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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2
votes
1answer
103 views

Diagonal Matrices with Zero on Diagonal

As far as I understand, a diagonal matrix is one whose non-zero elements are on the main diagonal. Am I correct in assuming that the diagonal can contain zeros as well? ie: $$ \begin{bmatrix} 0&0 ...
0
votes
1answer
234 views

Limit of the determinant of a series of matrices

Given the $N\times N$ matrix $A$, consider the series: $$B=\sum_{k=1}^{N}(A^k)^{-1}$$ where the symbol $o^{-1}$ means the inverse of $A^k$ is it possible and if yes how, to find all the matrices for ...
1
vote
1answer
61 views

Understanding Jordan blocks all in the same $GL_n(\mathbb{C})$-conjugacy class

This may be an elementary question and I haven't been able to find if such a question has been asked on Math StackExchange but here it is: suppose $M$ is an $n\times n$ matrix over $\mathbb{C}$ that ...
0
votes
1answer
26 views

Looking for general solution

A table of n rows and n columns is created such that value of ith row and jth is given by i + j + 1 . Some numbers are selceted in such a way that no other number is selected from corrosponding row or ...
1
vote
1answer
22 views

Why the final equality in this verification of positive semi-definiteness?

In showing that a particular Hessian is positive semidefinite the author writes: $$ \frac{2}{y^3} \left[ \begin{array}{cc} y^2 & -xy \\ -xy & x^2 \end{array} \right] = \frac{2}{y^3} \left[ ...
2
votes
1answer
79 views

Eigenvalues of hermitian matrix

I am just a bit confused, so: Is determining eigenvalues of hermitian matrix basically the same as how we determine eigenvalues of real-number matrices?
2
votes
1answer
226 views

positive definite of hessian matrix around a point

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^2 $ function and $x^*$ be a point such that $\bigtriangledown^2f(x^*)$ is positive definite.Is it always true that,there exists a neighborhood ...
1
vote
2answers
2k views

How to Find a Finite-Difference Matrix

I have read several websites trying to explain finite-differential equations, but I haven't been able to find one that explains how it's put into the matrix form. $f(x) = -\frac{d^2u}{dx^2}$ where ...
7
votes
1answer
470 views

A problem on skew-symmetric matrix

If $A∈M(n;\mathbb{R})$, let $A^t$ denote its transpose. A matrix $S\in M(n;\mathbb{R})$ is said to be skew-symmetric if $S^t = −S$. Pick out the true statements: a. If S ∈ $M(n;\mathbb{R})$ is ...
2
votes
4answers
138 views

Show that for vectors $\bf u$ and $\bf v$ in $ℝ^3$, $\bf u \times v = (-v) \times u$

How would I use the properties of determinants to show that for any two vectors $\bf u$ and $\bf v$ in $ℝ^3$ $$\bf u \times v = (-v) \times u$$
1
vote
2answers
132 views

Failure during calculating the matrix exponential, but where?

I have to calculate $e^{At}$ of the matrix $A$. We are learned to first compute $A^k$, by just computing $A$ for a few values of $k$, $k=\{0\ldots 4\}$, and then find a repetition. $A$ is defined as ...
1
vote
5answers
2k views

How to multiply a vector of scalars with a vector of vectors in Matlab?

This sounds a bit strange, I'll explain it further. Assume we have a row vector $c = (c_1,c_2,\dots,c_n)$ and we have $n$ column vectors $v_i\in\mathbb R^4$ for $i\in\{1,\dots,n\}$. The $c$ is stored ...
8
votes
0answers
147 views

Uniqueness of an infinite system of linear ODEs

How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices? More specifically, let $Q$ be a bounded infinitesimal generator, i.e. ...
0
votes
1answer
52 views

If $A$ is symmetric show that $(BA^{-1})^T(A^{-1}B^T)^{-1}=I$

If $A$ is symmetric show that $(BA^{-1})^T(A^{-1}B^T)^{-1}=I$ I can see that: $$ (BA^{-1})^T(A^{-1}B^T)^{-1}\\ (A^{-1})^TB^T(B^T)^{-1}(A^{-1})^{-1}\\ A^{-1}B^T(B^T)^{-1}A\\ ...\\ A^{-1}...A=I $$ I ...
1
vote
1answer
303 views

How to calculate the determinant of this 6x30 matrix?

I have this matrix: ...
0
votes
0answers
102 views

Let $a$ be any nonzero vector, If $v=a-\alpha e_1$, where …

Let $a$ be any nonzero vector, If $v=a-\alpha e_1$, where $\alpha =\pm \left\|a\|_2\right.$ and $H=I-2 (v v^T)/(v^T v)$, Show that $Ha=\alpha e_1$
1
vote
1answer
500 views

Sylvester's criterion and function extrema

If we have a function of three variables, $\ f(x, y, z)$ and want to look for its extrema, we can use Sylvester's criterion by creating a matrix: $$\begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \\ ...
1
vote
2answers
256 views

Ordered quadruples in a grid

Moderator Note: This question is from a contest which ended on 22 Oct 2012. Consider $(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ such that the ordered quadruple satisfies the following: ...
3
votes
4answers
807 views

diagonal dominance versus positive semi-definitenesss

I know that for a symmetric matrix $A$, diagonal dominance, i.e. $|A_{ii}|\ge \sum_{j}|A_{ij} |$ implies positive semi-definiteness, right? How about the other way? Does positive semi-definiteness ...
1
vote
0answers
203 views

The norm of a diagonalizable matrix is its largest eigenvalue?

In relation to the Euclidean norm... What are the conditions for when this occurs? Is it only real symmetric matrices? When is this not the case?
1
vote
1answer
42 views

Eigenvalues vs.matrix sparsity

For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity? Is there a closed-form expression that states this ...
0
votes
1answer
40 views

T/F Adding Linear Algebra Solutions

T or F If $\mathbf{A}\vec{x} = \vec{b_{1}}$ has no solution, $\mathbf{A}\vec{x} = \vec{b_{2}}$ has many solutions, then $\mathbf{A}\vec{x} = \vec{b_{1}} + \vec{b_{2}}$ has many solutions. This is ...
3
votes
2answers
54 views

T/F Underdetermined Nullspace

True or False: Let $A$ be an $m\times n$ matrix. If $m<n$ then $\dim(\ker(A)) > 0$ (i.e the dimension of the null space of is positive). This is true? Because the $n$-$m$ extra ...
1
vote
2answers
51 views

How can I prove the IFF condition for this group of matrices?

I'm tasked with showing that matrix $A$ commutes with every $2\times2$ matrix if and only if $A = \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$ for some a. I was able to prove in the first ...
2
votes
1answer
928 views

How to remove linearly dependent rows/cols

How would one remove linearly dependent rows/columns from a rank-deficient matrix. For example, (from wikipedia): $$ A = \begin{bmatrix} 2 & 4 & 1 & 3 \\ -1 & -2 & 1 ...
1
vote
2answers
977 views

Find the inverse a matrix with trigonometic entries

What is the inverse of \[ \begin{pmatrix} 1&0&0\\0&\cos x &\sin x\\ 0 &\sin x &-\cos x \end{pmatrix} \] Please help me to solve the above problem.
2
votes
2answers
159 views

Can we assume left and right pseudo-inverse to be close enough?

As we know, the left pseudo-inverse of matrix $A$ is $$A^{\dagger}_l = (A^TA)^{-1}A^T,$$ and we have $A^{\dagger}_lA = I$. The right pseudo-inverse of matrix $A$ is $$A^{\dagger}_r = A^T(AA^T)^{-1},$$ ...
1
vote
1answer
46 views

T/F Linear Algebra Uniqueness

Q: If $A$ is a $3 \times 3$ matrix and $y$ is a vector in $\mathbb{R}^3$ such that $Ax = y$ does not have a solution, then there exists no vector $z$ in $\mathbb{R}^3$ such that the equation $Ax = z$ ...
1
vote
2answers
309 views

Condition for solvability of a linear system

Given a linear system $Ax=b$, for $A\in\mathbb{R}^{n\times n}$, does the exact solution $x$ exist if $b~\bot~ Ker(A^T)$, ie. vector $b$ is orthogonal to each vector of the null-space of $A^T$? If one ...
1
vote
2answers
168 views

Row reduction over any field?

EDIT: as stated in the first answer, my initial question was confused. Let me restate the question (I have to admit that it is now quite a different one): Let's say we have a matrix $A$ with entries ...
2
votes
1answer
577 views

Multiple choice question on matrix with complex entries

Let $A$ be an $n$-th order square matrix with complex entries. Which of the following statements are true? (a) $A$ is always similar to a diagonal matrix. (b) $A$ is always similar to an ...
1
vote
1answer
178 views

Solve $Bx = [ 1 2 1 ]^T$ for each $k \in \{ 1, 2, -3 \}$?

Where $B$ is the matrix: $$ \begin{pmatrix} 1 &1 &k\\ 2 &k &1 \\ k &2 &2 \end{pmatrix} $$
1
vote
0answers
123 views

How does adding extra row and column of ones affect a matrix's inverse?

I'm working on a homework problem, and I'm stuck. I guess my linear algebra still needs some work... I've arrived at $\mathbf{D}= \left[ \begin{matrix} \mathbf{C} & \mathbf{1}^T \\ \mathbf{1} ...
7
votes
1answer
507 views

Algorithm to find conjugacy classes of subgroups/elements (in matrix groups)?

I'm looking for a simple (=doable to implement by myself) algorithm to compute the conjugacy classes of elements and subgroups of a given subgroup of $\text{P}{\Gamma}\text{L}(n,q)$. So given a group ...
8
votes
3answers
235 views

Minimize $||Ax-b||$ but for $A$, not $x$

I have a machine learning regression problem. I need to minimize $$ \sum_i||Ax_i-b_i||_2^2 $$ However I am trying to find matrix $A$, not the usual $x$, and I have lots of example data for $x_i$ and ...
2
votes
1answer
481 views

Solving a system of equations in a non-square matrix

Mrs Brown has pet cats, pet parrots, pet fish and pet rocks. Each pet rock has three eyes, no legs and no tail. The other pets have as many of these features as you would expect. Let the number of ...
2
votes
2answers
449 views

Intuition for the Product of Vector and Matrices: $x^TAx $

When I took linear algebra, I had no trouble with the mechanical multiplication of matrices. Given the time to write things out and mumble a bit about ith and jth rows, I can do the products no ...
3
votes
0answers
174 views

Solving a system with present pre-multiplication

Suppose a symmetric matrix $L\in\mathbb{R}^{n\times n}$ is given, and a rectangular matrix $A\in\mathbb{R}^{n\times m}$, $m<n$. A solution to the system $$LAx=b, \tag 1$$ is sought, for known ...
6
votes
3answers
472 views

Image of determinant on symplectic/orthogonal matrix group

Let $\mathbb K$ be a field, $n\geq 1$, and $G=GL_n({\mathbb K})$ be the group of invertible $n \times n$ matrices with coefficients in $\mathbb K$. For $J\in G$, we can define (in analogy to ...
3
votes
2answers
1k views

By using properties of determinants show that

$$\begin{vmatrix}1+a^2-b^2&2ab&-2b\\ 2ab&1-a^2+b^2&2a\\ 2b&-2a&1-a^2-b^2\end{vmatrix}=(1+a^2+b^2)^3$$ I have been trying to solve the above determinant. But unfortunately my ...
6
votes
1answer
678 views

Additivity of the matrix exponential of infinite matrices

It is well known that the matrix exponential of finite dimensional matrices is additive if the exponents commute: $AB=BA\implies e^Ae^B=e^{A+B}$ (cf. e.g. Bernstein, Corollary 11.1.6). Under what ...
5
votes
1answer
2k views

Generalized rotation matrix in N dimensional space around N-2 unit vector

There is a 2d rotation matrix around point $(0, 0)$ with angle $\theta$. $$ \left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right] $$ Next, ...
0
votes
1answer
281 views

Complex number matrix calculation

How do I find a complex number $\lambda $ such that $\pmatrix{ 3&-2\\2&3}\vec v$ = $\lambda\vec v $ where $ \vec v $ is non-zero. Yes, this is a homework problem, I didn't learn complex number ...
2
votes
1answer
249 views

Matrix product and linear independence.

In my problem, $R$ is a commutative ring with identity. Suppose $A$ is an n by n matrix with entries from R. Let $\operatorname{rk}(A)$ be the largest $m$ such that some $m$ by $m$ submatrix of $A$ ...
1
vote
1answer
249 views

Similar matrix proof

$A$ and $B$ are similar matrices, if $B=PAP^{-1}$ holds for a square, non-singular matrix $P$. Now am wondering if $S^{-1}T$ and $S^{-1/2}TS^{-1/2}$ are similar matrices? Am looking for a proof for it ...
2
votes
1answer
622 views

How do I prove that in every commuting family there is a common eigenvector?

The proof given by my textbook is highly non-satisfying. The author adopted some magic-like "reductio ad absurdum" and the proof (although is correct) didn't reveal the nature of this problem. I made ...
1
vote
1answer
51 views

A linear algebra problem - matrix equation

Let $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n$ column vectors, each with the same $n$ components. So: \begin{equation} \mathbf{v}_i = \left[\begin{array}{c}v_i\\ v_i \\\vdots \\ ...
2
votes
1answer
339 views

Properties of diagonal and permutation matrices.

I've been reading about equivalent codes, and the topic of monomial automorphisms came up. These are the set of monomial matrices (square matrices with exactly one nonzero entry in each row and ...
13
votes
2answers
2k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
1
vote
0answers
116 views

Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...