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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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
28 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 ...
1
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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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votes
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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votes
0answers
18 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.
0
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1answer
22 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
19 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
41 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
votes
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
51 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?
1
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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
votes
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
votes
1answer
47 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
votes
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 ...
0
votes
1answer
48 views

Matrix inverse and Hermitian transpose

Could anyone help me to prove the following the equation? $\large ( G_2^H G_2 + K_w^{-1} )^{-1} = Q$ which leads to $\large K_w = Q - Q G_2^H ( G_2 Q G_2^H - I )^{-1} G_2 Q$ Here ...
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votes
0answers
32 views

Eigenvector proving question [Math engineering]

I just have no idea how to do this advance question in my textbook . Can some body give me hint of how to achieve it ?. Have good weekend to all of you . :) cheer.
0
votes
1answer
25 views

Linear Algebra — Finding The Inverse

We are asked to find the inverse of the following matrix, provided the inverse exists: $$ \left[\begin{array}{rrr} 1 & a & b+c\\ 1 & b & a+c\\ 1 & c & a+b ...
1
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2answers
31 views

Can we prove properties of linear maps, just by proving the same properties on their associated matrix?

For example, If we want to prove than a operator is unitary. It is enough to prove than their associated matrix is unitary?. From my point of view, it is correct since there's an isomorphism between ...
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0answers
16 views

How to write linear combination of vectors in a matrix in terms of preceding vectors [closed]

How would I go about say writing several vectors that make up a matrix in say any real space, take 3 for example in R^3 as linear combinations of the preceding ones? I am not sure how to do this on ...
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votes
0answers
40 views

Vector Space, Basis, Matrices [closed]

Let $S_n$ denote the vector space of all $n\times n$ symmetric matrices (i.e. $M = M^T$). Let $A_n$ denote the vector space of all $n \times n$ anti-symmetric matrices (i.e. $M^T = - M$). (a) Find a ...
0
votes
2answers
43 views

Solving a matrix equation.

Questions in this general form have been asked a lot here, but I've searched for hours and I haven't found any that I can generalize to my problem, so I've asked it again: I've been given three ...
0
votes
4answers
44 views

Proving properties about matrix $A$ s.t. $A^2 = -I$

$A$ is $nxn$ matrix, and $A^2=-I$. Show $A$ is non-singular and show that $n$ is even. (Further questions, show that $A$ has no real eigenvalues. Show det(A) = 1. But I'm not there yet.) So if $A$ ...
0
votes
1answer
18 views

Terminology confusion - basis of an operator

I have a bit of a dumb question. I'm currently taking a linear algebra course. Right now we're working with operators and just finished up diagonizability. So I know this definition: An operator ...
4
votes
4answers
109 views

$tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$ implies $A$ nilpotency

Let's consider a $n \times n$ matrix and the sequence of traces $tr(A)=tr(A^{2})= \ldots = tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u ...
0
votes
2answers
30 views

Show a matrix is similar to a lower triangular matrix

$A = \left(\begin{array}{cc}2 & -1 \\0 & 2\end{array}\right)$ $B = \left(\begin{array}{cc}\lambda & 0 \\1 & \lambda\end{array}\right)$. I know that the $\lambda = 2$. And $r(1,0)^t$, ...
0
votes
1answer
76 views

Proving the continuation of the Cayley-Hamilton theorem from Schur's triangularization theorem

The Cayley-Hamilton theorem says that every square matrix can satisfy its own characteristic equation, $p(\lambda) = 0$, or $p(\mathbf{A}) = \mathbf{0}$. The question is to show how the ...
1
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0answers
23 views

How do I find a vector given by the reflection of a vector with respect to a plane in R^3?

$L: \Bbb{R}^3 \to \Bbb{R}^3$ is reflection with respect to a plane. Two vectors on the plane are given and I calculated their cross product to get the basis $\mathcal{B} = ...
1
vote
1answer
10 views

Multivariate gaussian distribution in imm filter

I'm try to implement an Imm filter. In one step I need to perform the multivariate gaussian probability function. The problem is that the covariance matrix S of the filter has a negative determinant. ...
0
votes
1answer
24 views

Trying to find the markov chain and adjacency matrix of this graph?

This is graph of the problem: Suppose animal x is at node 3 of the graph. It chooses small path labelled s with 2 times probability then long path l. If length is same then probability is same ...
1
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3answers
36 views

Why is this theorem (on eigenvalues and invertibility) important?

Theorem 5.1.5 in this book says A square matrix $A$ is invertible if and only if $\lambda=0$ is not an eigenvalue of $A$. I'd like to know if this theorem is important for practical purposes. ...
0
votes
2answers
25 views

Finding a non-singular matrix $C$ such that $C^{-1}AC$ is diagonal

$A = \left(\begin{array}{cc}1 & 0 \\1 & 3\end{array}\right)$. I find the eigenvalues = 1,3. The eigenvector corresponding to 1 = $t(1,-2)^t$. The eigenvector corresponding to 3 = $r(1,0)$. ...
1
vote
2answers
31 views

I'm having a conceptual issue with similarity matrices

So I know that $A = T^{-1}AT \implies T \text{ is a similarity transformation matrix}$. Say $A = \begin{pmatrix}9 & 13 \\ -3 & -3\end{pmatrix}$, then how would I go about finding T without ...
4
votes
1answer
283 views

Inverse of the sum of the inverse of two matrices

I need to compute $ (A^{-1} + B^{-1})^{-1} $. Both $A$ and $B$ are symmetric and $A$ is invertible and PSD. I already know $B^{-1}$ and $A$, but I don't have $A^{-1}$ and $B$. Is there a formula to ...
1
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0answers
73 views

Determine over $\mathbb{Q} $ and $\mathbb{C}$ all $3 \times 3$ matrices with $A^4 = I$

I know that if $A$ is a $3 \times 3$ matrix whose minimal polynomial divides $x^4 -1$, then the minimal polynomial has the restriction that has at most degree $3$. On the other hand, we have: $$x^4 ...
1
vote
1answer
19 views

Continuity on kronecker product

Let $\{A_k\}_{k\in{\mathbb{N}}}$ a sequence of $n\times n$ matrices. Suppose that $A_k \to A$, and consider $B$ another $n\times n$ matrix. Its true that... $A_k \otimes B \to A \otimes B$ ?
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votes
1answer
32 views

How do you find the subset of $2\times3$ matrices that forms a basis for a subspace $V$?

I know know how to find a seubet of vectors that forms a basis but how can you from a basis with matrices?
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0answers
23 views

Prove that a transformation is linear [closed]

Prove that if $M$ is an $m \times n$ matrix then the transformation $T:{\mathbb{R}^n} \to {\mathbb{R}^m}$ given by $x \mapsto T(x) = M \cdot x$ for any $x$ that exists in ${\mathbb{R}^n}$ is linear. ...
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0answers
18 views

Creating a matrix so its Euclidean norm (p=2) is < 1

I have to convert $Ax=b$ to the form $x=Bx+b$, so $\|B\|_2 < 1$. I'm having serious trouble with picking/creating the matrix $B$ and it's making me nuts. Can someone give me an example or a hint? ...
0
votes
0answers
14 views

4 D rotation matrix

I have 2 vectors with 4 elements each and they are perpendicular to each other. z = [ -0.0310 -0.0894 -0.9451 -0.3128] and w = [0.9451 0.3128 -0.0310 -0.0894] how do i compute the 4x4 rotation ...
1
vote
1answer
48 views

A matrix similar to a basis change matrix is also a basis change matrix [closed]

Please tell me whether this statement is right or not. I think it's right but I can't prove it. A matrix similar to a basis change matrix is also a basis change matrix.
1
vote
1answer
27 views

How should I build a SU(4) matirx with a C4 vector?

I have a complex vector $S=[S_1,S_2,S_3,S_4]$ with $|S_1|^2+|S_2|^2+|S_3|^2+|S_4|^2=1$. My question is how to bulid a matix $C\in SU(4)$ while \begin{equation}C= \left( \begin{array}{cccccc} S_1 ...
0
votes
1answer
29 views

Need help with a better understanding of change of basis matrix and corresponding theorems

I'll try to summarize here what I understand so far about the concepts of change of basis matrix etc. Let $\beta$ and $\gamma$ be two different ordered bases for the vectorspace $V$, and let $P ...
1
vote
1answer
28 views

Bound on the eigenvalues of PSD matrix [duplicate]

Given That A and B are two PSD (positive semi-definite) real matrices and the following holds $$ A \leq B $$ (meaning that $$ B-A $$ is also PSD) can I bound the eigenvalues of A using the ...
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votes
0answers
12 views

$Mod 3$ or $mod 5$ for for an $N5(3, 5)$ is a $[5, 2, 4]5$ Reed Solomon code.

I am trying to reduce a matrix to the form $G = [I|-B^T]$ but I just cannot get my solution to match the solution in my notes. I am reducing it Mod 3 as it is part of a coding theory question in ...