -2 \end{pmatrix}, \quad n = \begin{pmatrix} -1 \\ 4 \end{pmatrix} $$ Given that $$sm+3n = ...
0
votes
1answer
42 views

Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.
0
votes
0answers
34 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
0
votes
1answer
23 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
0
votes
2answers
21 views

Identify Orthogonal Proj. and Reflection within given choice of Matrices.

The problem states that out of five given matrices, one represents an Orthogonal Projection onto a line and another a Reflection about a line; I'm supposed to identify them. Rather than list the ...
4
votes
2answers
327 views

Eigenvalues and Eigenspace Question

Thank you ahead of time for the help, I am having a problem with part $4$. I understand parts $1$ and $2$ and $3$ and have solved them but I cant seem to understand $4$. If someone could help me out, ...
1
vote
1answer
36 views

Are these two matrices equivalent?

I am supposed to row reduce a matrix to reduced row echelon form. $$ \begin{bmatrix} 1 & 2 & 4 & 8\\ 0 & 0 & 1 & 4\\ 0 & 0 & 0 & 0 \end{bmatrix} $$ I have tried ...
6
votes
1answer
99 views

Existence of $p \times p $ matrices $A$ and $B$ over the field $\mathbb F_p$, $p$ prime, such that $AB-BA=I$. [duplicate]

Let $p$ be a prime number. Prove or disprove that there exists $p\times p$ matrices $A$ and $B$ over a field $\mathbb F_p$ with $AB-BA = I$. With the aid of MAPLE i was able to find out that ...
0
votes
1answer
13 views

Question about Joint spectral radius.

Given a bounded set $\mathcal A\subset \Bbb R^{n x n}$. The joint spectral radius is given by: $\sigma(\mathcal A)$=$limsup_{m\to\infty}(sup_{A\in\mathcal A^m} \rho(A))$ where $\rho$ is the normal ...
0
votes
1answer
35 views

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$?

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$? For example, Can I say that $M^R_{2x2}$ is a subspace of $M^R_{2x3}$ so it can be isomorphic to $R_4[x]$ ? (because they have ...
0
votes
1answer
28 views

Gauss-Jordan Method

I keep getting the wrong set of solutions can someone help me. I know that when using the Gauss-Jordan method, the rules that I must follow can be applied in a variety of different procedures then why ...
1
vote
1answer
45 views

Matrix with unknown coefficients, finding another basis

let $(e_1,e_2,...,e_5)$ canonical basis of $R^5$, $V=(a,b,c,d,e)\in R^5$ with $V\neq(0,0,0,0,0)$. we consider $f:R^5\to R^5$ and its matrix : $$Mat(f) = M= \begin{pmatrix} ...
2
votes
2answers
40 views

M is real anti-symmetric matrix, prove that exp(M) is isometry

M is nxn real anti-symmetric matrix.I need to prove that exp(M) is isometry. Could anyone give me any hint , I don't have any approach to this question. thank you
1
vote
1answer
32 views

What's the connection between rank of matrix and $0$ eigenvalue?

My matrix B is nxn and know nothing about if diagonalizble, but I know that rank B = 1. Therefore the geometric multiplicity of λ=0 as an eigenvalue is n-1. But by knowing the rank is 1, can I say ...
3
votes
3answers
46 views

$A$ matrix, $+i, -i$ are eigenvalues.

If matrix $A$ is a square matrix. And $A$'s characteristic polynomial is $p(t) = t^2 + 1$. It's not necessary true that $A$ is non-singular right? Because, the eigenvalues are $i,-i$. and if $\dim(A)= ...
1
vote
0answers
25 views

Ax = B, group of columns of A.

Is there a matrix $A (n \times n)$ Over field $F$ and $b \in F^n$ has non-trivial solution to the equation $Ax=B $ ? Well, In the answer it is written that because that the set of the columns of ...
0
votes
0answers
12 views

Interpreting & Analysing a Transitional Matrix

How do you interpret such a problem Are we expect to add the rows, and that would be the one with larger number of goats in the long term. Therefore A(row 1) and b(row 2)... therefore the answer is ...
2
votes
1answer
23 views

If the first r columns of U are linearly independent, then so are the first r columns of A?

Let $U$ be a row echelon form of a square matrix $A$. If the first $r$ columns of $U$ are linearly independent, then should the first $r$ columns of $A$ be linearly independent? In my opinion, "Yes" ...
3
votes
1answer
22 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
1
vote
1answer
47 views

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$

How can I find $T^{-1}(x,y,z)$ (inverted matrix) of a linear operator $T:V_3 \to V_3$, which matrix relative to the basis: $A=\{ (1,0,0), (1,1,0), (1,1,1)\}$ is: $$T_A= \begin{bmatrix} 2 &0 ...
3
votes
6answers
382 views

How can I prove that a square matrix is invertible if it satisfies this polynomial equation?

For a 3x3 matrix $C$, it is given that $$C^3+I=3C^2-C$$ I am then required to prove that $C$ is invertible. I have attempted a proof, below, but I am not sure it is valid or if there is a better ...
1
vote
1answer
20 views

The similarity of the block matrices

Let $\mathbb{F}$ be a field,and let $A,B,C$ be matrices over $\mathbb{F}$ of respective sizes $n\times n , k\times k, $and $n\times k$. put $M=\begin{bmatrix} A&0 \\ 0&B ...
-1
votes
2answers
51 views

A question on Rank and trace of a special matrix [closed]

I want to share the following question which was asked in a competitive exam: For a fixed positive integer $n\geq 3$, let $A$ be the $n\times n$ matrix defined by $A=I-\dfrac{1}{n}J$, where $J$ is ...
2
votes
1answer
77 views

Some questions on Nilpotent matrix [closed]

Q & A style. Just wanted to share the following question which came in a competitive exam and so college level maths students may find it useful: A non-zero matrix $A\in M_n(\mathbb{R})$ is said ...
0
votes
1answer
41 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
0
votes
0answers
19 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
0
votes
1answer
47 views

Diagonalizable A, computing fast.

I have $A =$ $ \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix} $ I know that this matrix $A$ is diagonalizable when ...
1
vote
2answers
17 views

Scalar product, undefined matrix multiplications and order of operations

I am curious about the order of operations in linear algebra. Let $\mathbf{u},\mathbf{v} \in \mathbb{R}^{n}$ and $\mathbf{A},\mathbf{B}\in \mathbb{R}^{m \times m}$, with $m\neq n$ and $\mathbf{u} ...
0
votes
1answer
56 views

Tips on how I would find the transition matrix for the following phenomenon?

how would I go about finding the transition matrix for the following phenomenon (which can be modeled as a Markov process)? Any hints or advice is appreciated! During a study break, a student's ...
1
vote
4answers
369 views

Linear algebra, power of matrices

$P^{-1}AP = \begin{pmatrix} -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 2 \\ \end{pmatrix} $ with $P= ...
1
vote
2answers
35 views

Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
0
votes
0answers
40 views

Division by zero while generating the matrix

I got the formula and I need to create the lineral-equation system matrix using it and then solve it. I decided to use wolfram mathematica to do this. I wrote the code. And when N and M are $> ...
3
votes
1answer
90 views

Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by ...
1
vote
1answer
81 views

For which $x,y,z,w$ is matrix $A$ orthogonal/unitary?

given is $A = \frac{1}{2} \begin{pmatrix} x & 1 & 1 & 1 \\ y & 1 & -1 & 1 \\ z & 1 & -1 & -1 \\w & 1 & 1 & -1 \end{pmatrix} $ How do I have to chose ...
0
votes
1answer
40 views

Solve Matrix: How many trips can be made?

We have the matrix: $$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$ The matrix ...
12
votes
5answers
273 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
1
vote
2answers
45 views

Prove that if $B=P^{-1}AP$, then $q(B)=P^{-1}q(A)P$

Is it possible to prove this using a similarity invariant? For example showing that $$\det(q(B))=\det(q(A))$$
0
votes
1answer
47 views

Finding the representing matrix with respect to the standard basis.

Let $B=[(1,0,0),(1,2,0),(1,2,3)]$ be the basis for $\mathbb{R}^3$. Let $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear transformation, such that its representing matrix with respect to basis $B$ ...
0
votes
1answer
29 views

what does this question about a matrix mean?

here is a question says : what does that mean ? I did my best to solve this question myself but i didn't find a way to solve it is this question possible or there is something else that i don't ...
1
vote
1answer
45 views

Find the last column of a matrix. Find the matrix.

$A\left[\begin{matrix} 1 & 0 \\ 0 & 0 \\ 1 & 1 \end{matrix} \right]$ = $\left[\begin{matrix} 2 & 3 \\ -1 & 0 \\ 5 & -7 \\ 0 & 6 \end{matrix} \right]$ (1)Find the last ...
4
votes
0answers
81 views

Diagonalization

I'm having difficulty understand some questions. I will highlight the terms I do not understand. Question 1: Let $A =\begin{pmatrix} 1& -2 \\ 1& 3 \end{pmatrix}$ For the matrix $A$, ...
0
votes
0answers
21 views

A question on companion matrix

Let $A^*$ be the companion matrix of matrix $A$, where $A$ is $n\times n$ type. Then $$|-A^*|=(-1)^{n}|A^*|.$$ Is it right? I'm not sure. Thanks for your help! Thanks ahead:)