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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1answer
39 views

Diagonalization of a Toeplitz matrix

Let $0<\lambda\leq1$ so that the $n \times n$ matrix $$\Sigma = \begin{pmatrix} 1&1-\lambda& \cdots &1-\lambda\\ 1-\lambda&\ddots&\ddots& \vdots\\ \vdots ...
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1answer
22 views

Question regarding trasposes and norms

I was pondering my book of linear algebra and I found this solution to question 3.2.2 here; but the author of the solution follows a path that I am not sure it is correct... when he takes the ...
1
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1answer
121 views

Prove that $Im(A)+Ker(A)=R^n \iff Ker(A^2)=Ker(A)$

$\def\Im{\operatorname{Im}}\def\Ker{\operatorname{Ker}}$How to prove that for any squared matrix such that $ \Im(A)+\Ker(A)=\mathbb{R}^n$ if and only if $\Ker(A^2)=\Ker(A)$. It is evident to me that ...
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1answer
40 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 ...
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1answer
255 views

Notation for matrix and sum of matrix rows

I have a table that describes the influence of sources (columns) on sinks (rows) where rows=$(A,B,C)$ and columns=$(A,B,C,D,E)$. So my table looks like: ...
3
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1answer
66 views

Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)

Let $\mathbb K$ denote the skew field of quaternions and $A \in M^{n \times n}(\mathbb K)$ and $X\in M^{1\times n}(\mathbb K)$. Let $L_A : \mathbb K^n \to \mathbb K^n$ be defined as $L_A(X) = ...
4
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2answers
95 views

Determinant and trace as conjugations?

For real matrices $A$ it holds that $$\det\,\big(e^A\big)=e^{\mathrm{tr}\,A}$$ so we can write $$\mathrm{tr}=(\exp)^{-1}\circ \;\det\;\circ\;(\exp).$$ Is this interpretation of trace as the ...
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2answers
119 views

Group does not contain any elements of order $p^2$?

I am trying to show that the group of $3 \times 3 $ upper triangular matrices over the field $ \mathbb{F}_p $ with diagonal entries 1 does not contain any elements of order $p^2$ when $ p \geq 3$. ...
0
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1answer
36 views

Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
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3answers
126 views

Finding ker, im, dim of a linear transformation

1Ok, I am a student trying to wrap my head around some of these concepts and need help understanding how to approach some problems. Question: Let $\alpha:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the ...
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0answers
36 views

Computing Integrals over space of Hermitian Matrices.

I am currently working on an example for my research, and I'm getting stuck. Without going over the details of what things mean explicitly, I would like some help computing this integral. I will ...
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1answer
182 views

When is the equation $Ax = b$ solvable in the integers?

Let $A$ be an $m\times n$ matrix with integer entries, $b$ a column-vector with $m$ integer entries. Suppose the equation $Ax = b$ has infinitely many solutions. It is clear that the general ...
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1answer
162 views

can someone help me to prove rank(P A) = rank(A).?

is that correct and we should use the hint but how we use it correctly??
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0answers
36 views

Matrix almost similar to identity?

I wonder if I can find for any matrix $B\in \mathbb{R}^{n\times n}$ a regular matrix $A\in \mathbb{R}^{n\times n}$ that minimises $$ || A^{-1} B A - Id ||_F $$ where $||.||_F$ denotes the frobenius ...
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1answer
234 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
5
votes
1answer
70 views

Eigenvalues of a $4\times 4$ parameters matrix

Let $a,b,c,d\in\Bbb{C}$ and $B =\begin{bmatrix} a & b & c & d\\ d & a & b & c\\ c & d & a & b\\ b & c & d & a\\ \end{bmatrix}$ I ...
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0answers
128 views

what does autocorrelation matrix signifies in image processing?

I am trying to find out corners using auto correlation matrix in an image,what does it signifies?
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0answers
39 views

Find characteristic polynomial

Suppose $A$ and $B$ are $n \times n$ complex matrices such that $$AB-BA=aI+A,$$ where $a \in \mathbb{C}.$ Find the characteristic polynomial of $A$. If $A$ happens to be a Jordan block, this would be ...
0
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1answer
19 views

Matrix norm can assume infinite values?

Given a real-valued matrix $A=a_{i,j} \in M_{n,n}$ When: $||A||_2= \sqrt {\sum_{i=0}^n a^2_{i,j}} < + \infty$ ? Why?
3
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0answers
38 views

what is the significance of Eigen values of autocorrelation matrix?

I am trying to find auto correlation matrix of an image to get Harris corners.Paper I am referring suggest that if eigen values of auto correlation matrix are large the point will be corner point.so ...
1
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1answer
55 views

Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$ Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant. How can write it as in roots ...
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1answer
40 views

Finding the canonical form of a matrix

$$A= \begin{bmatrix} 2 & 0 & -1 \\ -5 & 3 & 3 \\ \end{bmatrix}$$ I have to find two invertible matrices $P(2\times 2)$, $Q(3\times 3)$ such that $P^TAQ$ is a canonical matrix. I know ...
3
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2answers
200 views

Linear algebra calculus trick.

I have a matrix and a vector: $$ A=\begin{bmatrix} a &b\\ c&d \end{bmatrix}, $$ $$ \vec v=\begin{bmatrix} a+b\\ c+d \end{bmatrix} $$ Is there an algebraic operation that produce the ...
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1answer
41 views

Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
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2answers
158 views

Hadamard Matrix

Prove that if $H$ is a (normalized) Hadamard matrix, then so is the matrix $\pmatrix{ H& H\\\ H& -H}$. I have been working on this and I know this statement is true. My book just simply says ...
2
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1answer
71 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
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1answer
103 views

Why diagonal matrix SVD sorted from largest to smallest value?

Why diagonal matrix SVD sorted from largest to smallest value? D is diagonal matrix, $D=(d_1 \ge ,d_2 \ge ,..., \ge d_L)$. Whether there is a journal that could explain this?
0
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1answer
79 views

Matrix inverse and Change of basis

I have 2 Change of Basis Matrices $ S_{A,B} $ and $ S_{A,C}$ I want determinate $ S_{C,B} $ We know that $$ S_{A,B} S_{B,C} = S_{A,C} $$ $$ S_{B,C} = S_{A,B}^{-1} S_{A,C} $$ Now i'm quite not ...
3
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3answers
223 views

Matrices as generators of free group.

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing... In my words: The matrices $\begin{pmatrix}1 & 0 \\ 2 ...
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3answers
60 views

$A$ is not similar to a diagonal matrix over the reals

Let $A = \begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 & -3 \end{bmatrix} $ then $A$ is not similar to a diagonal matrix over the reals and it is not similar to a ...
0
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1answer
205 views

Inverse of a block 2x2 matrix

How to solve this type of problem: We've got a block 2x2 matrix : $$A=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}$$ If matrices $A$ and $A_{22}$ are invertible, show that a ...
0
votes
1answer
46 views

Vector proportional to column of cofactors

Let $A$ be a skew-symmetric $n\times n$ matrix over real numbers with rank $n-1$, and let $v$ be a vector such that $Av=0$. Let $p_{i}$ be the cofactor at position $(i,1)$. Suppose that $p_i> 0$ ...
2
votes
1answer
173 views

Principal ideal ring

Let $K$ be a principal ideal ring. How to prove that for any $ x= (x_1, x_2)^t \in K^2 $ there exists a matrix $G \in SL_2(K)$ such that $Gx = (\gcd(x_1, x_2),0)^t $ ?
4
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2answers
143 views

Do cyclic permutations of rows and column entries generate all permutations?

Background: I am interested in the group of permutations of the entries of a general $m\times n$ matrix. In particular, I am interested in (1) interesting sets of simple generators for this group ...
0
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2answers
123 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
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0answers
79 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
2
votes
2answers
149 views

Prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$

I want to prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$ If $A\in \mathbb{F}^{m\times n}$ and $B\in \mathbb{F}^{n\times m}$ It is easy to show that $0$ has algebraic multiplicity of at least $m-n$ ...
0
votes
1answer
57 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 ...
2
votes
1answer
70 views

A simple question in matrix theory, but I was puzzled:

If matrices $AA^\dagger=BB^\dagger$, then there exists some unitary matrix V, s.t. A=BV. Is it true? If it is, how to prove it?(Here, $\dagger$ denotes transpose conjugate)
1
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1answer
115 views

Show that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite.

Prove that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite. If $A$ is symmetric then there exists an orthogonal matrix $S$, such that $S^TAS$ is a ...
1
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1answer
63 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} ...
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2answers
76 views

How can det(B)=-det(A) when this happens?

There's a property that says when you interchange two rows/columns from a matrix A, the resulting determinant B will have its determinant equal to the original one, but with its sign inversed: ...
2
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2answers
45 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
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1answer
108 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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2answers
64 views

Eigenvectors of a hermitian matrix to the same eigenvalue

Probably, this question has already been answered, but I did not find an answer. If a matrix A is hermitian and an eigenvalue $\lambda$ has multiplicity k, are there always k pairwise orthogonal ...
0
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2answers
68 views

If $A$ a $3 \times 3$ matrix, find $e^A$

Let $$A = \begin{pmatrix} -1 & -1& -1\\ -1 & -1&-1\\-1&-1&-1\end{pmatrix}$$ Find $e^A$. Any ideas?
0
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0answers
52 views

Matix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
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0answers
63 views

Maximum Eigenvalue and a corresponding Eigenvector of an infinite Hilbert matrix

I have the following matrix $$H=\begin{bmatrix} 1 & \frac{1}{2} & \cdots & \mbox{ad}\ +\infty\\ \frac{1}{2} & \frac{1}{3} & \cdots & \mbox{ad}\ +\infty\\ \vdots & \vdots ...
1
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1answer
124 views

If A and B are real orthogonal matrices how to prove that either A-B or A+B is singular?

Degree of matrices is odd $n$-th degree. I figured out all eigenvalues of matrices A and B have to $1$ or $-1$. Now I assume I have to prove $\det((A-B)(A+B)) = 0$ and from that either $\det(A-B)$ or ...
1
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1answer
67 views

Golden-Thompson inequality and Lieb's theorem

On the [Wikipedia article][1] on "matrix exponential", they draw a relation between the Golden-Thompson inequality and Lieb's theorem. My questions are: It mentions that Lieb's thoerem "accomplishes ...