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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24 views

question related to eigen value of matrix

Let $A$ belong to $M_2(R)$ be a matrix which is not a diagonal matrix. If $A^3 = I$, then why is $\operatorname{tr}(A) = -1$ and $\det(A) = 1$? I am trying to solve it as follows: let $x$ be an ...
0
votes
1answer
333 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 $B$ ...
1
vote
0answers
39 views

Product of two matrices with simple spectrum

We are given two square matrices $A$ and $B$ of the same size over the field of complex numbers and $\epsilon > 0$. Then it can be shown that there exist non-singular (even diagonalizable) ...
0
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1answer
200 views

Regular matrix and regular stochastic matrix

We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the ...
2
votes
1answer
45 views

is some of matrice with it's transpose positive definite? when eigenvalues of matrix is positive

Suppose M = A+ A^T , and we know that all of eigenvalues of A are real and positive, is M positive definite? or semi positive definite?
2
votes
1answer
105 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
3
votes
2answers
181 views

Eigenvectors of the Zero Matrix

Given the following matrix: $ \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix} $. I have to calculate the eigenvalues and eigenvectors for this matrix, and I have calculated that ...
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 ...
0
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1answer
44 views

find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question: let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$ find the characteristic and minimal polynomial of $T$. What I'm trying to do is the following: I ...
7
votes
1answer
334 views

Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$ a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right). $$ The first ten terms are: $0.75$ ...
0
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1answer
40 views

Is this rank $1$ matrix is semidefinite?

I have a matrix, $X = xx^T$, where $x \in \mathbb{R}^n$. Is the matrix $X$ semidefinite?
2
votes
3answers
71 views

Is there any way to check wheter the determinant of a matrix $A$ with $|\text{det }A|=1$ is positive or negative?

Let $A\in\text{GL}(n,\mathbb{R})$ with $|\text{det }A|=1$. Is there any way to check wheter $\text{det }A$ is positive or negative without computing it?
2
votes
4answers
101 views

how to prove that this determinant is a polynomial? [closed]

Given two square matrices $A$ and $B$ of size $n\times n$. I am wondering how to prove that $\det(A+xB)$ is a polynomial function of $x$ ? does anyone has a (simple if possible) proof of this fact? ...
0
votes
1answer
18 views

Find which points are unchanged under this reflection?

I am doing revision and I can't understand how to get the points here. The Question is The matrix $F = \pmatrix{0& -1& 0& 10 \\ -1& 0& 0& 10& \\ 0& 0& 1& ...
5
votes
2answers
107 views

The periodic nature of the fibonacci sequence modulo $m$

Let $x_n$ denote the $n$-th element of the fibonacci sequence and $$A:=\begin{pmatrix} 0&1\\1&1 \end{pmatrix}$$ It's easy to show, that it holds: $$A^n=\begin{pmatrix} ...
1
vote
2answers
66 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
1
vote
1answer
29 views

Matrix equality has a certain solution

I am wondering about the following matrix equality $$ \begin{pmatrix} 0 \\ 1 & \lambda_{1} \\ & 1 & \lambda_{2} \\ && \ddots & \ddots \\ &&& 1 & \lambda_{k} \\ ...
0
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2answers
73 views

$\ker A \subset \ker A^2$? or backwards? $Im~ A \subset Im~ A^2$? or backwards?

Be $n \in \mathbb{N}$. Which of the following statements are true? (i) $\ker A \subset \ker A^2 ~~ \forall A\in \mathbb{R}^{n \times n}$ (ii) $\ker A \supset \ker A^2 ~~ \forall A\in \mathbb{R}^{n ...
5
votes
2answers
100 views

${\rm rank}(BA)={\rm rank}(B)$ if $A \in \mathbb{R}^{n \times n}$ is invertible?

I'm having some trouble with the following question: Let $A, B \in \mathbb{R}^{n \times n}$ and let $A$ be invertible. Is it true that in this case $rank(BA)=rank(B)$? I think that this statement is ...
1
vote
1answer
59 views

Matrices and Complex Numbers [duplicate]

Given this set: $$ S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\} $$ Part I: Why is this set equivalent to the set of all complex numbers a+bi (when both are ...
2
votes
1answer
73 views

Is it possible for a system of equations to have a non-zero determinant and no solution at the same time?

I am quite confused by the solution I was given for the following problems: a) Solve the following system of equations using Gauss elimination only: $2x - y = 5$ $-x + 2y = -4$ $3x - y = -1$ b) ...
0
votes
1answer
76 views

How are these two statements equivalent?

I'm having trouble understanding that if a matrix $A$ is invertible, then it's the same thing as saying the equation $Ax=0$ has only the trivial solution. Any answers are much appreciated.
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1answer
43 views

Orthonormal matrix by diagonal matrix by orthonormal matrix transpose. [closed]

Let $D$ be a diagonal $n \times n$ matrix and $V$ an $n\times n$ matrix whose columns are orthonormal. Is it true that $$VDV^T = D?$$ If so, why?
0
votes
1answer
60 views

Row reducing a matrix with determined pivots through Gauss-Jordan

In Algebra we've been given this matrix: $P=\begin{pmatrix} 3 & 2 & 6 & 10 \\ 8 & 4 & 9 & 5 \\ 7 & 3 & 12 & 4 \end{pmatrix}$ I'm asked to row-reduce it with ...
1
vote
1answer
47 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 ...
0
votes
1answer
41 views

Is the operation associative

Is it known that the multiplication of matrices is a associative operation ? So,is the relation $(A \cdot B) \cdot C=A \cdot (B \cdot C)$ true?? ($A,B,C$ are matrices)
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vote
0answers
160 views

Matrix for orthogonal projection with respect to ordered and canonical bases

Orthogonal projection onto the line $y = 2x$ gives a linear transformation $T: R2 → R2$ such that $$T(1,2) = (1,2)$$ and $$T(−2,1) = (0,0)$$ Then the matrix of T with respect to the ordered basis ...
3
votes
1answer
68 views

What is the maximal torus in the Lorentz group $O(m,n)$?

I'm close to certain it's just the product of the maximal tori of $O(m)$ and $O(n)$, but I can't quite prove it. I've tried the following: ...
1
vote
0answers
107 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix ...
1
vote
0answers
37 views

$A$ matrix is diagonalisable if $\exists S : S^{-1}AS $ is a diagonal matrix, how can I find S?

Per definition a matrix $A$ is diagonalisable if there exists a matrix S such that $S^{-1}AS$ is a diagonal matrix. My question is how do I find the matrix $S$? Is it always the combination of the ...
1
vote
1answer
70 views

matrix representation of linear transformation

For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis ...
4
votes
1answer
170 views

Can these two matrices be represented as diagonal matrices with respect to an orthonormal basis?

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 $A$, ...
2
votes
1answer
50 views

Is this a theorem (is it correct?)?

My instruction notes have specified a theorem of matrix transpose that be there two compatible matrices $A$ and $B$ in respect of their sums and products, then: $(AB)^T =A^TB^T$ So I set on to ...
0
votes
0answers
63 views

Matrix Induction proofs using $\rm mod$.

Please could someone advise me on where to even start with this question. Thanks in advance Let $A$ be an $n\times n$ matrix, for some $n\geqslant 1$. Given any prime number $p\gt1$, write $A_p$ for ...
0
votes
0answers
186 views

Looking for a simple algorithm to scale/resize a matrix, or an image.

I am looking for a simple algorithm to scale a matrix of any size. Given matrix A of dimensions [w1,h1]. Given a scaling factor (or resizing factor) SF, which is a real number (not necessarily ...
0
votes
0answers
47 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
4
votes
1answer
92 views

Left and right eigenvectors perpendicular to each other

I just read in a textbook on numerical methods that you can always have that the right eigenvectors of a matrix can be taken as orthonormal to the left eigenvectors for a diagonalisable matrix. This ...
1
vote
2answers
436 views

Proof of Cayley-Hamilton Theorem for Diagonalisable Matrices [Lay P326 Ch 5 Sup Q7]

Proof for Diagonal Matrices from Page 2 of 7: Let $A \in M_{n}(C)$ be diagonal, to wit, $A _{ii}=\lambda_{i}$. Then $ p_{A}(t) = \det(tI-A)= \det \begin{bmatrix} t - \lambda_1 & ~ & ~ \\ ...
2
votes
3answers
594 views

$\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$

How can I prove that $\operatorname{adj}(AB) = \operatorname{adj} B \operatorname{adj} A$, if $A$ and $B$ are any two $n\times n$-matrices. Here, $\operatorname{adj} A$ means the adjugate of the ...
2
votes
2answers
314 views

When Dim eigenspace = 1, any $2\times 2$ complex matrix A is similar to $\left(\begin{array}{ll} \lambda & 1\\ 0 & \lambda \end{array}\right)$.

$\bbox[5px,border:2px solid gray]{ \text{ Case 3 } }$ If $\dim E_{\lambda}=1$, take a nonzero $v\in E_{\lambda}$, then $\{v\}$ is a basis for $E_{\lambda}$. Extend this to a basis $\mathfrak{B}=\{v,\ ...
3
votes
2answers
80 views

$X$, find $A$ such that $A^m=X$

I encountered a problem as folows: Show a $3\times 3$ real matrx $A$, such that $$A^4=\left(\begin{array}{ccc}3&0&0\\0&3&1\\0&0&0\end{array}\right)$$ well, this problem ...
0
votes
1answer
90 views

finding an inner product so that matrix is self-adjoint

Given the endomorphisms $$A= \left( \begin{matrix} 1 & 2i & 3i\\ 0 & 2 & i\\ 0 & 0 & 3 \end{matrix} \right) \in Mat(3 \times 3, \mathbb C), $$ and $$B= \left( ...
0
votes
1answer
51 views

How to find a upper triangular matrix similar to A.

Given a matrix A how do you find a matrix Q such that Q inverse A Q is upper traingular, in complex. I know that it is possible and the proof uses a induction argument but i don't see how to find such ...
1
vote
2answers
87 views

b such that Ax = b has no solution having found column space

$A:=\begin{bmatrix} 2 & 6 & 0 \\ 3 & 1 & 3 \\ 1 & 0 & 0 \\ 4 & 8 & 1 \end{bmatrix}$ I've found the basis for the column space by doing row reduction (i.e. ...
0
votes
1answer
79 views

Nonsingular Z-Matrix <==> Nonsingular M-Matrix?

Here I'm considering only M-matrices that are also Z-matrices, so all the off diagonal elements are negative in all matrices I consider. If I have a Z-Matrix with real, positive eigenvalues, is it ...
2
votes
1answer
54 views

Find a Hermitian Matrix

We are given two column matrices A and B. Can we find a Hermitian matrix $H$ such that $ A_{4 \times 1} = H_{4 \times 4} B_{4\times 1} $ ? We tried to solve it by multiplying a $1\times4$ row matrix ...
5
votes
4answers
207 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
1
vote
0answers
51 views

iterates of generalized matrix system

This may be somewhat of an underspecified question but I'll nonetheless give it a try. In the context of applied work, I've recently come across systems of the form $$ ...
1
vote
0answers
101 views

About Jordan-Chevalley decomposition

I have this problem: Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$. I saw that this ...
1
vote
0answers
27 views

Which is the starting and ending basis? - Matrix of a linear transformation [Lay P294 Q 5.4.28]

Denote some arbitrary linear transformation as $L.$ When a question asks "to find a matrix of $L$ with respect to S and T", does this denote $[L]_{T \leftarrow S}$ or $[L]_{S \leftarrow T}$ ? How can ...