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
4answers
56 views

Finding a matrix U such that B=UA.

A= \begin{bmatrix} 1 & 0 & 1 \\ 2 & 3 & 1 \\ \end{bmatrix}B= \begin{bmatrix} 1 & 3 & 0 \\ 4 & 3 & 3 \\ \end{bmatrix} How does one go about solving this problem? ...
0
votes
0answers
18 views

Norm Calculation Problem

I received this problem on an old homework assignment as extra credit, the period for getting credit is long passed but I'm frustrated that I can't even seem to know where to begin this problem. Let ...
0
votes
2answers
14 views

Solve the following matrix equation

What is the simplest way to solve such an equation? $\left[\begin{array}{cccc}0&1&2&3 \\ 1&2&0&1\end{array}\right] \cdot \left[\begin{array}{c}x \\ y \\z \\ ...
0
votes
2answers
17 views

Find the solution of binary xor operator equation

I am working in binary xor operator $\mathbb Z_2$. I have to resolve my problem such as $$\begin {cases} x_1+x_2+x_3=1\\ x_1+x_2=0\\ x_1+x_3=1\\ \end {cases}$$ Could you suggest to me any method to ...
0
votes
2answers
52 views

Lights Out with custom rules set

I'm trying to understand how to use linear algebra to solve a custom Lights Out puzzle with the following rules: There are 8 lights, all the lights are off at the starting point, I need to turn on ...
0
votes
0answers
31 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
0
votes
2answers
27 views

$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?

$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?
3
votes
1answer
32 views

Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,…,\lambda_n$ are the eigenvalues of $A$.

Title restated: Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A^*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$. This question comes from "Matrices ...
1
vote
1answer
44 views

Determinant of a 2nd rank tensor help and inverse!

I have the following 3x3 matrix $$U_{ij} = g_{ij} + \epsilon_{ijk}u_k$$ and I want to find its inverse using the fact that it can be written as the linear combination of its symmetric part and its ...
0
votes
2answers
29 views

Prove $ A^-=\dfrac{1}{4}(-A^2+4A+I)$

Let $$ A=\begin{bmatrix} 1 & 1 & 2\\ 1 & 2 & 1\\ 2 & 1 & 1 \end{bmatrix}$$ Show that $ A^-=\dfrac{1}{4}(-A^2+4A+I)$ I have absolutely no clue how to do this. Could someone be ...
2
votes
0answers
30 views
+100

Covariance of $Z'Vb$ given that the rows of V are i.i.d.

Suppose that we have the following entities $$ \underbrace{Z}_{n\times k},\quad\underbrace{V}_{n\times L},\quad \underbrace{b}_{L\times 1}. $$ $Z$ and $b$ are nonstochastic whereas we assume that the ...
1
vote
1answer
18 views

How to solve this matrix for h and k?

I am going through my mathproblems, to check up on what was done during class. The TA had us solve this augmented matrix, but during awnsering he mixed up the h and k. So my awnser is incomplete, and ...
0
votes
3answers
36 views

How do I define this matrix in matlab?

$$A = \begin{pmatrix} 2 &-1 & 0 & \dots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & \ddots & \ddots & \ddots & 0 \\ \vdots & \ddots & -1 & 2 ...
0
votes
2answers
32 views

Which of the three matrices will the powers remain bounded?

Let $A = \begin{pmatrix} 2 & 1\\ -1& 0\end{pmatrix},$ $B = \begin{pmatrix} 0 & 1\\ -1& 0\end{pmatrix},$ $C= \begin{pmatrix} 1.98 & .99\\ -.99 & 0\end{pmatrix}$ Consider the ...
0
votes
2answers
31 views

Determine matrix of linear transformation

Let $T:R^2\rightarrow R^2$ by $$ T \left( \begin{bmatrix} x_{1} \\ x_{2}\end{bmatrix} \right) = \begin{bmatrix} x_{2} \\ x_{1}\end{bmatrix} $$ Let A be the matrix of T. What is A. I'm having trouble ...
2
votes
0answers
29 views

Eigenvalues of 5x5 matrix given equation involving matrix

I have been given the matrix $A$ and we are told it is a $5\times 5$ matrix s.t. $A^4=A^2\neq A$. I want to find the eigenvalues so I tried $A^2(A-I)(A+I)=0$ so the eigenvalues are $0, 1, -1$ but I ...
0
votes
1answer
23 views

Optimization problem $L(R, PQ) \rightarrow \min$

Suppose we have some $n \times m$ matrix $R$ and we want to find non-negative decomposition on matrices $P$ of dimension $n \times d$ and $d \times m$-matrix $Q$. But since exact decomposition usually ...
-1
votes
0answers
23 views

Matrix determinant, eigenvalues [closed]

"...has the determinant $x*y*(1-ab)$. Since $L<bK<b(aL)$, $1-ab<0$ and the fixed point is a saddle. So I know that for the fixed point to be a saddle, one eigenvalue must be positive and one ...
3
votes
1answer
28 views

What is the limit of the rank of the power of a matrix?

The problem is about $rank (\mathbf{A}^k)$ when $k \rightarrow \infty$ for a $n\times n$ matrix $\mathbf{A}$. I know that for a nilpotent matrix, $\mathbf{A}^k=0$ when $k$ is big enough, which means ...
0
votes
1answer
15 views

The number of symmetric matrices of order 5 with each element either 0 or 1

Question is to find The number of symmetric matrices of order 5 with each element either 0 or 1 . What i am trying is If i take matrix of order 2 $$A=\left[\matrix{ A & B \\ B & C \ ...
0
votes
0answers
21 views

Degenerate eigenvalue and minimal polynomial

I'm learning about elementary linear algebra and I am confused on a specific point related to minimal polynomial. When we have non degenerate eigenvalues it is just equal to the characteristic ...
0
votes
2answers
17 views

Can we say that the columns of the given matrix always lies in its range space

Can we say that the columns of the given matrix always lies in its range space. For example, suppose we have a square matrix $A$ of order $n\times n$ then can we claim that its columns say $c_1, c_2 ...
1
vote
4answers
159 views

Given a matrix $A$ . Calculate $A^{50}$

I have given matrix with me as follows . I need to calculate $A^{50}$ . Hint is to diagonaize it ,but since it has repeated eigen values so can't be diagonalized.Can any1 help me with this ...
1
vote
0answers
5 views

Number of scalar addition required to compute P(QR) ,where P,Q,R be matrices or order $3× 5,5×7,7×3$

Firstly i don't understand what is meant by scalar additions here ? This is my main concern here ,computation comes after .If anyon can help it will be great
1
vote
0answers
52 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
1
vote
2answers
33 views

Are all symmetric and skew-symmetric matrices diagonalizable?

I know by theorem that every hermitian and skew-hermitian matices is similar to diagonal matrices. But, is this fact also true for symmetric and skew-symmetric matrices? And, Symmetric matrices ...
0
votes
0answers
16 views

What's the rank of a matrix that has constant number ones in each col/row over $F_2$

Let $A$ denote a $n\times n$ matrix over $F_2$, which means $A \in \{0,1\}^{n\times n}$. Also assume that each row and each column only has exactly 3 ones. 1) What is the upper bound and lower bound ...
2
votes
1answer
38 views

How to decompose a matrix into an antisymmetric matrix plus a multiple of the identity

I was given a problem to solve earlier that I couldn't figure out. I don't still have it, but it was basically: Given the invertible matrix $A$, find the invertible matrix $P$, such that ...
3
votes
1answer
48 views

Bilinear functional to be elementary

Let $V$ be a $n$ dimensional vector space over $\Bbb C$. We say a functional $f:V\times V\to \Bbb C$ is bilinear if $f$ is linear in each variable if the other variable is fixed. And $$f$$ is called ...
0
votes
3answers
29 views

A quick way to generate 3x3 matrices with determinant equal to 1?

Perhaps a formula involving the row number and column number of an element or just some parametric equations for each element. I know that I can just multiply two of these matrices together to get ...
6
votes
2answers
84 views

Proving if A and B are matrices such that A, B, and AB are normal, then BA is also normal.

If A and B are matrices such that A, B, and AB are normal, then BA is also normal. I've seen this statement around, although I've only seen the site/publication/etc... state that it was proven by ...
-1
votes
4answers
47 views

True or false? Prove it.

If $A$ is an $n\times n$ invertible matrix and $B$ is an $n\times m$ matrix, then $\operatorname{rank}(AB) = \operatorname{rank}(B)$. Is this true or false? I've tried proven that if $B=0$, then ...
0
votes
0answers
19 views

Find the values of x,y,z so that the 3 x 3 matrix is singular?

Find the values of x, y, z that the matrix is singular? With an explanation.
2
votes
2answers
29 views

Use the cayley hamilton theorum to work out high powers of matrices

Let Matrix $$A= \left( \begin{array}{ccc} 1 & 2& 3 \\ 0 & 1 & 0 \\ 0 & 5 & -1 \end{array} \right) $$ Compute $A^{25}$ using the cayley hamilton theorum I know i use ...
0
votes
3answers
25 views

Determine if a set is linearly independent or dependent.

If $S = \{r,u,d\}$ and $S$ is a set of linearly independent vectors. and if $x = r + 4u + 2d$, determine whether $T = \{r,u,x\}$ is a linearly independent set as well. Not sure how to go about solving ...
0
votes
1answer
13 views

Why use transpose in finding if a subset is also a subspace.

I had a homework question in my linear algebra course that asks: Are the symmetric 3x3 matrices a subspace of R^3x3? The answer goes on to prove that if A^t = A and B^t = B then (A+B)^t = A^t + B^t ...
-1
votes
0answers
21 views

Upper bound on Lyapunov equation solution

We know from literature on the Lyapunov equation that there exists a unique, symmetric, positive definite solution for the matrix $P$ in the equation $$A'P+PA=-I,$$ where $A$ is a Hurwitz matrix. Is ...
2
votes
1answer
48 views

Embed $1$-dimensional torus in $SO(2)$

Let $k$ be an algebraically closed field, and let $k^*$ be the one dimesional torus. We want to embed it in $SO(2)$ , the group of matrices $A$ such that $\det A=1$ and $A^tA=Id$. My first attempt ...
0
votes
1answer
32 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
0
votes
0answers
24 views

Quadratic Sieve, matrix problem

I read this: Quadratic Sieve Matrix Reduction and I am basically stuck. My Gaussian elimination says the answer is v= 0,0,0. Although you can clearly see that the correct answer is (1,1,1). How does ...
1
vote
3answers
197 views

Sum of eigenvalues of a symmetric matrix

Problem to calculate the sum of eigenvalues of a matrix: $$ \begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \\ \end{pmatrix}$$ I can ...
0
votes
1answer
30 views

Norm of Triangle Matrix

How to find the norm of the following matrix, please? Thank you! $$T := \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix},$$ and $$\|T\| = \sqrt{n^2+1}.$$
1
vote
2answers
43 views

Prove the matrix $ \left( \begin{array}{ccc} B & A^T \\ A & 0 \\ \end{array} \right)\ $ is nonsingular [closed]

Suppose the matrix $A\in\mathbb{R}^{m\times n}$, $m\leq n$, and has full row rank $m$, $B\in\mathbb{R}^{n\times n}$ is a symmetric, $Z\in\mathbb{R}^{n\times(n-m)}$ is the matrix whose columns span ...
1
vote
2answers
24 views

Finding the $\lim_{n\to\infty}(A^{n})$ of a complex matrix

Let $$ A = .5\begin{pmatrix} 1+\alpha & -1+\alpha\\ -1+\alpha & 1+\alpha\end{pmatrix}. $$ where $\alpha$ is a complex number. For which values of alpha does the limit ...
0
votes
1answer
15 views

$A\in\mathcal{S}^+_n,A={}^tMM$

How can I prove that any symmetric positive matrix $A\in\mathcal{S}^+_n(\mathbb{R})$ can be written $A={}^tMM$ where $M$ is an invertible matrix ? This is probably a duplicate, but I have not been ...
0
votes
5answers
100 views

Determine $\lim_{n\to\infty}A^{n}$

For matrix $$ A = \begin{pmatrix} 7/5 & 1/5\\ -1 & 1/2\\\end{pmatrix}. $$ Determine $\lim\limits_{n\to\infty} A^{n} $ Is the limit related to the eigenvalues? Using Matlab it appears that ...
2
votes
1answer
32 views

How to take the derivative of Matrices

I was browsing the derivation of the Least Squares estimates and stumbled about this problem. It said that: $$E = (Y + XB)^2$$ $$\frac{dE}{dB} = -X^TY + X^TXB$$ It is to my understanding that the ...
0
votes
3answers
59 views

Unit Eigenvalue if Determinant of an Orthogonal matrix is 1 [closed]

For a (2n+1)x(2n+1) orthogonal matrix M, det(M)=1. Show M has a unit eigenvalue.
0
votes
3answers
77 views

Can product of two singular matrices be invertible?

Suppose $A,B$ are square matrices of size $n\times n$. Can $AB$ be invertible, even though both $A$ and $B$ are singular (not invertible)? And if not, does it follow that if $A_1 \times A_2 \times ...
0
votes
0answers
9 views

Representing the sum of squared diagonal elements of a matrix with trace function

Assume that we have a $n\times n$ vector called $A$. I am interested in computing $\sum_{i=1}^n A_{ii}^2$ (i.e., sum of the squared diagonal elements). However, I want to do so as trace function and ...