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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1answer
20 views

Matrices, Transition matrix

I have a matrix $B:= \begin{bmatrix}0 & 1\\-1 & -\lambda\end{bmatrix} $ I need to diagonalise it and work out the transition matrix. I have worked out that the eigenvalues are $ \mu_± = ...
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0answers
13 views

If $\vec{v}$ is any non-zero vector perpendicular to $\vec{u}$, show that $\vec{v}$ is an eigenvector of $S$ [on hold]

I have the following problem.. I solved the first one however i can't find out how to solve the second (b). Suppose $\vec{u}$ is a unit row-vector in $\mathbb R^n$ , and $A=uu'$ matrix. (a) ...
2
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1answer
22 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
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0answers
17 views

Bound spectral radius of a certain matrix.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. ...
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1answer
14 views

$2\times2$ Matrix Problem and Recurrence relationship

For the $2\times2$ Matrix $A$, $a=1, b=1,c=1$ and $d=0$ Find a diagonal matrix $D$ and an invertible matrix $T$ such that $A = TDT^{−1}$. Hence solve the recurrence relation $f_{n+1} = f_n + ...
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0answers
19 views

Ways to prove that a matrix is nihilpotent/invertible

What are the ways to prove a matrix to be nilpotent/invertible? Showing that det(A) =! 0 is not possible and I can't find a way to have the polynomial in a recursive way. The dimension of A is ...
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1answer
28 views

How to rewrite a derivative w.r.t. tensor as w.r.t. vector

I'm stuck on a (probably very simple) problem I've come across. Take a function $f(A)$ where $A$ is a 2-tensor. Now suppose $A=vv^T$ for an $\mathbb{R}^n$ vector, $v$. I want to rewrite the object ...
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0answers
15 views

The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the ...
0
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1answer
22 views

Diagonalisation and Kronecker Product

If $A$ is a $n\times n$ matrix with complex numbers for elements, and $C$ the $2\times2$ matrix defined by $$\begin{bmatrix} -2&4\\-3&5 \end{bmatrix}.$$ How do you prove that the Kronecker ...
0
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0answers
16 views

Derivative of the detrminant map

Question : For $ v = (v_1, v_2) \in \mathbb R^2$ and $ w = (w_1, w_2) \in \mathbb R^2$, consider the determinant map $det : \mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R$ defined by $det ...
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1answer
34 views

Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
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2answers
56 views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
1
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1answer
27 views

“a matrix is positive semi-definite” not necessarily equavalent to “all leading principle minors are nonegative”?

Have a look at this matrix: $$ H = \left( {\begin{array}{*{20}{c}} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{array}} \right).$$ All the leading ...
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0answers
9 views

How to get relative rotation matrix from two orientation values in android?

Following http://www.codeproject.com/Articles/729759/Android-Sensor-Fusion-Tutorial , I get two orientation values. Then, I transform those values to rotation matrices R1, R2. I think the relative ...
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0answers
11 views

meaning of principal eigenvector of the normalized link matrix (pagerank)

The PageRank algorithm of a page is sometimes describe as: principal eigenvector of the normalized link matrix. What is the meaning of principal eigenvector and how does it relate to pageRank? This ...
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1answer
13 views

$M_n(D)$ is left and right-simple?

Is it true that if $D$ is a division ring and $n\in\mathbb{Z}_{\geq1}$, then the only left and right ideals of the ring $M_n(D)$ are the trivial ones? I know that $M_n(D)$ is simple, and the ...
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0answers
13 views

Matrix multiplication homomorphism

θ:M_2x2 (R)→R^+ defined by θ(A)=A_11 A_22+A_12 A_21β is this a homomorphism? How can you determine and if it is, how do you find the kernel?
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1answer
19 views

Basis and dimension of the subspace of solutions to $A\mathbf{x}=\mathbf{0}$

Consider $$ A =\left( \begin{matrix} 1 & -1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ \end{matrix} \right) $$ and find a basis and the dimension of $S(A,0)$, where $S(A,0)$ is the ...
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0answers
18 views

How many permutations do we need before we're in $SU\left( n\right)$?

Let $\mathcal{L}\subseteq \mathfrak{su}\left( n\right)$ be a Lie algebra for $n \geq 2$ with Lie group $G = e^{\mathcal L}$, and let $X \in G$ be represented by an $n\times n$ matrix (I prefer fixing ...
1
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1answer
34 views

Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
0
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1answer
21 views

Show matrix is element in eigenspace

Let $A$ be an $n\times n$ matrix such that $A^2=A$. a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show ...
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votes
4answers
123 views

Rank of a matrix $A^2$ without calculating the square

I have a matrix $A=\begin{bmatrix} 2 & 0 & 4\\ 1 & -1 & 3\\ 2 & 1 & 3 \end{bmatrix} $ with rank 2. How do I prove that the matrix $A^2$ has also rank 2 without actually ...
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0answers
12 views

path connectedness of space of almost commuting matrices

Let $R$ be a topological ring which is a domain. Let $n$ be an integer and let $\zeta_n$ be a $n$-th root of unity. Denote by $X$ the set of $m$ by $m$ invertible matrices with coefficients in $R$ ...
1
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1answer
27 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
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0answers
17 views

Tensor operator

I have come across the following expression: H:E where, H = e(levi-cita symbol)*a constant which means a 3rd order tensor with 27 components E = 2nd order tensor, now, what does H:E mean? I know ...
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2answers
54 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
1
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1answer
25 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...
0
votes
1answer
14 views

principal eigenvectors of an unknown matrix

Do you have any idea about how we can find the principle eigenvectors of an unknown matrix ${H}$. The only information that we have is that $H$ has only a few (up to 3) dominant eigen modes regardless ...
0
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2answers
26 views

Normal matrices connected?

Is the set of all normal matrices connected in $M_n(\mathbb{R})$, where the metric is the usual metric of $\mathbb{R}^{n^2}$? ($A$ is normal iff $AA^{t}=A^{t}A$.)
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3answers
66 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
0
votes
1answer
38 views

Inverse of a 3x3 matrix error!

I have this 3x3 matrix $$E_{ij} = g_{ij} + \bar{\epsilon}_{ijk}z^k$$ and want to derive its inverse. I know that its inverse is given by $$(E^{-1})^{ij} = \frac{1}{1+z^2}(g^{ij} + z^{ij} - ...
2
votes
1answer
44 views

Reduce matrix to Smith Normal form.

I've been given the finitely generated abelian group: $$\langle x_1, x_2 \mid 6x_1-6x_2, -6x_1-12x_2, 4x_1-8x_2\rangle$$ and written the corresponding matrix: $$A=\begin{pmatrix} 6 & -6 \\ -6 ...
3
votes
2answers
42 views

If $A^2$ is diagonalizable, must $A$ be such as well?

Given a diagonalizable matrix $A^2$, must the matrix $A$ be diagonalizable as well? I can prove that this is true for when $A\in M_{n\times n} (\mathbb{C})$ by using the theorem that the Minimal ...
4
votes
2answers
39 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
1
vote
2answers
15 views

Efficient inversion of a symmetric, positive definite matrix

I have to invert a symmetric, positive definite matrix in order to execute an extended Kalman Filter. I know quite some matrix decompositon methods like Cholesky or QR, but the question is what is the ...
1
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0answers
10 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
3
votes
0answers
106 views

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
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0answers
16 views

Are isometry matrices connected [on hold]

Let A be set of isometry matrix such that they are not square matrices. Is this set connected? I know that invertible matrices is a connected set
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0answers
38 views

Proving that elementary row operations are preserved after multiplication

If $E$ is an elementary $n \times n$-matrix, show that if $A$ is any $n\times n$-matrix, then $EA$ is a matrix obtained by carrying out a single elementary row operation on $A$, and that $AE$ is a ...
2
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1answer
48 views

Proof that an involutory matrix has eigenvalues 1,-1

I'm trying to prove that an involutory matrix (a matrix where $A=A^{-1}$) has only eigenvalues $\pm 1$. I've been able to prove that $det(A) = \pm 1$, but that only shows that the product of the ...
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0answers
54 views

Formula for Area of parallelogram induced by linear operator

I'm given that the linear operator $L: \mathbb R^2\to\mathbb R^2$ is invertible. The set (u,v) is a linearly independent set in $\mathbb R^2$. I must find a formula for the area of the parallelogram ...
0
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1answer
40 views

If $A$ is skew-symmetric, then a fixed row/column operation produces a new skew-symmetric matrix

Suppose $A$ is a skew-symmetric matrix. Fix an elementary row operation. If we carry out this row operation on $A$, and then carry out the corresponding column operation on the resulting matrix, do we ...
2
votes
1answer
24 views

Exponential of Matirx

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc) to calculate this exponential e^At with A (0 9) (-1 0) I'd ...
2
votes
1answer
47 views

Show that $EA$ is obtained from an elementary row operation on $A$

Suppose $E$ is an elementary $n \times n$-matrix. Prove that if $A$ is any $n\times n$-matrix and $E$ is any elementary matrix, then $EA$ is a matrix obtained by carrying out a single elementary row ...
1
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1answer
36 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
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1answer
20 views

Number of possible graphs from a reachability matrix?

I need to know how to work out how many possible different digraphs can be drawn from a given reachability matrix. It needs to be with the minimum number of arcs between the nodes within the graph ...
0
votes
1answer
22 views

Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), ...
1
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0answers
22 views

Can I reform this to a tensor/matrix product?

so I have the following vector matrix product: $$v = A w$$ Now I have this $n$-times: $$v^{(n)} = A^{(n)} w^{(n)} \quad \forall n$$ Is there any way to write this without $\forall$. Maybe somthing ...
1
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0answers
37 views

Eigenvalues of a matrix with special form

Let $p,a_1,...,a_n\in(0,1)$ and $\sum_{i=1}^na_i=1$. Now consider the following matrix: $$ \left(\begin{array}{ccccc} (1-p) & \sqrt{p(1-p)}a_1 & \sqrt{p(1-p)}a_2 & ... & ...
0
votes
2answers
19 views

Computing orthogonal projection

The question asks: A vector u and a line L in R^2 are given, compute the orthogonal projection w of u on L. u=[3,4] and y=-x In one example i was given two ...