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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2answers
63 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 \\ ...
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1answer
46 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\\ ...
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3answers
159 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. ...
1
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1answer
738 views

Proof of a matrix is positive semidefinite iff it can be written in the form $X'X$

I know the fact that a matrix is positive semidefinite if and only if it can be written in the form $X'X$. But how to prove it? Thanks in advance.
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2answers
98 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 ...
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2answers
2k 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 ...
1
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2answers
886 views

Derivative with respect to a matrix

How do we start with the matrix differentiation of this kind of equation? $$ V = [y_t-Cx_t]^TR^{-1}[y_t-Cx_t] $$ here $x_t$ and $y_t$ are vectors and $C$ and $R$ are matrices. R is a covariance matrix ...
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1answer
49 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 ...
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3answers
1k 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 ...
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2answers
58 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?
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5answers
439 views

Prove $BA - A^2B^2 = I_n$.

I have a problem with this. Actually, still don't have the right way to start :/ Problem : Let $A$ and $B$ be $n \times n$ complex matrices such that $AB - B^2A^2 = I_n$. Prove that if $A^3 + B^3 = ...
3
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1answer
235 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 $B$ ...
0
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1answer
53 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 ...
1
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1answer
43 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
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1answer
194 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
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1answer
54 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
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1answer
102 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 ...
3
votes
2answers
460 views

Finding all possible $n\times n$ matrices with non-negative entries and given row and column sums.

What I am ultimately attempting to do is find the solution that maximizes a given equation so I need to find all possible solutions so I can check them. I need all possible solutions to an $n \times ...
0
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3answers
795 views

Rotation matrix in 3-dimensional space with two angles.

I am trying to find a description of a rotation in a three-dimensional space with a matrix that uses only 2 angles. It is easy to find one which uses three angles, since I can always consider the ...
1
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2answers
115 views

Presentation of finite rings (fields)

One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. Can every finite ring be represented by a ring of matrices, i.e., is every ring ...
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1answer
60 views

Smart way $2\times 2$ JNF

I wanted to find a fast way to construct the JNF (with basis transformation) of a $2\times 2$ Matrix which is not diagonalizable, which means that we need to have one eigenvalue with algebraic ...
2
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0answers
40 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 ...
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2answers
448 views

Division matrix by polynomial [closed]

there is a polynomial: $$p(x)=1\cdot x^3+bx^2+cx+d$$ And there is a matrix of form - Toeplitz matrix with coeffcients of $p(x)$ on main diagonal: ...
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1answer
46 views

eigenvalue and independence

Let $B$ be a $5\times 5$ real matrix and assume: $B$ has eigenvalues 2 and 3 with corresponding eigenvectors $p_1$ and $p_3$, respectively. $B$ has generalized eigenvectors $p_2,p_4$ and $p_5$ ...
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4answers
78 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 $2n$ ...
2
votes
2answers
791 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
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1answer
122 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 $\R^k_{++}$ ...
2
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2answers
594 views

Partial derivative with respect to a vector x for $F(x) = x^TA(x)x$

I have the next function $F(x) = x^TA(x)x$, where $x$ is a real vector with dimension $n$, and $A$ is a square real matrix $n \times n$ depending on the components of $x$. How can I compute the ...
1
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1answer
176 views

How find this vmatrix of this $a_{n}=\frac{(2n-1)!!}{(2n)!!}$

find this value $$B_{n\times n}=\begin{vmatrix} a_{1}&a_{0}&\cdots&\cdots&0\\ a_{2}&\cdots&&\cdots&\cdots\\ \cdots&\cdots&&\cdots&\cdots\\ ...
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1answer
94 views

Need some help to understanding the formula

This is pinhole camera model (I don't get, is there [R t], or (R, t)) This formula is used to model the projection from a space point M to an image point m. Projection drawing Tilde over vector, ...
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0answers
50 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 ...
2
votes
2answers
205 views

Given factorization A = QR where Q's columns are pairwise orthogonal, but not orthonormal, how do i normalize Q's columns?

My questions is: Given a factorization A = QR where Q's columns are pairwise orthogonal, but not orthonormal, how do i normalize Q's columns while transforming R so result is still equal to A ? I ...
0
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1answer
36 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 ...
0
votes
2answers
163 views

Multiplication of square matrix that results in the component form $A_i \times A_j$

Suppose that there are some $n$ matrices, $A_1, A_2, ..., A_n$ One wants to form a square matrix $B$ that contains all aforementioned $A$ matrices as entries (this does not mean that all components of ...
3
votes
1answer
37 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.
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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
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1answer
306 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.
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1answer
134 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 ...
1
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2answers
114 views

Eigenvalues of $(A+B)^{-1}$

Suppose I know the eigenvalues of $A$ and $B$. Is there a way to write eigenvalues of the following? (1). $(A+B)$ (2). $(I+A)$ (3). $(I+A)^{-1}$ (4). $(A+B)^{-1}$ where $A, B$ are matrices in ...
1
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2answers
82 views

$\det (A^{-1})$ from eigenvalues of $A$

Suppose I have invertible square matrix $A$ in the complex field and I know all of its eigenvalues and they may be assumed to be non zero. Is there a way to write $\det(A)$ and $\det (A^{-1})$? PS. ...
2
votes
3answers
290 views

Conjugate elements of $GL_2(\mathbb{R})$

Decide whether or not the two matrices $A= \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$ and $B= \begin{pmatrix} 1 & 1 \\ -2 & 4 \end{pmatrix}$ are conjugate elements of the general ...
7
votes
5answers
479 views

Determine the value of a second determinant based on the first

I know the theory of determinants, but I have no idea how to apply it to this problem. Suppose $$\det\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} = 6$$ What is the value ...
2
votes
2answers
190 views

eigenvalues of two positive commutative matrices

Let $A$ and $B$ be two positive commutative matrices. I am going to prove $$\lambda_{j}(A+B)\leq \lambda_{j}(A)+\lambda_{j}(B)$$ for $j=1,2,\ldots n$, where $\lambda_{j}$ are eigenvalues of matrix ...
0
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1answer
65 views

Matrix inequalities

We know that for a complex number $x=a+ib , \forall a,b\in \mathbb{R}$ we have the following inequality $$|x|\leq |a|+|b|$$ Question: Do we have a matrix version of the above inequality? i.e. can we ...
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2answers
78 views

How can I prove $P^{-1}ABP=BA$?

Let $A$ and $B$ be $n \times n$ square matrices, with $\operatorname{det}(A) \ne 0$. Prove that $AB$ is similar to $BA$. Do I need to prove there is a matrix $P$ such that $P^{-1}ABP=BA$?
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2answers
2k views

Proving that similar matrices have identical ranks

Prove that if $A$ and $B$ are similar $n\times n$ matrices, then $\text{rank}A=\text{rank}B$. I can't seem to think of any relation between rank and similar matrices. The book does not have a ...
0
votes
1answer
50 views

Equality with norms of matrices

I have a problem with prooving of following equality: $$\|E(I-\frac{ss^T}{s^Ts})\|_F^2=\|E\|_F^2-\frac{\|Es\|^2_2}{s^Ts},$$ where $E\in\mathbb{R}^{n\times n}$ and $0\neq s\in\mathbb{R}^n$. I tried to ...
3
votes
1answer
137 views

Equation for finding maze solvability

I am programming a game where users can edit the state of a maze. The state of each vertical and horizontal wall (present/not present, on/off, 1/0, etc...) is stored in a database and then referenced ...
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0answers
57 views

Asymptotic behaviour of sum of inverse matrix elements

I was wondering if anyone knows if some theory exists on the following problem. I'm considering the minimization problem $h^2\boldsymbol{v}M_N\boldsymbol{v}$ subject to ...