Tagged Questions

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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6 views

A question about the properties of the pseudospectrum

Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by $$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$ ...
2
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1answer
24 views

Inverse 4x4 matrix

If I have a 4x4 matrix (consists of a lot of irrational numbers) and want to calculate the inverse, what is the easiest/fastest way? The calculator I am allowed to use (casio fx991 es plus) can handle ...
1
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0answers
35 views

Rank of a matrix $A$ such that $A + A^T = 0$ [duplicate]

I need to prove (using only elementary operations and induction) that rank of a matrix $A\in \Bbb C^{n\times n}$ such that $A + A^T = 0$ is an even number. I know that elementary operations doesn't ...
0
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1answer
19 views

To find a complex symmetric matrix

I need to find a complex symmetric matrix $A$ such that there is no unitary matrix $P \in U_n(\mathbb{C}) $ with $PAP^{*}$ diagonal. I couldn't find one easily.
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2answers
20 views

Cardinality of a set of matrices

Consider the set $S$ of $3\times3$ matrices with binary coefficients, that is the coefficients are integers modulo 2. Compute $|S|$ I am not sure what is this question trying to ask. Am I right to ...
0
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2answers
47 views

Construct an example matrix such that $\mathbf A \mathbf A^T$ is not invertible

One theorem says: A matrix $\mathbf A \in \mathbb R^{m \times n}$ is: full column rank iif $\mathbf A^T \mathbf A$ is invertible full row rank iif $\mathbf A \mathbf A^T$ is invertible (proof ...
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1answer
16 views

Use row operation to find the determinant?

Use row operations to find the determinant: Can someone give me a full answer please? Also can anyone tell me if the sign of the determinant matters ?
0
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0answers
19 views

Using generalized eigenvectors as a basis for eigendecomposition?

As I understand, the eigenvectors of a Diagonalizable matrix form a basis. Hence, for a diagonalizable matrix A, any vector x can be written as: $x=\alpha_{1}v_{1}+\alpha_{2}v_{2}..+\alpha_{n}v_{n}$ ...
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0answers
14 views

Rational Canonical Form of matrix

I have the following matrix below: $$A=\begin{pmatrix}0 &0 &1 \\1& 0 & 0\\ 0 & 1&0\end{pmatrix}.$$ I have found the characteristic and minimal polynomial to be the same x^3 - ...
1
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1answer
29 views

Linear Transformation and Matrices

I have been studying linear algebra for a while now, and I still can't understand the basic concept of linear transformation and the easy ''translation'' of them the matrices. I understand that every ...
0
votes
1answer
21 views

existence of LU factorization

I am given a matrix $M=E-\alpha X$, where $E$ is an identity matrix, $0<\alpha<1$, and $0 \le X_{i,j} \le 1$ and the sum of every column of $X$ is 1. Does the matrix $M$ always exist a LU ...
0
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1answer
11 views

Interpretation of market completeness: full row rank payoff matrix

Suppose that there are $K$ assets and $S$ states of nature. The assets' payoff is represented by the matrix $$ \underbrace{R}_{S\times K}=\begin{pmatrix} r_{11}&\cdots& r_{K1}\\ ...
1
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0answers
41 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Could you point out some references (undergraduate level) that give a geometric understanding of vector spaces, matrices, and linear applications? As far as I know, many textbooks start with ...
0
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1answer
12 views

Creating a matrix out of a column vector by matrix multiplication

Is there a way to transform a $(n*n) \times 1$ column vector into a $n \times n$ matrix that contains the entries of the vector in its columns using matrix products? Example: I want to transform ...
0
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1answer
24 views

Showing that the eigenvectors (when eigenvalue is 1) can be chosen to be integer valued

Suppose $A$ is an $d \times d$ matrix with integer entries. If there exists $\underline{n} \neq 0 \in \mathbb{R}^d$ such that $(A^T)^k \underline{n}= \underline{n}$. How can you show/justify that ...
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1answer
17 views

Show that if one of the diagonal elements of a symmetric matrix is positive, then the matrix has at least one positive eigenvalue [on hold]

Show that if one of the diagonal elements of a symmetric matrix is positive, then the matrix has at least one positive eigenvalue.
1
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1answer
19 views

$A=(a_{ij})_{m\times n}$ real matrix, $n>m$ then I need to say which of the following are correct statements.

$A=(a_{ij})_{m\times n}$ real matrix, $n>m$ then I need to say which of the following are correct statements. $Ax=0$ has a solution $Ax=0$ has no nonzero solution $Ax=0$ has a nonzero solu ...
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0answers
8 views

Finding an overgroup or a subgroup in PGL

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$. Let $G=PGL_4(k)$. Let $H=\{ \small\left[\begin{array}{cccc} x & ...
0
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0answers
43 views

an interesting matrix [on hold]

I have a tri-diagonal infinite block matrix, as follows: $$S=\begin{pmatrix} 1 & & & &\\ & A_0 & & &\\ & & A_1 & &\\ & & & A_2 &\\ ...
0
votes
1answer
5 views

Show that the entries of the square of diagonal matrix are equal to the square of the entries of the diagonal matrix.

The question seems trivial which is why I have some trouble coming up with a proof that is mathematically correct. BTW I cannot yet use eigenvalues as we have not yet covered them in class. If ...
1
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1answer
26 views

Linear combination of vectors in orthogonal set

If an orthogonal set is linearly independent how can we get the linear combination of these vectors to form another vector that is in the orthogonal set? I thought linear independence meant you cant ...
1
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1answer
34 views

Matrix Norm of a Symmetric Matrix

A paper is claiming that for any matrices $A,B$ of appropriate dimensions, say $n \times n$, then $$ \|A^TA-B^TB\|^2 = (\|Ax\|^2-\|Bx\|^2)^2. $$ where $x$ is the largest eigenvector, $\|x\|=1$, for ...
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0answers
7 views

Matrix transfrom $AXB=P$ to $A'X=P'$

I have a series of linear equations: $P = aS^0+bS^1+cS^2+dS^3$ $a = a_0R^0+a_1R^1+a_2R^2+a_3S^3$ $b = b_0R^0+b_1R^1+b_2R^2+b_3S^3$ $c = c_0R^0+c_1R^1+c_2R^2+c_3S^3$ $d = ...
1
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1answer
26 views

Is it possible to show that the determinant of the following matrix is greater than one?

Is it possible show that the determinant of the following matrix is greater than one? $\det\left(I+AB+CD\right)\geq1$ where $A, B$ and $C$ and $D$ are positive semi-definite.
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3answers
48 views

What is the meaning of “Hermitian”?

Google search-bar gives the definition of Hermitian as: Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal ...
2
votes
1answer
33 views

Weed out numerical artifacts from matrix inversion

I am working with the inverses to a set of large sparse matrices (in Matlab). A key indicator for my application is the number of non-zero entries in each row, and I recently discovered that I was ...
1
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2answers
43 views

sum of three inverse matrices

The following Searle identity computes the sum of two inverses: $A^{-1}+B^{-1} = A^{-1}(A+B)B^{-1}$. Is there any generalisation of this for the sum of three inverses? $A^{-1}+B^{-1}+C^{-1} = ? $
1
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1answer
18 views

Determinant of the Kronecker product involving the identity

Let $A$ be a square matrix and $I$ the $k \times k$ identity matrix. Then the identity $$ \det(A \otimes I) = \det(A)^k,$$ holds as can be seen from a general result on the determinant of block ...
0
votes
4answers
54 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
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0answers
16 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
46 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 ...
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0answers
28 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
25 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
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1answer
29 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
39 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 ...
0
votes
0answers
13 views

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
13 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
2answers
31 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
25 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
21 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 ...
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votes
0answers
23 views

Matrix determinant, eigenvalues [on hold]

"...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
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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
150 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 ...