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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0
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1answer
90 views

matrix positive semidefinite

If $n \times n $ matrices $A, B$ are positive semi-definite, matrices $P$ and $Q$ are $n\times p$ and $n\times q$ matrices and their column vectors are orthogonal, which is to say $$P^{T}P=I_{p\times ...
1
vote
1answer
225 views

What guarantees that joint diagonalization is possible?

According to the Wikipedia article on the common spatial pattern algorithm, one can find the following matrices by joint diagonalization of a pair of covariance matrices $R_1$ and $R_2$: $$ P = ...
2
votes
2answers
187 views

$A$ is a real $n\times n$ matrix such that $AA^T=A^TA$ and all eigenvalues real. Then A must be symmetric

Once again, this is from a past qualifying exam I am trying to work on. Here is the problem. True or False? Let $A$ be a real $n\times n$ matrix such that $AA^T=A^TA$ and all eigenvalues of $A$ ...
2
votes
3answers
239 views

Find all the linear involutions $f: E \to E$, where $E$ is a finite-dimensional real vector space

Can someone help me? I've been thinking about this question for a while and got stuck. At first I only found the Identity transformation ($I$) and the anti-Identity transformation ($-I$). But then I ...
4
votes
1answer
129 views

Inequalities for Differences of Absolute Values of matrices

Let $A$ and $B$ be two real symmetric $n\times n$ matrices. Let $A=USU^T$ be the eigen-decomposition $A$ and let $|A|=U|S|U^T$ where $|S|$ just denotes elementwise absolute value of the diagonal ...
8
votes
5answers
9k views

Prove that if $AB$ is invertible then $B$ is invertible.

I know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible ...
2
votes
1answer
74 views

Can this matrix inverse be re-written?

I have a matrix inverse of the form $(\mathbf{AB}+\mathbf{C})^{-1}$, where each matrix is $2\times 2$ and each of the subelements below are known a priori. $$ \left( \left[ \begin{array}{cc} ...
1
vote
1answer
38 views

Conditions under which $BA = I_{n}$, where $A\in\mathbb{C}^{m\times n}$ and $B\in\mathbb{C}^{n\times m}$

Let $A\in\mathbb{C}^{m\times n}$ . I want to to know what conditions can I apply on the matrix $B\in\mathbb{C}^{n\times m}$ such that product $BA = I_{n}$ or matrix $B$ is the left inverse of the ...
1
vote
0answers
83 views

Certain matrix inequalities

I want to solve the following inequalities: \begin{equation} \left| Tr\left( \frac{(X\otimes Y).A.(X\otimes Y)^*.B}{Tr((X\otimes Y).A.(X\otimes Y)^*)}\right)\right|>2, \quad\text{given} \quad ...
3
votes
1answer
176 views

To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
1
vote
1answer
184 views

Optimization problem to find an optimal matrix

I need to find a $n\times m$ matrix $N$ with binary values $(0,1)$ which will maximize an objective function. N(i,j)=0 or 1 indicates whether jth offer is made to ith customer $m$ represents number ...
1
vote
1answer
64 views

Matrix Decomposition?

How do you prove that $$ GL(n,\mathbb{C})/ B(n, \mathbb{C}) \sim U(n)/T(n) $$. Where $ B(n,\mathbb{C}) $ is the group of invertible upper triangular matrices, and $ T(n) $ is the group of diagonal ...
1
vote
1answer
171 views

How to generate algebraic span of a set of matrices (how many multiplications?)

I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all? Suppose you have ...
2
votes
1answer
55 views

Solve for symmetric matrix

I have the following equation: $$ \left( \begin{array}{c} \mathbf{I}\\ \mathbf{K} \end{array} \right)\mathbf{x} = \mathbf{y} $$ where $\mathbf{K}$ is a symmetric $2\times 2$ matrix, $\mathbf{I}$ is ...
2
votes
1answer
790 views

Orthogonal projection onto an affine subspace

If we want to find the distance from a vector $x$ to a subspace $S$, we take $\| (I-P_S) x\|$, where $P_S$ is the orthogonal projection onto the subspace $S$. Obviously we could do the same thing for ...
2
votes
2answers
450 views

How rank is related to characteristic polynomial of the adjugate matrix?

I am trying to solve an exercise to prove if $A$ is semi-positive definite matrix, then its adjugate matrix $A^{*}$ is also semi-positive definite. The proof comes to that, if $A$ is not full rank ...
0
votes
1answer
54 views

What kind of/family of (real) $N\times N$ matrix has unique (real) $n$ eigenvalues (spectral theorem does't guarantee uniqueness)

I'm wondering what kind of/family of $N\times N$ (real) matrix has unique(distinct) $N$ (real) eigenvalues. The spectral theorem doesn't guarantee uniqueness.
5
votes
1answer
170 views

Positive semidefinite matrix problem

This is a simple question, at least, looks like. Let $x\in\mathbb{R}^n$ and consider the matrix $C$ such that $C_{ij}=|x_i|+|x_j|-|x_i-x_j|$, show that $C$ is positive semidefinitive. I could prove ...
1
vote
0answers
141 views

Convergence of the Jacobi iteration method

I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm $||I-Q^{-1}A||_\infty < 1$ and ...
1
vote
1answer
48 views

Combinations of matrix with 1's and -1. Where Row and Col product is 1

Suppose you have a $5\times 5$ matrix where each element is either $1$ or $-1$. How many unique matrices are there such that each row and each column multiplies to $1$? How to adapt this to a ...
5
votes
1answer
95 views

$\|A-B\|^2 = ?$

We know that if $x,y \in \mathbb{R}$ \begin{equation} (x-y)^2 = x^2 -2xy + y^2 \end{equation} If $x,y$ are vectors in $\mathbb{R}^n$ we have \begin{equation} |x-y|^2=|x|^2 - 2 \ x \cdot y +|y|^2. ...
4
votes
1answer
999 views

Pseudo inverse of a product of two matrices with different rank

Let $V$ be an $n \times n$ symmetric, positive definite matrix (of rank $n$). Let $X$ be an $n \times p$ matrix of rank $p$. Define $A^- = (A^\top A)^{-1} A^\top$ as the pseudo inverse of $A$ when ...
2
votes
1answer
88 views

Divergent Mercador Series of Matrix Logarithm

The logarithm of a matrix $$ \ln(I+A)=\sum_{k=1}^{\infty}{(-1)^{k+1}\over k}A^k$$ converges when ${\rho}(A)<1$ Suppose $n>{\rho}(A)>1$. Can one use the following transformation ...
0
votes
2answers
1k views

Finding reflection transformation matrix

I have two 3 dimensional points. $A [x_1, y_1, z_1]$ and $B [x_2, y_2, z_2]$. I need to find a transformation matrix which when multiplied to $A$ will give me $B$ and when multiplied by $B$ give me ...
7
votes
4answers
637 views

$\det(I+A) = 1 + tr(A) + \det(A)$ for $n=2$ and for $n>2$?

Let $I$ the identity matrix and $A$ another general square matrix. In the case $n=2$ one can easily verifies that \begin{equation} \det(I+A) = 1 + tr(A) + \det(A) \end{equation} or \begin{equation} ...
0
votes
2answers
85 views

Matrix; Linear transformations

Let $ ( x , y ) $ be the co-ordinates of a point P referred to a set of rectangular axes $OX$, $OY$. Then its co-ordinates ($x^{'}$,$y^{'}$) referred to $OX^{'}$, $OY^{'}$, obtained by rotating the ...
3
votes
2answers
2k views

Any good proof for Strictly diagonally dominant matrix are non singular

Hi try to find a good proof for strictly diagonally dominant matrices, There a proof for on the paper but I'm wondering whether are better proof such as using deterinant etc to show SDD is non ...
0
votes
0answers
922 views

Getting non-singular (invertible) matrix from a singular one

I am struggling with specific matrices that are singular, but I would like to make them invertible (non-singular). My first ideas are: to eliminate certain rows and columns; to find "healthy" ...
3
votes
2answers
3k views

How to construct magic squares of even order

Could someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?
1
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0answers
63 views

Why the following observations regarding lattices hold?

The following is an excerpt of a recent paper on lattice cryptography: Let $n$ and $q$ be integers [...], and let $\beta > 0$ . Given a uniformly random matrix $A \in \mathbb Z^{n \times m}_q$ ...
5
votes
2answers
250 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
3
votes
2answers
154 views

Problem with matrix and vector norms

I already try to multiply by a orthogonal matrix in both sides, multiply by $q$ and $d$, factor, expand...nothing works. This problem comes from Demmel's book, Applied Numerical Linear Algebra. Let ...
4
votes
1answer
223 views

$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant

Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to $1$. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the ...
1
vote
1answer
708 views

Finding the inverse of a matrix using elementary matricies

Can somebody help me understand what exactly is being asked here? I understand how to construct elementary matrices from these row operations, but I'm unsure what the end goal is. Am I to assume that ...
0
votes
1answer
180 views

Is there fast approximation of the n-th power of diagonalizable matrix A?

My thoughts on the subject. Because of diagonalizability $A$ can be written as $A = PDP^{-1}$ and then $A^n = PD^nP^{-1}$ here $P$ is matrix of eigenvectors and $D$ is diagonal matrix with ...
1
vote
2answers
136 views

If $MM^{'}$ is positive definite, $M$ is invertible? [duplicate]

Supposing I have a square real matrix M. If $MM^{T}$ is positive definite, is $M$ invertible? I came up with the proof $MM^{T}=M\times I \times M^{T}$, that is equal to say $MM^{T}$ is congruent to ...
3
votes
5answers
302 views

How to generate unique id from each element in matrix?

I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the ...
1
vote
1answer
75 views

Let $R$ be a unitary noncommutative Artinian ring, how can I prove that $M_n(R)$ is Artinian?

I found a topic here Matrix Rings over Artinian commutative Rings, and know that Artinian is a Morita invariant condition. But I'm looking for a direct proof of it. Now let $\rho_1 \supset \rho_2 ...
15
votes
1answer
943 views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $ A_{ij}=\frac{1}{i + j - 1}$. I need to show that $A$ is invertible and the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
1
vote
0answers
219 views

Diagonal elements of a matrix multiplication

I have two $n \times n$ matrices, $Q$ and $P$, with $n > 2$. These matrices have the following properties. All their elements are non-negative real numbers. Their diagonal elements are zeroes. ...
1
vote
1answer
397 views

Cholesky/LU decomposition from matrix and its inverse?

Usually, we have a matrix $A$ and want to calculate the $LU$ (or sometimes Cholesky, depending on $A$'s properties) decomposition. This is often the hard part. Now, if we have the $LU$ decomposition ...
2
votes
3answers
80 views

eigenvalues properties

So I have a question which I do not know how to solve. $A \in M_{m}( \mathbb{C})$ and $ \exists r > 0 $ such that $A^{r}=I$ $\lambda$ is eigenvalue of $A$ and there is no other eigenvalues. ...
2
votes
2answers
94 views

the strategy about a $0-1$ matrix game

Given a $4\times4$ binary matrix as following, $\left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 ...
1
vote
2answers
466 views

Find a vector $\mathbf x$ whose image under $T$ is $b$.

I am having trouble with this question and how to get the answer. With $T$ defined by $T(\mathbf x)=A\mathbf x$, find a vector $x$ whose image under $T$ is $b$. $$ A = \begin{pmatrix} 1 & -3 ...
1
vote
3answers
157 views

using the spectral decomposition of a matrix

if I have $$ A= \begin{bmatrix} \hphantom{-}1 & 0 & 0\\ -1 & 1 & 1 \\ -1 & 0 & 2\\ \end{bmatrix} $$ I am given the characteristic values are $$ \lambda_{1}=1 \text{ and } ...
1
vote
0answers
42 views

Looking for a “Neat” Transform to Yield a Convex Set

Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly ...
2
votes
1answer
289 views

Property for Norms of Matrices

I am having trouble with the following problem: Show that the vector norm $||x||_1$ gives the subordinate matrix norm: \begin{equation} ||A||_1 = \max_{1\leq j\leq n}\sum_{i=1}^n|a_{ij}| ...
1
vote
2answers
36 views

Finding the Matrix of a linear map $T$.

Let $T\colon \mathcal M_{22}(\Bbb R) \to \mathcal M_{22}(\Bbb R)$ be defined by: $ T\left(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\right) = \begin{bmatrix} 2c & a+ c\\ b-2c & ...
1
vote
1answer
111 views

Norms and invertibility of a summation

I need to show that if $X,Y$ are matrices with $X$ invertible and $$\lVert Y-X\rVert < \lVert X^{-1}\rVert^{-1}$$ then $$Y^{-1} = X^{-1} \sum_{k=0}^\infty (I - YX^{-1})^k,$$ where $I$ is the ...
2
votes
1answer
2k views

Gaussian Elimination with Scaled Row Pivoting for numerical methods

I am solving a system first with basic Gaussian Elimination, and then Gaussian Elimination with scaled row pivoting (used in numerical methods) Basic Gaussian Elimination on the system $Ax=b$: ...