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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0answers
22 views

2D convolution: how to eyeball it?

I have a question of doing simple convolution in 2d by just "eye-balling" it without doing the actual computation. In 1D discrete time, when we have a simple input ...
2
votes
3answers
65 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
0
votes
0answers
24 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...
1
vote
3answers
48 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
-2
votes
1answer
36 views

Domain/codmain + range/kernel for linear mappings [on hold]

Consider the linear mapping: $$L(x_1,x_2)=(2x_1-3x_2,4x_1+5x_2,2x_1-x_2)$$ Solve for: (a) Domain and codomain of L (b) Standard matrix of L (c) Basis for the range of L (d) Basis for the kernel ...
0
votes
1answer
19 views

Is the derivative of the characteristic polynomial equal to the sum of characteristic polynomial of principle submatrices?

Let $A$ by an $n \times n$ matrix over the complex numbers and let $\phi(A,x) = \det(xI-A)$ be the characteristic polynomial of $A$. Let $B_i$ be the principal submatrix of $A$ formed by deleting the ...
1
vote
3answers
64 views

Show that a set of vectors is linearly dependent

Show that the set $S = \{(3, 2), (−1, 1), (4, 0)\}$ is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. (Use $s_1$, $s_2$, and ...
-1
votes
1answer
31 views

Linear Algebra - Prove trival solution eigenvalue

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$. Prove that $(2A+5I)x=0$ has only trival solution. I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that ...
0
votes
1answer
31 views

Can't understand matrix based derivation

$\beta(k,d)=(X'X+kI)^{-1}(X'y+kdB_L)$ $=[I+k(X'X)^{-1}]^{-1}(X'X)^{-1}(X'y+kdB_L)$ $=[I+k(X'X)^{-1}]^{-1}(B_L+kd(X'X)^{-1}B_L)$ $=[I+k(X'X)^{-1}]^{-1}(B_L- dB_L)+dB_L$ $B_L=(X'X)^{-1}Xy$ X=n*p ...
1
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0answers
17 views

median eigenvalue

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
0
votes
2answers
82 views

Question concerning cross product and orthonormal vectors

Assume we have a vector $u= (u_1.u_2, u_3) \in R^3$ My problem is to find vectors $\vec w, \vec v$ such that $u= v \times w$ All vectors should be orthonormal. If $u= (u_1, u_2, u_3)$ ,is there a ...
1
vote
2answers
98 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
6
votes
1answer
59 views

What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
0
votes
0answers
17 views

Is any triangular matrix with positive diagonal elements a Cholesky factor?

I'm having a hard time finding information about Cholesky factors, and I'm sure it's a very simple question if it was asked to the right person. I need to create positive semi-definite matrix using ...
1
vote
1answer
34 views

Gram matrix to be cancelled

Let $V$ be a $n$ dimensional Euclidean space with inner product $<\cdot,\cdot>$, with basis $e_1,\cdots,e_n$. Then the Gram matrix is $A=(a_{ij})$ with $a_{ij}=<e_i,e_j>$. It is well-known ...
1
vote
2answers
29 views

Positive definite matrix to be cancalled

From $ax\geq 0$ for $a>0$, we have $x\geq 0$. So I suggest that if $Ax\geq 0$ for $A$ positive definite matrix, $x$ a column vector, $0$ is the column vector with $0$ as elements, then $x\geq 0$, ...
0
votes
0answers
17 views

Understanding the difference in (double) diagonally traversing trough a square matrix

I have been struggling with an algorithm to solve the PE Problem #149. I was able to find a solution (algorithm) for this problem on the internet, which can be found here. I do understand this ...
1
vote
0answers
22 views

Determinant of specific infinite matrix

What is the limit, as n approaches infinity, of the determinant of an n x n matrix where each cell has the value cos(n * row + column)? My friend and I believe the answer to be 0, but can't figure ...
4
votes
1answer
46 views

Finding unknown matrices in a set of simultaneous matrix equations

I've come across a thorny problem in my research, which is too complicated and specific to ask here. However, it bears some similarity to the following problem, and understanding how to solve this ...
2
votes
1answer
22 views

Is there a 3D equivalent of a 2D matrix?

Just thinking, is there a 3D 'equivalent' of a matrix. I know it's possible to get matrices that only have one row or column (i.e. vectors) thus making there a sort of 1D equivalent, but is there a 3D ...
5
votes
0answers
82 views
+50

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
0
votes
1answer
39 views

Linear Algebra vs Matrix Algebra [closed]

Hi I don't know if this would be a proper question to ask here or not Anyway, I am an undergraduate electrical engineering student and I am considering taking another math course. What is the ...
1
vote
0answers
40 views

Diagonalization of Hermitian matrix

I would like to perform diagonalization of a Hermitian matrix $A$ and I know the steps but at the end I am not getting diagonal matrix with eigenvalues on the main diagonal, can anyone help me why? ...
1
vote
2answers
37 views

Some Matrix product $A \odot B$

I'm confronted with the following problem: Let $G=(V,E)$ be a directed graph with edge costs $c:E\rightarrow \mathbb{R}$ (Negative cycles do not matter). Let $V=\{v_1,\dots,v_n\}$. For Matrices $A$ ...
0
votes
2answers
35 views

Two Matrices with Negative Eigenvalues of Each Other?

I have two matrices, $A$ and $B$. I was (perhaps naively) expecting them to be more-or-less similar ("more-or-less" because this is in a numerical setting), but instead of having exactly the same ...
0
votes
1answer
32 views

On Neumann-series of matrices

Let $A \in \mathbb{R}^{n \times n}$ and we denote with $\rho(A)$ the spectral radius of $A$ and with $I_n \in \mathbb{R}^{n \times n}$ the identitiy matrix. Applying Carl Neumann's result on matrices ...
3
votes
2answers
57 views
+50

Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the cholesky ...
5
votes
1answer
222 views
+200

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
2
votes
1answer
48 views

about the power of a matrix

Assume that matrix $A$ contains only 0 or 1 elements. Could anyone give me some condition, under which the matrices $A^i$ (for $i=1,2,3,...,k$) still contains only 0 or 1 elements. For example, I ...
0
votes
0answers
36 views

Two matrices with the same row space are row equivalent

If $\operatorname{Row} A = \operatorname{Row} B$, then $A $ is row equivalent to $B$. So far I have So $\operatorname{Row}(B) = \operatorname{Row}(A)$. That is rows of $B$ belong to ...
0
votes
1answer
35 views

Find the standard matrix of the linear transformation

Suppose there is a linear transformation $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $$T\left( \begin{array}{ccc}2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc}1\\ 4 \end{array} ...
1
vote
1answer
49 views

Solve quadric equation system

How to solve this? For given real and symetric matrices $A_1,A_2,A_3,A_4\in\mathbb{R}^{4\times4}$ find $x\in\mathbb{R}^4$ $$x^TA_1x=0$$ $$x^TA_2x=0$$ $$x^TA_3x=0$$ $$x^TA_4x=0$$
0
votes
0answers
16 views

Reduce to diagonal form.

Problem is to reduce $5X^2+3Z^2+4XY-4YZ+6ZX$ into diagonal form over $\mathbb{R}$. With my knowledge, We ned to make a non-singular variable transformation so that above form comes into a form like ...
-2
votes
1answer
29 views

Diagonalizing a block matrix

Suppose the matrices $A,B,C,D$ are $M\times M$, $M\times N$, $N\times M$ and $N\times N$, respectively. Then I give you the following block matrix: $$ G = \begin{pmatrix} A & B \\ C & D ...
1
vote
2answers
79 views

Linear Algebra - Prove $AB=BA$

Let $A$ and $B$ be any $n \times n$ defined over the real numbers. Assume that $A^2+AB+2I=0$. Prove $AB=BA$ My solution (Not full) I didn't managed to get so far. $A(A+B)=-2I$ ...
0
votes
0answers
24 views

Solve for a with the given equations

Given: $x+2y-3z=4$ $3x-y+5z=2$ $4x+y+(a^2-14)z=a+2$ Find $a$ so the system has: a)no solutions b)exactly one solution c)infinitely many solutions Should the system of equations ...
0
votes
1answer
22 views

Show that $P(A) = 0\implies P(B) = 0$ if matrix $A$ is similar to matrix $B$ and $P(x)$ is any polynomial.

Note: This is a homework problem. I've been working on this one for a while now and seem to be a bit stuck. The one thing for sure I know we can say is that $P(A) = P(M^{-1}BM) = 0$ Is there some ...
0
votes
0answers
15 views

Finding coefficients in matrices making the matrix inconsistent, and consistent (infinite/unique soln.)

I need help in figuring out this question I have seen some solutions to other questions but they involve finding determinants of matrices which we are not allowed to use because we've only learned ...
2
votes
3answers
25 views

Solving for a matrix in equation form

Solve for $X$ assuming all matrices are n x n and invertible as needed. $$B(X+A)^{-1}=C$$ I solved this the following way: Multiply both sides by $(X+A)$ Multiply both sides by the inverse of $C$ ...
0
votes
0answers
22 views

Linear transformation on finite-dimensional vector space such that T²=T, how is the matrix?

I'm trying to have a picture of how the matrix of such a linear transformation looks like and why? I can't really find anything but that applying the map once can change the input into a new output ...
0
votes
2answers
27 views

Solving $CT = PC$ for transforms in $SE(3)$

I have three transforms: $C$, $T$, and $P$. Each of these transforms consists of 3D rotations and translations. I know $T$ and $P$, and I would like to solve for $C$. They are related by $T = C^{-1} P ...
1
vote
2answers
47 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots ...
3
votes
1answer
29 views

Is this argument on positive definite matrices correct?

Let $A$ be a $N\times N$ positive definite matrix. Then, there exists a $N\times 1$ gaussian random vector $a$ such that $A=E[aa^T]$ where $E[.]$ denotes expectation. Then for any given vector $x$, ...
6
votes
1answer
53 views

What's an easy way to show that $GL(n,\mathbb C)$ is connected? [duplicate]

I think I've to show it's path connected, but can't figure out the path functions explicitly. Can anyone give these path maps?
1
vote
1answer
32 views

Are the spaces of real orthogonal, complex unitary, hermitian or symmetric matrices connected?

I want to know which of these are connected and which are not. I think I've to take some continuous map from the set of matrices to $\mathbb R$ or $\mathbb C$ and interpret these matrix sets as ...
1
vote
0answers
32 views

Fast Cholesky Factrorization for Tree Laplacians

Suppose $T_1$ and $T_2$ represent two Laplacian matrices of two spanning trees of $n$ vertices. Since the Cholesky factorization needs $O(n)$ time for each $T_i\ (i=1,2)$ due to the tree structure, ...
1
vote
1answer
50 views

Proof that if $A$ is similar to $B$, then $B$ is similar to $A$

$A$ is similar to $B$ if there is an invertible matrix $S$ such that $B = S^{-1}AS$. Prove that if $A$ is similar to $B$, then $B$ is similar to $A$. So if $A$ is similar to $B$ then $B = ...
0
votes
0answers
5 views

Matrix with highly correlated adjacency entries

I am learning about SVD from this book. One of the exercise questions asks me to create matrix with highly correlated adjacency entries and then conduct some experiments to discover the nature of the ...
2
votes
3answers
66 views

What is inverse of $I+A$ given that $A^2=2\mathbb{I}$?

I have the next problem: Let $A$ be a real square matrix such that $A ^ 2 = 2\mathbb{I}$. Prove that $A +\mathbb{I}$ is an invertible matrix and find its inverse. I tried with the answers given ...
0
votes
0answers
12 views

Eigenvalues of Hankel matrices

Let $\mathbf{A}$ be a $4-$ dimensional symmetric matrix with real entries, whose elements are given as \begin{equation} \mathbf{A} = \left( \begin{array}{cccc} a & b & c & d \\ b & c ...