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), ...

learn more… | top users | synonyms (2)

1
vote
1answer
15 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
1answer
21 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
1answer
21 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 ...
5
votes
6answers
1k views

Why can't you divide matrices?

I was just wondering that because one can multiply and add and subtract matrices, why can't one divide them?
1
vote
1answer
57 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 ...
1
vote
1answer
300 views

If integral matrices $A,B$ commute, is $B = f(A)$, where $f$ is an integer polynomial?

Let $A\in M_n(\mathbb C)$ be a square matrix of order $n$. Suppose that the characteristic polynomial of $A$ equals its minimum polynomial. It is well known that every matrix that commutes with $A$ ...
0
votes
0answers
9 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
15 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 ...
3
votes
2answers
48 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 ...
0
votes
0answers
7 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 ...
4
votes
3answers
5k views

Proving: “The trace of an idempotent matrix equals the rank of the matrix”

How could we prove that the "The trace of an idempotent matrix equals the rank of the matrix"? This is another property that is used in my module without any proof, could anybody tell me how to prove ...
-2
votes
2answers
41 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?
0
votes
1answer
11 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 ...
3
votes
1answer
36 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
10 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
8 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
15 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 ...
1
vote
2answers
31 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
26 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 ...
1
vote
2answers
55 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
14 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
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 ...
7
votes
1answer
187 views

Why is the kernel of this strange polynomial homomorphism what it is?

I've been trying to delve a little further into linear algebra, but I'm not following something I think is supposed to be obvious. Suppose $M_{m,n}(\mathbb{C})$ is the set of rectangular $m\times n$ ...
0
votes
0answers
13 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
22 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
29 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
11 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
2answers
48 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
36 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 ...
1
vote
1answer
250 views

Is the set of all invertible $n \times n$ matrices a vector space?

I'm studying Algebra and I'm asked to prove or disprove the statement above. I found that is true, but I'm not sure how to prove it. My problem here is that this statement is too "broad", i.e. I ...
1
vote
1answer
21 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 ...
1
vote
1answer
25 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
0answers
29 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 ...
0
votes
1answer
14 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?
2
votes
0answers
156 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) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
1
vote
1answer
27 views

Prove that the product of two positive semidefinite and symmetric matrices has non-negative eigenvalues

How can I prove the following fact: If $A$ and $B$ are two positive semi-definite and symmetric matrices then all eigenvalues of $AB$ are non-negative.
0
votes
2answers
25 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 ...
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 ...
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
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
3answers
54 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 ...
2
votes
0answers
39 views

How to find nonnegative solutions of a linear system?

I have a $M$ equation and $N$ variables like this : $ \begin{bmatrix} 3 & 0 & 1 & 0 & -1 & -3 & 2\\ 1 & 2 & 0 & 4 & 0 & 0 & -1\\ 1 & 1 & 0 ...
1
vote
0answers
12 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
24 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
2answers
70 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 ...
0
votes
0answers
30 views

Derivations on the space of triangular matrices

I have started to research matrices and have been asked the following. If $d$ is a a derivation on $T_n(\mathbb R)$ and $d(e_{ij})=0$, with $1\le i \le j \le n$, Show that for every $r \in ...
10
votes
2answers
1k views

Integral of matrix exponential

Let us be given a square $n \times n$ matrix $A$. For a system \begin{align*} \dot{x}(t) = A x(t), \hspace{0.3 cm} x(0) = x_0 \end{align*} the solution is given by $x(t) = e^{At} x_0$. I am ...
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
51 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 ...