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)

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
25 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
34 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
25 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
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 ...
0
votes
1answer
17 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
63 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
16 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
65 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
24 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)$
1
vote
0answers
13 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
0
votes
1answer
15 views

Matrix representation of rotation proof?

C is for Cos, S is for sine To find the matrix representation, we just apply R n to each of the standard basis vectors, as in Equation 3.3, and then place the resulting vectors into the rows of a ...
1
vote
0answers
12 views

Recomputing the Gram Matrices

Recompute the following Gram matrix for the weighted inner product $\langle x,y\rangle=x_1y_1+\frac{1}{2}x_2y_2+\frac{1}{3}x_3y_3+\frac{1}{4}x_4y_4$: ...
0
votes
1answer
15 views

How to prove $V*V^T=I$ in SVD? [duplicate]

How to prove $V*V^T=I$ in SVD: $M=U*S*V^T$? It's easy to understand $V^T*V=I$. It seems $V*V^T=I$, but how to prove it?
1
vote
1answer
31 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.
2
votes
0answers
48 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 ...
0
votes
0answers
41 views

How to show that e.g. $E(\mathbf{w}) = \ldots \Rightarrow \frac{1}{2}(\Phi\mathbf{w} - \mathbf{t})^T(\Phi\mathbf{w} - \mathbf{t}) $

We have to show for a few formulas that they can also be written in matrix notation. For example: For $\mathbf{x}=(x_0,x_1,\ldots,x_n)$,xi∈R $\sum_{n=1}^ix_n^2 = ...
1
vote
1answer
31 views

Why is there two versions of the rotation matrix?

Why is there two versions..for example, I got some matrices for the x axis rotation This is for the X AXIS the other one is for the x axis also, it is.. I think it could be from going ...
1
vote
1answer
24 views

Optimisation over matrix entries

I was looking to write the KKT conditions to solve this optimisation problem. $$\min_{\substack{\sum_j x_{ij}\le k_i \\ i=1,2,\ldots N}} a^\top (I-X)^{-1} b $$ Since there are $N^2$ decision ...
0
votes
0answers
22 views

Help with a matrix problem

I'm stuck with the following matrix problem: Consider $A = $$\{ X \in \mathcal{M}_2(\mathbb{C})\ \mid X = \left( \begin{array}{ccc} a & 0 \\ 0 & b \end{array} \right); a, b \in \mathbb{C}; ...
0
votes
0answers
14 views

Bounds on the coherence of very flat matrices (that are more tight than the Welch bound)

I am studying the coherence of matrices in the context of sparse recovery. Let us say I have a matrix $\mathbf \Phi$ of size $M \times N$ with, say, unit Euclidean norm columns ${\mathbf \varphi}_n$. ...
1
vote
2answers
49 views

Find the $n^{th}$ power of a $2$x$2$ matrix.

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. Using Lagrange's interpolation compute $A^n$ for $n\in\mathbb{N} $ So far I've worked out the minimum polynomial of $A$ to be $(x-2)(x+1)$ but ...
0
votes
1answer
23 views

finding the inverse of a matrx

In order to decrypt a cipher text using hill cipher, we must first find the inverse matrix of a given matrix. From this link http://en.wikipedia.org/wiki/Hill_cipher, ...
1
vote
2answers
32 views

How to calculate row sums of a power of a matrix

Let $P $ be an $n\times n$ matrix whose row sums $=1$.Then how to calculate the row sums of $P^m$ where $m $ is a positive integer?
0
votes
1answer
18 views

Derivative involving inner product

How would I take the derivative of a function $$f(x) = < x,x >=x^{T}x?$$ The answer seems to be 2x but I don't know how to explicitly show this other than saying "there are 2 x's being operated ...