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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3
votes
2answers
206 views

Eigenvalues of a tridiagonal stochastic matrix

I've tried to calculate the eigenvalues of this tridiagonal (stochastic) matrix of dimension $n \times n$, but I had some problems to find an explicit form. I only know that 1 is the largest ...
2
votes
2answers
141 views

Solving a system of equation modul0 5

Consider the system of linear equations $$\begin{pmatrix} 6 & -3\\ 2 & 6 \end{pmatrix}\begin{pmatrix} x_1\\ x_2 \end{pmatrix}=\begin{pmatrix} 3\\ 1 \end{pmatrix} $$ a) Solve the system in ...
2
votes
2answers
79 views

Maximum of two positive operators

Let $A,B$ be two positive operators in $B(H)$. Does there exist, in general, an operator $C$ such that for each $T$, if $A \leq T$ and $B \leq T$, then $$A\leq C \leq T\quad \text{and}\quad B\leq ...
0
votes
1answer
120 views

Linear System where Coefficient Matrix is a Kronecker Product

I have a system of linear equations where the coefficient matrix and right hand side is given by a Kronecker product: $(A_1 \otimes A_2) u = f_1 \otimes f_2$ My question: Is the solution simply ...
3
votes
0answers
36 views

Why must this kind of matrix be positive definite? [duplicate]

$A$ is an $n\times n$ matrix, whose elements can be expressed as, $a_{ij}=a^{-|i-j|}$, or, $a_{ij}=a^{-(i-j)^2}$, where $a$ is a constant and larger than $1$. Why is this kind of matrix positive ...
0
votes
2answers
92 views

Prove: if $A$ is an $n\times n$ real matrix such that $A^3 = A$, then $\det(A) = 0, -1,$ or $1$.

If $A$ is an $n\times n$ real matrix such that $A^3 = A$, then $\det(A) = 0, -1,$ or $1$. I'm not sure how to go about proving this. Please help. Thanks.
2
votes
1answer
210 views

Binary Operations for grouping

Which of the following binary operations are closed? subtraction of positive integers division of nonzero integers function composition of polynomials with real coefficients multiplication of ...
0
votes
0answers
48 views

Show that the product of complex matrices is a continuous mapping

Let $\mathbb{M}(n, \mathbb{C})$ be the space of all complex $n\times n$ matrices with operatornorm. On $\mathbb{M}(n, \mathbb{C}) \times \mathbb{M}(n, \mathbb{C})$ we define the product metric, ...
0
votes
1answer
48 views

$||AB||_{op} \leq ||A||_{op} ||B||_{op}$

How to prove the following? $||AB||_{op} \leq ||A||_{op} ||B||_{op}$ Research effort: $||A||_{op} = \sup_{||v||=1}||Av||$. So it's my task to show that: $$ \sup_{||v||=1}||Av|| \cdot ...
0
votes
3answers
101 views

Prove that $T(K) = a\ \mathrm{trace}(K)$ if $T(XY)=T(YX)$

Let $T: M_{n,n}(\Bbb R) \to \Bbb R$ be a linear transformation such that $T(XY)=T(YX)$ for any two $n\times n$ matrices $X,\,Y$. How do you prove that $T(K)=a\ \mathrm{trace}(K)$ for some scalar $a$ ...
1
vote
1answer
90 views

Eigenvalues of $A$, $B$ when $A-B\geq 0$?

If $A\geq B$ means $A-B$ is positive semi/definite can we say all eigenvalues of $A$ are greater than all eigenvalues of $B$? My actual problem is $A$ is positive semi/definite then I know $aA-A$ is ...
0
votes
1answer
49 views

How is $\det(I+aXY)=\det(I+aX^{\frac{1}{2}}YX^{\frac{1}{2}})$?

How can I prove that $$\det(I+aXY)=\det(I+aYX)=\det(I+aX^{\frac{1}{2}}YX^{\frac{1}{2}})$$ for positive semi/definite matrices $X, Y$ in complex plane and real positive $a$? Thanks a lot, PS. NOT ...
2
votes
2answers
65 views

Find all real matrices $A$ such that $A^2 = \mathrm{diag}(1,1,2,3,5,8,13)$

Let $A \in \mathcal{M}_{7 \times 7} (\mathbb{R})$ such that $$A^2= \begin{pmatrix} 1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&2&0&0&0&0 ...
1
vote
2answers
1k views

Show that the diagonal entries of symmetric & idempotent matrix must be in [$0,1$]

Show that the diagonal entries of symmetric & idempotent matrix must be in [$0,1$]. Let $A$ be a symmetric and idempotent $n \times n$ matrix. By the definition of eigenvectors and since $A$ is ...
2
votes
0answers
6k views

How to compute homography matrix H from corresponding points (2d-2d planar Homography)

I went through this thread Mapping Irregular Quadrilateral to a Rectangle If i know the 4 corresponding points in image say p1->p1' p2->p2' p3->p3' p4->p4' then how to compute pi(x,y) from pi'(x,y) ...
0
votes
1answer
774 views

Multiplying both sides of matrix equation by inverse

Say I have the following relationship between matrices: $AB = A^{2} + 2A$ If I multiply both sides of the equation by $A^{-1}$ is the resulting equation equivalent, meaning it doesn't change the ...
1
vote
2answers
66 views

Matrix Multiplication Problem

I'm working on the following problem and I can't seem to come up with the right answer. $$ \text{Let}: A^{-1} = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 1 \\ ...
1
vote
1answer
72 views

Matrices over a ring: does $PAQ=A'$ imply $\mathrm{Coker}A\cong\mathrm{Coker}A'$?

In A Singular Introduction to Commutative Algebra by Greuel & Pfister, there is written on p. 127: Let $R$ be a commutative unital ring and $A\in R^{n\times k}$, $P\in R^{n\times n}$, $Q\in ...
1
vote
1answer
48 views

Expressing elementary matrices in terms of each another

How can I express an elementary matrix of type 2 in terms of the product of elementary matrices of types 1 and 3? Just for clarity, here are the types: Type 1: \begin{bmatrix}1&a\\0&1\\ ...
0
votes
2answers
109 views

Echelon form of a system of equations?

My prof gave us this definition of an Echelon system: A system of m linear equations in n variables is called an echelon system if m ≤ n. Every variable is the leading variable of at ...
1
vote
1answer
55 views

What's the meaning of a formula in MatrixCookbook?

I'm learning the derivatives of matrices and vectors. In Matrix Cookbook Chapter 2(page 7), there is a formula as follows: $$\frac{\partial{X_{kl}}}{\partial{X_{ij}}}=\delta_{ik}\delta_{lj}$$ The ...
1
vote
2answers
3k views

Finding the inverse of a matrix by elementary transformations.

While using the elementary transformation method to find the inverse of a matrix, our goal is to convert the given matrix into an identity matrix. We can use three transformations:- 1) Multiplying ...
0
votes
3answers
2k views

for the following system to be consistent what must k not equal to

$−3x+5y+7z=7$ $−3x-7y+kx=8$ $15x+23y-19z=-40 $ by using echolon form I got to this \begin{bmatrix} -3 & -7 & k & 8 \\[0.3em] 0 & -12 & 5k-19 ...
2
votes
6answers
1k views

why is the square of this matrix with sin and cos equal to the identity matrix?

I have a question about why the square of the matrix Q, below, is equal to the identity matrix. Q = cos X -sin X sin X cos X My knowledge of ...
1
vote
2answers
59 views

Give positive integers $m$, $n$ and example of $m \times n$ matrix $A$ with the following property:

$Ax=b$ has no solutions for some $b \in \Bbb R^n$, and one solution for every other $b \in \Bbb R^n$. Can you please explain the reasoning behind your answer?
1
vote
1answer
250 views

How to express this in matrix notation (row-wise normalisation)

My questions are: How do I describe the row sum of a matrix? How do I describe the number of non-zero elements per row of a matrix in matrix notation? How do I divide a vector elementwise? To give ...
0
votes
1answer
56 views

Basic questions about Linear Mappings.

What is the difference between a change of basis and a simple linear transformation? Be A the matrix expressing the linear mapping T from U to W. Now, we know that columns of A represent the ...
0
votes
3answers
176 views

Book on Linear algebra/ Matrix analysis?

guys, I plan to learn more on the linear algebra/ matrix as I am going into convex optimizations. Basicly, I many need Matrix decomposition like SVD. Some contents on semidefinite/definite matrix. ...
3
votes
1answer
281 views

Eigenvalues of $AB$ from eigenvalues of $A$ and $B$

Is it possible to find the eigenvalues of $AB$ if we know the eigenvalues of $A$, say $\lambda_1, \lambda_2,...,\lambda_n$ and those of $B$ say $\lambda_1, \mu_2,...,\mu_n$ and given that $A$ and $B$ ...
1
vote
1answer
45 views

Convergence of an iterative process for matrix

$\{X^{(k)}\}$ is a sequence of $N\times M$ matrices given by $X^{(k+1)} = AX^{(k)}B+C$ where $A,B,C$ are $N\times N$, $M\times M$, $N\times M$ matrices repectively. How can I analysis the ...
0
votes
1answer
56 views

Can this matrix really be used as a preconditionner?

I've read Boxerman's thesis and I feel that there is possibly a mistake. We have to resolve $$Ax=b$$ $A$ is a positive-definite symmetric matrix and is very sparse so the conjugate gradient method ...
0
votes
1answer
222 views

What happens when solving a system of equations Ax=b for a matrix A that is nearly singular?

Which of the following are necessarily true when solving a system of linear equations Ax=b for a matrix A that is nearly singular? Note: the residual of a solution is defined here to be the Euclidian ...
0
votes
1answer
104 views

Bound of $\det$ of positive definite matrices

I need to know that if the following holds for complex vectors $x=[a\cdot A \mid b\cdot B]u$, and $y= [A \mid B]u$ $$\det(I+d \frac{yy^*}{rI})\leq\det(I+\frac{xx^*}{rI})\leq \det(I+c ...
0
votes
1answer
35 views

inverse of subelliptic matrix valued function

Let $M:\mathbb{R}^n\rightarrow GL(n,\mathbb{R})$ be such that for some $C>0$ $\frac{1}{C}\vert x\vert^2\leq x^TM(.)x\leq C\vert x\vert^2$ for all $x\in\mathbb{R}^n$. Does the map $x\mapsto ...
2
votes
0answers
42 views

Update SVD of multiplied matrices

I have a matrix $\Phi A \Phi^T$, where $A$ is a diagonal matrix that changes constantly and $\Phi$ is a $n\times m$ constant matrix ($n<m$) whose singular value decomposition is $UDV^T$. Question ...
2
votes
2answers
876 views

Is sum of two orthogonal matrices singular?

I am trying to solve following problem. Let $A,B\in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $det(A)=-det(B)$. How can be proven that $A+B$ is singular? I could start with implication: ...
1
vote
1answer
54 views

characterization of an infinite matrix mapping and continuity

Show that an infinite matrix mapping $A=[a_{ij}]$ $:l^{\infty}\to l^{\infty}$ is continuous iff $sup_{i\in \mathbb N}$ $ \sum_{k=1}^{\infty}{|a_{ij}|}=||A||<\infty$. Give a characterization of the ...
0
votes
1answer
149 views

Inverse of a certain block matrix

So I'm trying to compute the inverse of a block matrix that's a subset of a larger consideration I was attempting (this particular matrix comes from the normal and orthogonal equations for least ...
1
vote
1answer
127 views

Prove the following set is compact

$\def\R{\mathbb R}$Fix vectors $b\in\R^k_+$ and $D\in\R^k_{++}$, and a matrix $A\in\R^{N\times k}$. Here, $\R^k_+$ denotes the set of vectors in $\R^k$ whose entries are nonnegative, and $\R^k_{++}$ ...
0
votes
0answers
52 views

Does this type of matrix have a name?

I am using a square matrix that has the same values on the diagonal (say, alpha) and a different value for all diagonal elements (say, beta). Does this type of matrix have a name? It is such a ...
0
votes
2answers
137 views

Linear Algebra- Matrix derivative

I have a question related to finding derivatives of matrices. What is the derivative of $$(A.X)*(X^n)*(X.B)$$ with respect to $x_{11}$ ? . is the element wise product * is matrix product $x_{11}$ is ...
0
votes
1answer
163 views

Find a matrix (in a base $B$) the projection operator of space $V$ on the subspace $\ker(F-I_{v})$ along the subspace $\ker(f-2I_{v})$

$$ B = (v_{1}, v_{2}, v_{3}, v_{4})$$ - basis of the vector space V, F - linear operator on the $V$ $$M^{B}_{B} = ...
0
votes
1answer
38 views

Left inverse of a matrix with an extra condition

Let $A, B\in\mathbb{R}^{m\times n}$ be matrices with $m>n$ and suppose that $A$ and $B$ have orthonormal columns. Is there a matrix $C\in\mathbb{R}^{n\times m}$ with orthonormal columns such that ...
1
vote
1answer
30 views

Characterize if a triangular matrix is diagonalizable by the values on the diagonal.

Consider the following square matrix: $A= \left( \begin{matrix} \lambda_1 & a_{1,2} & & ... & a_{1,n} \\ 0 & \lambda_2 & & ... & a_{2,n} \\ . & & . & & ...
3
votes
1answer
41 views

A question about Golden - Thompson inequality

Given two hermitian matrices $A$ and $B$, the Golden - Thompson inequality states: $$tr\left(e^{(A+B)}\right)\le tr\left(e^Ae^B\right)$$ My question is: when the two traces are equal? Thanks.
0
votes
4answers
82 views

Is this a Positive Definite Matrix

Matrix $A$, $B$ and $C$ are symmetric Toeplitz matrices with $n$ by $n$ size, where $A$ and $C$ are positive definite matrices and $B$ or $(-B)$ is positive definite matrix, too. $R$ is a $2n$ by $2n$ ...
1
vote
3answers
1k views

what happens to rank of matrices when singular matrix multiply by non-singular matrix??

I need to find the prove for any rectangular matrix $A$ and any non-singular matrices $P$, $Q$ of appropriate sizes, rank($PAQ$)= rank($A$) I know that when singular matrix multiply by non-singular ...
0
votes
1answer
400 views

rank of complex conjugate transpose matrix property proof

I have a question about complex conjugate matrix. Prove that for any rectangular matrix $A$, rank $A$=rank $A^*$ where $A^*$ is complex conjugate transpose of A.
4
votes
1answer
198 views

What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
1
vote
1answer
150 views

Jordan block and cyclic vector spaces

I am currently reading this article about companion matrices [wikipedia][1] [1]: http://en.wikipedia.org/wiki/Companion_matrix . This brought me to the following question: I guess every companion ...