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
0answers
7 views

Seeking for an operator on matrices similar to cartesian products of sets

Cartesian product is an operator on sets. What is the similar operator on matrices? That is, an operator, that receives two matrices, and the outcome is a third matrix, where the set of rows in new ...
0
votes
0answers
2 views

How to get the minimum singular value of some points covariance matrix?

I'm having trouble understanding the context of my question. I have a set of points which correspond to some 3D coordinates. I guess i need a minimum of two for my question. So the points would be ...
0
votes
2answers
23 views

Prove that $\mathrm{span}\{ I,A,A^2… \} = \mathrm{span} \{ I,A,A^2,…, A^{k-1}\}$

Let $A\in M_n(F)$ and $k=\deg(m_A)$ where $m_A$ is the minimal polynomial of $A$. Prove that $\mathrm{span}\{ I,A,A^2... \} = \mathrm{span} \{ I,A,A^2,..., A^{k-1}\}$ So we have that $m_A = a_0 ...
0
votes
1answer
24 views

What is the minimal polynomial of $A^2$?

Let $A\in M_n(\mathbb{C})$. The minimal polynomial of $A$ is $m_A = x^6 - 4x^4+3x^2 +1$. What is the minimal polynomial of $A^2$? I'd be glad for an hint/direction. Thanks!
0
votes
1answer
10 views

Setting up a matrix from a recurrence relation to find diagonal matrix?

Considering the recurrence $F_n= F_{n−1}+3F_{n−2}−2F_{n−3}$ where $F_0=0$, $F_1=1$ and $F_2=2$, use diagonalization to find a closed form of the expression. If the sequence is continued the numbers ...
1
vote
3answers
27 views

Tell if $A$ is diagonalized using it's characteristic and minimal polynomials

$$A= \left( {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right)$$ I already calculated that $f_A(x) = (x-3)(x-1)$. Also, the minimal polynomial must be also $(x-3)(x-1)$. How can I use ...
1
vote
2answers
15 views

Manifold of all 2x2 Hermitian Matrices

Is it true that the manifold of all $2\times 2$ Hermitian matrices is $\mathbb{R}^4$?
0
votes
1answer
16 views

Prove $A$ is scalar matrix

Let $A\in M_n(F)$ and let's assume $A$ has only one eigenvalue. Also, $A$ is diagonalized. Prove that $A$ is a scalar matrix. My Try: $${P^{ - 1}}AP = \left( {\matrix{ \lambda & {} & 0 ...
0
votes
2answers
34 views

Is this diagonal matrix unique? [on hold]

How can I determine if the diagonal matrix is unique? \begin{bmatrix} 1 & \frac{1}{2}-\frac{1}{2}i & \frac{1}{2}+\frac{1}{2}i \\ 2 & -i & i \\ 1 & 1 ...
-3
votes
4answers
54 views

How $a_{13}=0$ in $\begin{bmatrix} {2}&{1}&{0}\\ {1}&{3}&{5} \end{bmatrix}$?

I'm reading Artin's Algebra. $$\begin{bmatrix} {2}&{1}&{0}\\ {1}&{3}&{5} \end{bmatrix}$$ It says that $a_{ij}$ is the matrix entry such that $i$ is the horizontal coordinate and $j$ ...
1
vote
0answers
19 views

How to applied Gaussian Elimination for non-full rank matrix

I have a question about gaussian elimination. I want to find solution of $$Ax=b$$ as soon as possible using Gaussian Elimination. This is my matrix A ...
1
vote
2answers
25 views

Bounds of Sparse Matrix Multiplication

Does anyone know a good reference for bounds on sparse matrix multiplication? I'm interested in bounds of the number of scalar products required and bounds of the sparsity of the product. I know that ...
0
votes
1answer
21 views

Adjoint of a matrix and inverse of a matrix

As everyone know that we can use a matrix $A$ to represent an operator $T$. The adjoint of a matrix $A$ is denoted as $A^*$, which takes complex conjugate of $A$ and then transpose. My problem ...
0
votes
1answer
25 views

Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times ...
0
votes
0answers
8 views

Matrix derivatives for the HJB and ARE relationship

How does one take the derivative of these matrix equations? (Backround:{My professor used them in the proof showing that the Hamilton-Jacobi-equation equivalently solves the free end-point ...
0
votes
1answer
32 views

Obscure Rotation Matrix

The points of the shape after the shear are $(-0.25, 1)$, $(1.75, 1)$, $(0.25, -1)$, $(-1.75, -1)$. Other than that the only other information given is that the vertical edges of the original shaded ...
2
votes
1answer
53 views

How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
0
votes
0answers
13 views

Matrix coordinate blocks and null space of a vector

Suppose we have some non-zero $c \in \mathbb{R}^n$. Let $\{U_i\}_{i=1}^m$ be a collection of $n \times n_i$ matrices where each column is a standard basis vector. Suppose that each standard basis ...
0
votes
1answer
27 views

What does this determinant mean?

I have the following Jacobian matrix for an equilibrium of an SIR model $$J=\left( \begin{array}{cccc} -\text{$\alpha $N} & 0 & \zeta & 0 \\ \text{$\alpha $N} & -\beta -\rho & ...
0
votes
0answers
13 views

Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
0
votes
0answers
16 views

why are the residual sums of squares less than the true sums of squared errors? [on hold]

How would I go about showing this is true: $\sum \hat{e_i}^{2} \leq \sum e_i^{2}$ The LHS is SSE = Y(I-H)Y' and the RHS are the true squared errors.
1
vote
2answers
32 views

Find all 2 x 2 skew-symmetric matrices A [on hold]

Find all $2 \times 2$ skew-symmetric matrices $A$, if any, such that $A^2 + I_2 = 0$ Please help me! Thanks!!
1
vote
1answer
33 views

Give a general formula in terms of $n$ for the determinant of the following matrix.

Let $M_n$ denote the $n$ x $n$ matrix over $\mathbb{R}$ of which the entry in the $i$-th row and the $j$-th column equals $1$ if $|i-j|\leq 1$ and $0$ otherwise. For example: $M_6=$ \begin{pmatrix} ...
-2
votes
0answers
20 views

Help : Moore & Penrose : Find an $ 2\times 2$ matrix $A≠0$ where $A=A^*$ [on hold]

Could you please help me on how to solve that problem Find an $ 2\times 2$ matrix $A≠0$ where $A=A^*.\;\;$ (From Moore & Penrose) Thanks in advance!
1
vote
0answers
15 views

Compute new inverse when old inverse and new and old matrix known

Say I have a matrix $M$ and know its inverse $M^{-1}$. Then every element changes so that $M'=M+(M'-M)$. Is there a fast way to find $M'^{-1}$ from this information? That is without computing the new ...
0
votes
0answers
17 views

How to calculate this matrix in component-form? (Undergrad)

If ${A}_{ab} = \delta_{ab} + \varepsilon_{abc}n^c$ and $B^{ab} = \frac{1}{1+n^2}(\delta^{ab} + n^an^b - \varepsilon^{abc}n_c)$ what is the correct way to evaluate $$C^{ab} = (AB)^{ab} $$ Here, ...
0
votes
2answers
30 views

be J be a matrix so: JJ^-1 = I and A a matrix so: A^t JA = J. prove that A is invertible so that AA^-1 = I

be J be a matrix so: $JJ^{-1}$ = I and A a matrix so: $A^tJA^ = J$. prove that A can invertible so that $AA^{-1} = I$ the big question here is: what are the properties of A transpose, that allows ...
1
vote
1answer
21 views

A is normal if and only if every matrix unitarily equivalent to A is normal

I need to prove that if $A$ and $B$ are unitarily equivalent, then $A$ is normal if and only if $B$ is normal. The proof is as follows: Suppose $A$ is normal and $B = U^*AU$, where $U$ is unitary. ...
1
vote
1answer
24 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_± = ...
0
votes
0answers
19 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
votes
1answer
28 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 ...
1
vote
1answer
35 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. ...
0
votes
1answer
19 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 + ...
0
votes
0answers
23 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 ...
0
votes
1answer
41 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 ...
2
votes
1answer
32 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
votes
1answer
23 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
votes
0answers
18 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 ...
2
votes
1answer
37 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 ...
0
votes
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
vote
1answer
28 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 ...
0
votes
0answers
10 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 ...
0
votes
0answers
12 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 ...
0
votes
1answer
14 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 ...
0
votes
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?
1
vote
1answer
20 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 ...
2
votes
0answers
39 views
+200

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
vote
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
votes
1answer
22 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 ...
4
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 ...