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
54 views

How do I show the covariance matrix of a multivariate normal random vector is positive definite?

The question is as follows: Suppose the $n$-dimensional random vector $\textbf{Z}$ has mean vector $\mu$ and variance-covariance $V$. By considering $Var(x^{T}\textbf{Z})$ for $x \in \mathbb{R}^n$, ...
0
votes
2answers
83 views

Orthogonal matrix and eigenvalues

How can I find an orthogonal matrix that can diagonalize the next matrix: $$M = \begin{pmatrix} \ a & b \\\ b & a \end{pmatrix}, b\ne 0.$$ Another question is how can I find the eigenvalues ...
0
votes
1answer
26 views

linear algebra - eigenvalues/vectors & diagnalization

$$R(θ) = \begin{pmatrix} \cosθ & -\sinθ \\ \sinθ & \cosθ \end{pmatrix}$$ $0 < θ < π$ Now, I understand that there are not any eigenvectors/values over $\mathbb R$ (but do has over the ...
1
vote
1answer
32 views

Why is $\left( \begin{smallmatrix} x & y \\ y & t \\ \end{smallmatrix} \right)$ orthogonally similar to this?

When working on a problem, I encountered the following statement. Let $x,y,t \in \mathbb R$ $\left( \begin{smallmatrix} x & y \\ y & t \\ \end{smallmatrix} \right)$ is orthogonally ...
1
vote
1answer
45 views

linear algebra and matrices, dimension

Let $W = \{p(B) : p\ \text{is a polynomial with real coefficients}\}$, where $B = \begin{pmatrix}0 & 1 & 0\\0 & 0 & 1\\1 & 0 & 0\end{pmatrix}$. The dimension $d$ of the ...
1
vote
1answer
38 views

Prove that number of nonzero elements in inverted matrix is at least 2n

Let $A$ is invertible matrix $n\times n$ with $ a_{ij} > 0 $ for every $i,j$. Prove that number of elements that equal to zero in $A^{-1}$ is less or equal to $n^2-2n$. In other words, $A^{-1}$ ...
2
votes
0answers
52 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
1
vote
1answer
44 views

Is $\lVert Ax \rVert^2 - \lVert Bx \rVert^2 = \lVert AA^T - BB^T \rVert$?

For matrices $A,B\in\mathbb{R}^{m\times n}$ and for any unit vector $x$, is the following true, and if so why? $\lVert Ax \rVert^2 - \lVert Bx \rVert^2 = \lVert AA^T - BB^T \rVert$ Equivalently, is ...
10
votes
1answer
926 views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
4
votes
2answers
78 views

A question about invertible matrices, $A,B$ are invertible matrices, $AB+BA=0$, show that n is even

Let $A,B\in M_n(\mathbb R)$ be invertible matrices, and let $AB+BA=0$, show that n is even. I know what the solution is: $AB=-BA\Rightarrow |1|=(-1)^n|1|\Rightarrow \text{n is even}$. So we ...
0
votes
1answer
53 views

Find unitary matrix so that $ P^{-1}BP$ is diagonal.

given is the matrix $ B = \begin{pmatrix} 1 & i & -i \\ -i & 2 & 0 \\ i & 0 & 2 \end{pmatrix} $. I have to find a matrix $P \in U(3)$ (in unitary group, meaning that $P^{-1}$ ...
1
vote
0answers
11 views

Symbol or name for Basismatrix of Linear Programming

This question is about the Basismatrix in the context of Linear Programming. Basically (haha!) we have the Matrix of the standard (or normal) form, which consists of (A|E) with the coefficient matrix ...
0
votes
1answer
216 views

Geometric meaning behind matrix-vector multiplication

Consider a matrix $A (m , n)$ and a vector $x (n , 1)$. I understand what the equation $Ax = b$ means ($A$ is transformation matrix and so on). I know what happens to $x$ due to this linear ...
1
vote
5answers
9k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
7
votes
3answers
99 views

Solving $X+X^T=tr(X)M$

Let $M$ be a $n\times n$ complex matrix. Solve the equation $X+X^T=tr(X)M$ where $X$ is a $n\times n$ complex matrix. I've done some case-checking. Suppose $X$ is a solution. if ...
4
votes
3answers
105 views

Prove or disprove that trace of matrix $X$ is zero

I was trying to solve a question from a competitive exam paper. This is a part of that question. Let $I_n$ and $O_n$ be $n\times n$ identity and null matrices respectively.Let $S$ be $2n\times ...
1
vote
1answer
34 views

how to prove the existence of a solution from an infinite linear system?

I need to prove the existence of a solution for variables $x_j$ with $j=1,2,3,\cdots,\infty$ from the linear system $$\sum_{j=1}^\infty A_{i,j}x_j=b_i (i=1,2,3,\cdots,\infty)$$ Where $A$ is a ...
4
votes
0answers
19 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
0
votes
1answer
39 views

invert or transpose

Is this correct: When finding the diagonalization of a matrix $A$ of the form $QDQ^{-1}$ then if you normalize your eigenvectors instead of having to invert $Q$, you could just take $Q^t$. Just ...
1
vote
3answers
651 views

How does a row of zeros make a free variable in linear systems of equations?

I don't understand how a row of zeros gives a free variable when solving systems of linear equations. Here's an example matrix and let us say that we're trying to solve Ax=0: $$\left[ ...
0
votes
1answer
40 views

Inverse matrix - transformation

I am finding inverse matrix $A^{-1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A. $$A=\begin{pmatrix}1 &3 & 9& 27\\3 & 3 & ...
0
votes
1answer
22 views

Show there's no ordered basis $E$ with the following conditions

Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that: $$T\left( {\matrix{ x \cr y \cr } } \right) = \left( {\matrix{ 2 & 1 \cr 3 & 4 \cr } } \right)\left( {\matrix{ ...
1
vote
1answer
29 views

Equation of matrices

Let $V$, a 3d vector space above $F$. Let $T:V\rightarrow V$, linear transformation and $E$, an "ordered" basis such that: $$[ T ]_E = \left( \matrix{ 0 & 0 & a \cr 1 & 0 & ...
1
vote
1answer
48 views

Any name for a special matrix with only non-zero entry

Consider an $n\times n$ matrix $\mathbf{E}_{ij}$ which is 1 at entry $(i,j)$ and zero everywhere else. Is there any special name for this kind of matrices?
3
votes
1answer
41 views

Comparing polar- and Cartan decomposition

Comparing polar- and Cartan-decomposition, can we conclude that every positive definite symmetric matrix $A\in\text{Gl}(n,\mathbb{R})$ can be written as $A=\text{diag}(\lambda_1,..,\lambda_n)\cdot P$ ...
2
votes
0answers
21 views

Finding the matrix ${\left[ T \right]_E}$

Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where: $${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$ It's given that: $${\left[ T \right]_{B \to E}} = ...
0
votes
1answer
63 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
4
votes
6answers
235 views

Find maximal possible determinant value given constraint

Task is to find maximal possible determinant value for 2x2 and 3x3 matrices given following constraint: $$\sum_{i,j=1}^na_{ij}^2 \le 1$$ I was able to come up with solution, but I received the test ...
2
votes
1answer
53 views

How to calculate the determinant of a matrix?

There are three $2\times 2$ matrices A, B and C they satisfy the following relation \begin{equation} A_{ij}=B_{ij}+[CB+(CB)^T]_{ij}x+(CBC^T)_{ij}x^2, \end{equation} where $x$ is an arbitrary variable. ...
2
votes
1answer
112 views

Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...
2
votes
2answers
498 views

If the Jacobian matrix is positive definite, does that imply that the optimization problem has a unique solution?

My PhD adviser told me that if the Jacobian matrix of the optimality conditions is positive definite, then it implies that the optimization problem has a unique solution. I was wondering what is the ...
1
vote
1answer
390 views

Hermitian Matrix Unitarily Diagonalizable

I am having trouble proving that Hermitian Matrices ($A = A^{*}$) are unitarily diagonalizable ($A = Q^{*}DQ$, where Q is a unitary matrix, $QQ^{*} = I$ and D is a diagonal matrix). I also know that ...
0
votes
0answers
116 views

norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...
0
votes
1answer
50 views

Matrices and rank exercise

Let $A \in \mathbb C^{m \times n}$ with $m \geq n$, of rank $n$. Prove there is $B \in \mathbb C^{n \times m}$: $BA=Id_n$. I have no idea how to solve the problem, I would appreciate suggestions.
2
votes
1answer
157 views

How to find $ P $ such that $ A^\top = PAP^{-1} $?

Let $ A $ be a matrix such that $A \in \operatorname{Mat}_3(\mathbb{R}) $. How to find $ P \in \operatorname{Mat}_3(\mathbb{R}) $, without doing heavy calculus, such that $ A^\top = PAP^{-1} $, where ...
5
votes
2answers
197 views

What linear transformations preserve these conditions?

Main Question Let's define $\Gamma(n)$ as the set of real antisymmetric matrices of size $n$ ($n$ is an even Integer), fulfilling: $$ \forall \gamma\in \Gamma(n) \Rightarrow \gamma^2=-\mathbb I_n$$ ...
0
votes
2answers
122 views

How to find matrix $A$ given $Ax=b$. Also $det(A)$ & $sum(A)$ are known. [duplicate]

Here is an example: $A = \begin{bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix}$ $x = \begin{bmatrix} 3 \\ 8 \end{bmatrix}$ $b = Ax$ so $b = \begin{bmatrix} 30 \\ 23 \end{bmatrix}$ Now i want to ...
0
votes
1answer
51 views

Diagonalizable A, computing fast.

I have $A =$ $ \begin{pmatrix} a & 0 & 0 \\ b & 0 & 0 \\ 1 & 2 & 1 \\ \end{pmatrix} $ I know that this matrix $A$ is diagonalizable when ...
0
votes
1answer
46 views

Commutativity in 2 dimensional vector spaces

If A,B are two matrices of order $2\times 2$ then prove that $(AB-BA)(AB-BA)$ commutes with any $2\times 2$ matrix. Please also comment on $why$ this is true and why this is not true for matrices of ...
2
votes
1answer
71 views

linearly independent vectors and rows/cols space

Given $n$ vectors, we want to determine if those vectors are linearly independent. One way doing it is writing those vectors as columns of a matrix and row-reduce it. The vectors are linearly ...
2
votes
0answers
75 views

Finding transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ …

Is there a Linear Transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ so $$\operatorname{Ker}T = \{( x,y,z,t,w) \in \Bbb R^5 \; | \; x = 2y, \text{ and, } z = 2t = 3w\}$$ if so find an example of ...
0
votes
1answer
175 views

Matrix with constant row sum

It is well known (and shown several times on this site) that if we have a matrix so that each row sums to zero then the matrix must be singular. I am curious if the following partial converse is ...
0
votes
2answers
44 views

A is inversible if and only if $rank(A)=n$ complex matrice?

Let $A\in \mathcal{M}_n(\mathbb{R})$ I know that : A is invertible if and only if $rank(A)=n$ Does the conclusion remains valid for $A$ a complex matrix?
1
vote
1answer
24 views

Is this subset of $PSL(n,\mathbb{R})$ Zariski-closed?

For some non-identity element $[A]\in PSL(n,\mathbb{R})$ ($[A]$ being the class of $A\in SL(n,\mathbb{R})$) and linearly independent vectors $x,y\in\mathbb{R}^n$, let $[x],[y]$ denote the classes of ...
0
votes
1answer
29 views

how to do SVD using the covariance matrix

I have an $(N \times M)$ matrix $A$ with $M \gg N$, $M$ being millions and $N$ hundreds, and I want to do $SVD$ on the matrix $A$. Can I do this calculation using $A\cdot A$ (the covariance matrix)?
0
votes
1answer
141 views

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please ...
0
votes
1answer
107 views

Important topics in Matrix analysis

I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic ...
1
vote
1answer
42 views

Polar decomposition of $\textrm{diag}(\lambda_1,…,\lambda_n)$

Determine the polar decomposition of $$\textrm{diag}(\lambda_1,...,\lambda_n)\in\textrm{Gl}(n,\mathbb{C})$$
2
votes
1answer
59 views

Find the differential of $f(A)=det(A^{-1}-A)$ where $A$ is invertible.

The question is if $A$ is an invertible matrix with real entries of size $n$. Is $f(A)=det(A^{-1}-A)$ differentiable? and what is the differential. I think I managed to show it's differentiable. the ...
0
votes
1answer
51 views

Calculating Eigenvalues and vectors for higher dimensions

Sorry for these questions, I'm trying to understand and cannot seem to figure it out anywhere. The material I've read so far on Eigenvalues and eigenvectors have showed for ...