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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0
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1answer
17 views

Is the pair controllable/observable?

The matrices $Q\in\mathbb R^{n\times n}$ and $G\in\mathbb R^{n\times n}$ are both symmetric positive semidefinite, $A\in\mathbb R^{n\times n}$ is invertible. Moreover, $(A,G)$ is controllable, and ...
0
votes
5answers
39 views

Whether a matrix is a zero matrix

If a real square matrix $A $ is similar to a diagonal matrix and satisfies $A^n=0$ for some $n\in \mathbb N $,then can it be proved that $A$ must be a zero matrix?
1
vote
1answer
36 views

What are these tick marks after the x, y, and z called?

What are these marks called and what do they stand for? This is for a Affine Transformation.
1
vote
0answers
9 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
0
votes
0answers
21 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
0
votes
1answer
30 views

Know eigenvalues, get $Q$ of $A=QLQ'$

$A=\begin{bmatrix} 1 & -2 & 2\\ -2 & -2 & 4\\ 2 & 4 & -2 \end{bmatrix}$ I have calculated that the eigenvalues $\lambda=2,2,-7$. When $\lambda=2$, the eigenvector is ...
0
votes
0answers
11 views

Converting from X,Y,Z offset representing a rotation to matrices

I've been working on figuring out 3-dimensional rotations for graphics and I've reached a brick wall of understanding that I can't power through. Right now I have a function which calculates the ...
0
votes
1answer
21 views

Proof of upper triangular matrices

I am supposing that $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times n $ upper triangular square matrices. $\lambda \in \mathbb{R}$. So $a_{ij}=0$ whenever $i>j$. I am trying to prove that these are ...
0
votes
1answer
14 views

What is the change of basis in 2D?

I know how to apply a change of basis in 1D, but I was wondering: If I want to apply a change of basis to a nxn matrix, is it enough to apply the change of basis to every column of the matrix or is ...
5
votes
2answers
70 views

Row reduction and the characteristic polynomial of a matrix

Can you row reduce the matrix before computing $\det(\lambda I-A)$? Will this still give an equivalent characteristic polynomial?
-2
votes
2answers
43 views

Form a basis for R^3? [on hold]

This is a homework problem and I need help on. Consider the matrix with the given vectors as its columns. Do (1, -1, 3), (-1, 5, 1), (1, -3, 1) form a basis for R^3?
1
vote
0answers
15 views

Prove or disprove that the product $PVPVP$ is nonnegative

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a nonsingular symmetric M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is nonnegative. I know ...
0
votes
0answers
11 views

Determining matrix in terms of determinants of other matrices.

Determine |a+b e-f| |c+d g-h| in terms of the determinants of |a c| |b d| |a c| |b d| |e g| |e g| |h f| |h f| ...
0
votes
0answers
10 views

why is the covariance matrix of a bekk model always positive definite?

The BEKK(1,1) model is given by: $$\Sigma_{t}=A_{0}A_{0}'+A_{1}a_{t-1}a_{t-1}'A_{1}'+B_{1}\Sigma_{t-1}B_{1}'$$ where $a_{t}$ are serially uncorrelated, zero mean innovations, $A_{0}$ is a lower ...
0
votes
1answer
17 views

Coordinates of a vector under a basis in a Hilbert space?

Given an arbitrary basis $\{m_1, \dots, m_n \}$of a Hilbert space $H$ (or just think it as $\mathbb R^n$, and I think the methods should be the same) with given inner product, how can we find the ...
0
votes
1answer
33 views

Matrix raised to a power

Find $A^n$ for $n = 1,2,...$. Does $A^n$ tend to a limit? $$A= \begin{pmatrix} 4/5 & 2/5 \\ 1/5 & 3/5 \end{pmatrix}$$ I found the eigenvalues $\lambda=1,2/5$ and the eigenvectors ...
1
vote
2answers
34 views

Inversion of n x n matrix

A matrix F is given: $$ F = [e^{i\frac{2\pi kl}{n}}]_{k,l=0}^{n-1} $$ Find $$ F^{-1} $$ I know Gaussian method for inverting matrices but I suppose it doesn't apply to matrices with not given exact ...
2
votes
1answer
30 views

Diagonalizing the X and Z matrices

I've got two special matrices I'm trying to diagonalize : The Z matrix :$$\begin{bmatrix} 1&1&\cdots&1&1\ \\&&&1 \\&&\diagup \\&1 ...
0
votes
0answers
35 views

Isomorphism of vector spaces

Let $S$ be the space of all $3\times k$ matrices,$T$ be the space of all column vectors consists of seven components.If $S$ is isomorphic to a subspace of $T$ then what are possible values of $k$? I ...
0
votes
0answers
22 views

Stuck on kernel space question

I am completely stumped on what seems like a simple question. For a vector $v$ in $\mathbb R^n$, and $A$ being a $m \times n$ matrix with real entries. How do I show that $v-v$ is in $\ker(A)$? Any ...
1
vote
2answers
57 views

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true?

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true? $1.$ $A=I$ $2.$ $\det(A)=1$ $3.$ $\operatorname{trace}(A)=n$ I have counter example for the first option.For ...
0
votes
1answer
16 views

Invertibe matrix is a transition matrix?

It is true that all transition matrices are invertible, but does the converse hold: All invertible matrices are transition matrices? I'm asking with regard to matrices over a field, but more general ...
0
votes
0answers
15 views

Use the Kronecker delta matrix to answer question

So I have the Kronecker delta which is denoted as $\delta_{ij}$=$I$. Let $b_1, b_2, \cdots, b_n$ be a set of $n$ real numbers, I must show that: $\sum\limits_{i=1}^n b_i \delta_{ij} = b_j$ and ...
0
votes
0answers
24 views

Is there any simple way of finding a matrix which commutes with a given (say, more complicated) matrix?

Suppose I want to find the eigenvectors and eigenvalues of a hermitian matrix $A$, but $A$ is big and ugly. Is there an easy way to find another, nicer, hermitian matrix $B$, such that $AB=BA$ and so ...
0
votes
0answers
13 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
4
votes
1answer
40 views

Basis of the matrices with only non diagonalizable matrices

Is it possible to find a basis of $M_n(\mathbb{R})$ that only has non diagonalisable matrices ? I'm looking for a rather easy example, or a proof of the (non-)existence.
0
votes
0answers
38 views

Solving the characteristic equation $a^4+2a^3+5a+8=0$

I need to find the eigenvalues of a $4\times4$ matrix. I already determined the characteristic equation, which is $a^4+2a^3+5a+8$. Now I have to solve $a^4+2a^3+5a+8=0$, but I don't know how to ...
0
votes
1answer
26 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
0
votes
0answers
30 views

statements of matrix analysis

Let $y$ be fixed value. Let $A=a(x,y)$ be a matrix and $f_{t}(x)=\frac{\sum_{n=0}^{\infty}{a^{(n)}(x,y)(\frac{1}{t})^n}}{\sum_{n=0}^{\infty}a^{(n)}(y,y)(\frac{1}{t})^n}$ Show that ...
0
votes
0answers
30 views

Determine matrix from linear transformation

Let $T_{1}$ and $T_{2}$ be linear transformations given by $$T_{1}([x_{1}, x_{2}])=[3x_{1}+5x_{2}, 4x_{1}+7x_{2}]$$ $$T_{2}([x_{1}, x_{2}])=[2x_{1}+9x_{2}, x_{1}+5x_{2}]$$ Find a matrix A such that ...
1
vote
1answer
22 views

How to determine the Jordan form and give a Jordan base for a matrix?

given is $\begin{pmatrix} 3&0&-1&0&0 \\ 1&3&0&1&0 \\ 0&0&3&0&0 \\ 0&0&0&3&0 \\ 0&0&0&0&-3 \end{pmatrix}$ I have to ...
0
votes
1answer
18 views

Skew symmetric Matrix - Commutative property

If A and B are two odd size skew symmetric matrices(for example $3 \times 3 $). Let us say $X=AB,Y=BA$ Question What is the general relationship between X and Y? Can we write Y using X?
1
vote
1answer
70 views

Compute a 4 4 matrix M such that MA is the row-reduced echelon form of A.

Compute a 4 X 4 matrix M such that MA is the row-reduced echelon form of A. (Hint: M can be written as a product of elementary matrices.) A:= ...
0
votes
1answer
16 views

Calculating the images of transformations of matrices

$f\colon \mathbb{R}^2 \to \mathbb{R}^3$ by $f(a,b) = (a+b, 2a-b, a-2b)$. Find the kernel and the image. I found the kernel to be $\ker(f) = \{(0,0)\}$ but cannot get the right image. The book says ...
1
vote
1answer
64 views

How to find maximum of an inverse of a matrix?

If there is a square $~n\times n~$ matrix $~H~$ where ALL the elements of $~H_{i,j}~$ are variables between two bounds, $~H_{i,j})_{min}~$ and $~H_{i,j})_{max}~$. Is there any relation to maximize ...
0
votes
2answers
33 views

Sign pattern symmetric matrices

I am interested in sign pattern symmetric real matrices ($a_{ij} a_{ji} \ge 0$ for all $i \ne j$). I have seen a published proof that such sign-symmetric matrices cannot have purely imaginary ...
1
vote
0answers
14 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
0
votes
1answer
25 views

Prove True or false

if the rref of a has a row of 0', then the set of row vectors of a is linearly dependent. Please help me prove or give a counterexample
0
votes
1answer
16 views

Irreducibility of markov chains

Let $A=(a(x,y))_{x,y\in X}$ be a finite irreduzibel nonnegative matrix. Let $b,c >0$, and $\alpha=a^{(b+c)}(x,x)>0$. So $a^{(n(b+c))}(x,x)\ge \alpha^n$. And therefore $\lim ...
0
votes
1answer
52 views

Finding maximum number of solutions in a matrix

Given x+y+5z=2 x+2y+7z=1 2x−y+4z=a a) Determine the value of a which will make the given system have many solutions. Explain your answer. b) Choose a value of a which will make the given system ...
0
votes
1answer
21 views

Finding the transformation matrix R

Please help me in solving this problem, I am not sure what a transformation matrix R is and how to proceed.. Any help is appreciated. Find the transformation matrix R that relates the (orthonormal ) ...
0
votes
0answers
17 views

Proving the equality of matrix traces

I have to prove some traces given that A and B are $n\ .\ n$ matrices for a class assignment and would like to confirm that the following rules of associativity hold true. I don't want to post the ...
0
votes
1answer
20 views

How to determine if a set is a subspace of the vector space of all complex $2\times 2$ matrices?

I must determine if a each of the following is a subspace of the vector space consisting of all complex $2\times 2$ matrices. All matrices with real diagonals. All matrices for which the sum of the ...
-1
votes
1answer
31 views

Linear Algebra Subspace test

I'm currently studying Subspace tests in my linear Algebra module at uni, but am struggling to understand it, can anyone explain how to conduct a SubSpace test?
0
votes
1answer
28 views

finding if a linear transformation exists, and proving it.

We just started the topic of linear transformations and I have this hw question that I just don't understand. Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, ...
0
votes
1answer
23 views

Elementary row operations in matrices

This is really such a lovely math community, I am working on some differential equations hw and my teacher didn't teach this topic yet so I am a little confused. My first question is pertaining to ...
4
votes
2answers
68 views

Find a polynomial $f(Z)$ of degree less than 2 such that $e^{tA}=f(A)$

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. As the question says I need a polynomial $f(Z)$ of degree less than 2 such that $e^{tA}=f(A)$. Should my polynomial just be the first 2 terms ...
0
votes
0answers
25 views

Find a basis for the four fundamental subspaces. [on hold]

Find a basis for the four fundamental subspaces of: $$A=\begin{bmatrix}1 & -1 & 0 & 2 \\ 0 & 0 &1 &1 \\ 0 &0 &0 &0\\0 &0 &0 &0\end{bmatrix}$$ I'm ...
0
votes
0answers
11 views

Prove that the row sums of $T$ satisfy the following formula.

Consider the lower triangular matrix defined by the recurrence: $$T(n,1) = 1$$ $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = \sum _{i=1}^{n-1} T(n-i,k-1)+y \sum _{i=1}^{n-1} T(n-i,k) \; ...
1
vote
1answer
37 views

Prove that this statement about A and B is true.

$A,B \in \mathbb{R}^{2}$, If $AB - BA = A^2$ Prove that $ (B - A)^{2014} = B^{2013}(B-2014A)$