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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18 views

Generate duplicate element from a matrix by formula $b(i:j)=A(i:j,:) \times A^{-1} \times b$

I have an interesting question about generate duplicate elements from matrix. I assume that I have a matrix A (such as the bellow example $5 \times 5$) and vector $b$ is $5 \times 1$. My goal is make ...
2
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0answers
80 views

Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is equal to $\dim(\text{null}(T-\lambda I)^{\dim V})$

Without using induction, prove that the the algebraic multiplicity of an eigenvalue $\lambda$ is $$\dim (\text{null} (T-\lambda I)^{\dim V});$$ here, the algebraic multiplicity of an eigenvalue ...
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2answers
39 views

Matrix addition/multiplication with different sizes

I have the following two matrices: $$A=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ and }B=\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$ So I have two matrixes with ...
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0answers
40 views

How to calculate the inversion of a triangular matrix

Now I want to write a piece of code to calculate the inversion of a triangular matrix which do it in parallel. I know that the equation of the triangular matrix's inversion is like this: But I ...
1
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1answer
58 views

Properties shared by similar and unitary similar matrices.

We know that matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A=PBP^{-1}$ and they are unitarily similar if $P$ is unitary ($PP^*=P^*P=I$). I want to know : What ...
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1answer
28 views

If $A $ is a square matrix of size $n$ with complex entries such that $Tr(A^k)=0 , \forall k \ge 1$ , then is it true that $A$ is nilpotent ? [duplicate]

If $A$ is a square matrix of size $n$ with complex entries and is nilpotent , then I can show that all the eigenvalues of $A^k$ , for any $k$ , is $0$ , so $Tr(A^k)=0 , \forall k \ge 1$ . Now ...
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0answers
13 views

Extra Operation required for Smith Normal Form over PID-Theoretical Justification

Why does one need an extra operation for performing smith normal form over a PID? One might suspect and say that it is because of the lack of Euclidean algorithm or just say that we need the ...
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0answers
33 views

Inverse Matrix in 3D

Suppose we have the following matrix in three dimensions $$ M_{ij} = g_{ij} + e_{ijk}z^{k} $$ where $e_{ijk}$ is an antisymmetric density, i.e. $e_{ijk} = \sqrt{\det g}\cdot\epsilon_{ijk}$ and $z^{k}$ ...
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1answer
26 views

Square coefficient matrix, matrix transpose, and solvability of the corresponding system of equations

Let $\mathbb{F}$ be a field and $n \geq 2$. I would like to prove that, for every $n \times n$ matrix $A$ over $\mathbb{F}$, there is a $b \in \mathbb{F}^{n}$ such that $Ax = b$ is unsolvable if and ...
0
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1answer
33 views

Exist a eigenvector $v=\begin{bmatrix} x \\ y \end{bmatrix}$ such that $x,y >0$

Problem: Let $\begin{bmatrix} a &b \\ c&d \end{bmatrix}$ is a real $2 \times 2$ matrix such that $a,b,c,d>0$. Prove that exist a eigenvector $v=\begin{bmatrix} x \\ y \end{bmatrix}$ ...
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0answers
21 views

Prove a matrix is Hermitian, if its eigenvalues are real and satisfy an orthogonality relation

Prove a matrix is Hermitian, if: (a) Its eigenvalues are real, and (b) the eigenvectors satisfy $ r_{i}^\dagger r_{j} = \delta_{ij} = \left<r_{i}|r_{j}\right> $ I can see this is the reverse ...
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1answer
32 views

Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$?

$A$ is an $N\times N$ matrix with bounded row and column norms. Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$? I know this is true for ...
0
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0answers
18 views

Prove that minimum of the matrix norm is achieved at certain parametres

Given matrix $A\in R^{n\times m}$ prove that minimum of the $||A-xy^T||$, $||B||=tr(B^TB)$, is achieved when $x$ is an eigenvector of $AA^T$, corresponding to its greatest eigenvalue, and $y$ is an ...
0
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0answers
31 views

Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S How would one go about working S^-1/2 (inverse of the principle square)? Bearing in mind, I'm trying to understand a paper which states: I've ...
0
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2answers
36 views

Prove a matrix is not diagonalizable

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix. So, for example, if I want to prove that $$A=\begin{bmatrix} 0 & -1 ...
0
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1answer
69 views

How to prove that the matrix $A^k$ approaches $0$ as $k$ approaches infinity

First of all, what does it mean to say an eigenvalue is "less than unity"? I'm not exactly sure what this means. Secondly, how do I show that $\lim_{k\to\infty} A^k=0$ given that all eigenvalues of ...
0
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1answer
22 views

Eigenvalue Bound of Block Matrices

I have the following eigenvalue problem for block matrices A and B \begin{equation} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & ...
3
votes
2answers
48 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
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1answer
19 views

2D rotation matrix: express sin and cos in terms of the elements and the norm of an arbitrary non-zero vector

2D rotation matrix is used to derive the expressions for sin and cos in terms of $a_1 ,a_2 and ||\vec{a}|| $ with the following given I'm trying to figure out where the negative-sign comes from in ...
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0answers
23 views

Matrixproduct of A'A expressed as a sum

I have difficulties in proving (understanding, seeing) the following identity: $ \mathbf{A'A} = \mathbf{(a_1, a_2, ...,a_n)} \begin{pmatrix}\mathbf{a_1'\\a_2'\\ \vdots \\ a_n'}\end{pmatrix} = \sum ...
2
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2answers
55 views

Showing a linear combination of matrices is nilpotent for any constants

So I have three linear operators in a $3$-dimensional vector space $V$ over field $\Bbb k$ whose matrices w.r.t basis of $V$ are $$X= \left(\begin{matrix}1 & 0 & 1\\ 1 & 0 & 1\\ -2 ...
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1answer
48 views

If two invertible matrices A and B commute, then A^-1 and B^-1 must commute as well ??

If two invertible matrices A and B commute, so their inverse must commute as well or not ?
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1answer
24 views

Isomorphisms: $(Aut(V), \circ) \to (GL(n, \mathbb{R}), \cdot)$ and $(Or(V), \circ) \to (O(n), \cdot)$

Let $V$ be a $n$-dimensional $\mathbb{R}$-vector space. Let $Aut(V)$ be the set of the automorphism on $V$. I have shown that this is a group with respect to the composition of functions. However, I ...
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0answers
45 views

Characterizations of positive definiteness of a symmetric matrix of order $2$

Let $$M=\pmatrix{ a& b \\b&c }$$ be a symmetric matrix. In my textbook the following result is stated without proof, but I would like to know why it holds, but I cannot figure out what to do ...
0
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2answers
28 views

Some doubts and questions with the trace of a matrix

Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$. I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a formal proof?). ...
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0answers
24 views

Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...
0
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0answers
17 views

understanding relative matrices

$Let \:T:\:\mathbb{R}^4\rightarrow \mathbb{R}^2,\:T\left(x_1,\:x_2,\:x_3,\:x_4\right)=\left(x_1+x_2+x_3+x_{4\:},\:x_4-x_1\right)$ $v\:=\:\left(4,\:-3,\:1,\:1\right)$ ...
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votes
2answers
24 views

Convergence of a sequence of matrices

Let $A$ be a $n×m$ matrix with real entries, and let $B = AA^ t $and let $\alpha$ be the supremum of $x ^t Bx$ where supremum is taken over all vectors $x ∈ \mathbb R ^n$ with norm less than or equal ...
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2answers
46 views

When does $x^T (xy^T) y = x^T (x^Ty) y$?

$x$ and $y$ are column vectors. When does $x^T (xy^T) y = x^T (x^Ty) y$? After a few trial and errors, I found that if at least one of $x$ and $y$ is a zero matrix then the equality is true. The ...
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2answers
86 views

How to prove that tr(A) = tr(B) given that B is similar to A [duplicate]

If A and B are similar, how does one prove that tr(A) = tr(B)
1
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1answer
29 views

Determinant and submatrices

I have an m x n matrix that has the rank at most one. What I am trying to show is that the determinants of all 2 x 2 matrices is zero. My idea is that I can row reduce the main matrix to one row ...
0
votes
1answer
31 views

Deducing a formula for multiplying a tri-diagonal symmetrical matrix with vectors

This is more like a math-programming problem, dealing with memory efficiency, but I thought it would be nice to expose it here. Let $A \in \mathbb{R}^{n \times n}$ be a tri-diagonal symmetrical ...
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2answers
31 views

Finding Eigenvalues and Eigenvectors for Leslie Matrix

A Leslie Matrix is given by $$L =\begin{pmatrix}0 & (3/2)a^2 & (3/2)a^3\\1/2 & 0 & 0\\ 0 & 1/3 & 0\end{pmatrix}\cdot$$ Find the Eigenvalues and determine the dominant ...
0
votes
2answers
44 views

What is the characteristic polynomial?

Let $A\in M_4(\mathbb{F})$, such that the minimal polynomial is $m_A = (x-3)(x^2+6x+10)$. What is $f_A(x)$ (the characteristic polynomial)? I'd be glad for help. By the way, I just proved a ...
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1answer
32 views

Diagonalizing a matrix in C

A question for homework asked to show that the matrix $[T]^{\alpha}_{\alpha}$ is diagonalizable, and find a basis $\alpha$, for $[T]^{\alpha}_{\alpha}$, where $T:C^{3}\to C^3$ ...
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0answers
20 views

Derivative of a matrix function by a matrix

How can I obtain the derivative of a matrix function $f(X)=X^TX$ by matrix $X$? Does the derivative organized in matrix form? Thanks in advance.
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1answer
23 views

How do I change this basis for a transformation?

I have $$\left[ L\right]_\mathcal{B}^\mathcal{B} = \begin{pmatrix}2&2&-1\\7&4&-2\\8&5&2\end{pmatrix}$$ and I want to get $[L]_\mathcal{E}^\mathcal{E}$ where the ...
0
votes
1answer
49 views

Cayley–Hamilton theorem and the characteristic polynomial

Let $A$, an invertible matrix and $f_A(x)$ to be the characteristic polynomial. By Cayley–Hamilton theorem we know that $f_A(A) = 0$. More detailed: $$0 = f_A(A) = a_0 + a_1A + \ldots + ...
1
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1answer
39 views

Shortcut when finding D when diagonalizing matrices when encountered with tedious matrices

P is given as P = $\left(\begin{array}{rrr} 1 & 1 & 1\\ 1 & 0 & -2\\ 1 & -1 & 1 \end{array}\right).$ It is known that P is invertible. I is a 3x3 identity matrix Supposed ...
2
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0answers
22 views

Some maximization over stochastic matrix

I am writing some applied assignment which leads me to the following problem. I will be very grateful if anyone can provide a solution or even some thoughts. Thanks a lot! Consider a (row-)stochastic ...
0
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1answer
8 views

Notation to define a function mapping from a vector to a two-dimensional matrix

I have a set $\mathcal{D}$, and I'm trying to define a mapping from that set to a two-dimensional matrix where each location contains either a $1$ or $0$. The notation I am using is ...
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1answer
19 views

${x^*}Bx \in R$ for all $x \in {C^n}$. Why $B = {B^*}$.

If ${x^*}Bx \in R$ for all $x \in {C^n}$ and $B \in {M_n}$. Why $B = {B^*}$ ?
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2answers
43 views

maximizing a coordinate of $x^T A^T A x \leq r^2$

Given a vector $\mathbf{x} \in \mathbb{R}^n$, a scalar $r\gt 0$ and an invertible matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$, I'd like to maximize one of the components $x_\alpha$ constrained by ...
0
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0answers
19 views

Inverse Square Root Of Matrix

So let's say a matrix is A. Then how do you find A^-1/2? It seems to be different from finding the inverse of A. Could someone provide a simple example as ...
0
votes
1answer
56 views

Explain the diference between your solution of the system Ax = b and the MATLAB's solution. [closed]

if I solved system with infinitely solution by my hand and by Matlab program what will be the different between two ways?
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1answer
27 views

Is there any relation between these matrices?

$Q$ is $(0,1,-1)$ vertex edge incidence matrix of a simple directed graph. $M$ is $(0,1)$ vertex edge incidence matrix of a simple non directed graph. $A$ is vertex vertex incidence matrix of a graph. ...
0
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1answer
81 views

Eigenvalue-eigenvector advance question

Anyone have some hint of how to do this question :) ?, a small $10 iTunes gift card will give away who help me to understand the question. thanks guy ;)
0
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1answer
48 views

Computing the dimension of a vector space in terms of matrix rank

Let $V=\mathbb C^n$ be a complex vector space, and $A,B:V\to V$ two commuting endomorphisms. I am interested in determining the dimension of the vector space $$F_{AB}=\{(a,b)\in V\times V\,|\,A\cdot ...
0
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0answers
16 views

Approach for this Popular Algorithmic Problem

Given a matrix we have to select one value from each row so that the total value cost selected is minimum. Now the problem is we cannot select column "0" to "J" in "I"th row if we have selected ...
1
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2answers
53 views

Finding the matrix of a linear transformation from an upper triangular matrix to an upper triangular matrix.

The question that I am trying to solve is as follows: Find the matrix $A$ of the linear transformation $T(M)= \begin{bmatrix} 7 & 3 \\ 0 & 1 \end{bmatrix} M$ from $U^{2×2}$ to ...