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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1answer
366 views

Natural basis for set of m x n matrices, visualized

I am told to consider the $m \times n$ matrices $E_{pq}$ described by: $[E_{pq}]_{ij} = \left\{ \begin{array}{ll} 1 & \textrm{if } i=p\textrm {, }j=q \\ 0 & \textrm{otherwise} \end{array} ...
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1answer
1k views

Finding the position of a point after rotation: Why is my result incorrect

I am attempting to calculate the position of a point after it has been rotated I have been using an algorithm but I am getting incorrect values which makes me think I am using the incorrect algorithm ...
4
votes
1answer
657 views

Symmetrizing matrix properties

A symmetrizer $P$ is a $n\times n$ symmetric matrix such that for a $n\times n$ matrix $A$ it holds that $AP=PA^T$. There exists a symmetrizer for any square matrix, and in general it is not unique. ...
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1answer
195 views

Partial derivative involving trace of a matrix

Suppose that I have a symmetric Toeplitz $n\times n$ matrix $\mathbf{A}=\left[\begin{array}{cccc}a_1&a_2&\cdots& ...
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1answer
99 views

General transformation matrix

I am currently working through some of my maths assignment, and i have this question, and i can't work out what it means, and i am sure there is something to missing which would make this question ...
1
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1answer
337 views

Spectrum shift except for zero eigenvalue

Suppose a square, real and symmetric matrix $G\in\mathbb{R}^{n\times n}$ is given, and it is known to have one zero eigenvalue associated with all ones eingenvector, $1_n$. I'm aware that the ...
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1answer
94 views

A lemma about matrices

Can someone explain, or even prove the lemma? I've thought about it for about 3 hrs but no idea about the lemma. $$\begin{align*} R_1 &=\left(\begin{array}{rrrr} -1 & 0 & 0 & 0\\ ...
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0answers
598 views

General properties of eigenvalues of a Jacobian matrix when premultiplied by a symmetric, positive definite matrix?

For a particular engineering problem that I'm working on, I have computed a Jacobian matrix $J$ and there is another matrix $M$ associated with the problem. $M$ is known to be symmetric, real-valued, ...
4
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3answers
2k views

Integrating matrix exponential

I have a question about equation 6 in this paper. Simplifying somewhat, the authors state the following $$\int_0^{\infty} e^{-tL} dt = L^{-1}$$ $L$ here is a graph laplacian and therefore is a ...
3
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1answer
706 views

Convert from fixed axis $XYZ$ rotations to Euler $ZXZ$ rotations

Because I'm a new user, I can't post images or hyperlinks, there is a complete version with images here: ...
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3answers
4k views

Dimension of subspace of all upper triangular matrices

If $S$ is the subspace of $M_7(R)$ consisting of all upper triangular matrices, then $dim(S)$ = ? So if I have an upper triangular matrix $$ \begin{bmatrix} a_{11} & a_{12} & . & . ...
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1answer
201 views

Need a transformation matrix to convert to new base vectors

I was searching for a solution, but can't find anything I can use with my superficial knowledge. So, I have vector A, vector B & vector C. I want to convert the space to base vectors A & B ...
3
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1answer
354 views

$PGL(n, F)=PSL(n, F)$

$PGL(n, F)$ and $PSL(n, F)$ are equal if and only if every element of $F$ has an $nth$ root in $F$.($F$ is finite field) I can show that if $PGL(n, F)=PSL(n, F)$ then $|F|$ have to be even.I have ...
3
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1answer
430 views

Finding a basis for the columnspace of a matrix

Find a linearly independent set of vectors that spans the same subspace of $R^4$ as that spanned by the vectors - $$ \begin{bmatrix} 2 \\ -4 \\ -1 \\ -2 \\ \end{bmatrix} , \begin{bmatrix} ...
0
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1answer
103 views

Find 3 normal variables which are linear combinations based on 3 ind std normal variable given a correlation matrix

I am given $3$ normal random variables $X_1$,$X_2$,$X_3$ which are linear combinations of $Z_1$,$Z_2$,$Z_3$. $Z_1$,$Z_2$,$Z_3$ are mutually independent standard normal variables. I am given a ...
4
votes
1answer
2k views

Computing the derivative of a quadratic form and matrix chain rule

I'm working on using the Generalized Method of Moments to analyze some yogurt purchase data, and in the course of trying to implement the standard Hansen method (i.e. not an empirical likelihood ...
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3answers
142 views

How can we determine if every matrix of $\mathbb{R}^{2 \times 2}$ can be written as a linear combination of specific $A, B$ matrices

We have these two matrices: $$K = \left(\begin{matrix} 2 & 1 \\ 8 & 7\end{matrix}\right), \quad L = \left(\begin{matrix} 2 & 1 \\ 2 & 7 \end{matrix} \right)$$ We have been asked if ...
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2answers
142 views

What are the properties of this cousin to the characteristic equation: $\det (xA-I)=0$

The characteristic polynomial, defined for a matrix $A$: $ c(x; A) = \det (A-I x ) = 0 $ has nice properties related to the eigenvalues $\lambda$, of the matrix: $ c(x; A) = ...
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2answers
2k views

Same eigenvalues, different eigenvectors

I'm interested in the case of a specific matrix having different eigenvectors corresponding to two identical eigenvalues. The method I use for spectral decomposition returns different eigenvectors, ...
0
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2answers
217 views

What is the use of a transpose of a matrix in an equation and how to solve one?

I have the following equation to solve, $$g(x) = x^t W_i x + {W_i}^t x + v_{i_0}$$ In this equation why the need to use a $x^t$ and $x$? I feel $x$ and transpose of it both are the same ($x$ is a row ...
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1answer
142 views

Group theoretical characterization of diagonal matrices

Let $k$ be a field. Is there a group-theoretical characterization of the subgroup $D_n$ of diagonal matrices in $GL_n(k)$ ? For example, if $k = \mathbb{C}\;$ then $D_n$ is a maximal torus, but, of ...
0
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1answer
456 views

Jordan Normal form — Complex matrices

Suppose we are given the  characteristic polynomial and minimal polynomial of a matrix $(x-a)^4(x-b)^2$ and $(x-a)^2(x-b)$ say. Then I can tell what the largest Jordan blocks are, and hence work out ...
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1answer
95 views

How to solve the matrix from known data?

If $A$ is a $n \times n$ nonsingular matrix, and $\det \left( {\begin{array}{*{20}{c}} {{a_{ij}}} & {{a_{ik}}} \\ {{a_{lj}}} & {{a_{lk}}} \\ \end{array}} \right) = {c_{ijkl}}$ for ...
8
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1answer
1k views

Nilpotent Matrix

So I saw this problem: Is there an upper triangular matrix $A$ such that $A^n\neq 0$ but $A^{n+1}=0$? Prove or disprove. I said no, and my reasoning was that the matrix must have a zero diagonal ...
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4answers
21k views

Diagonalizable Matrices: How to determine?

I am trying to figure out how to determine the diagonalizability of the following two matrices. For the first matrix $$\left[\begin{matrix} 0 & 1 & 0\\0 & 0 & 1\\2 & -5 & ...
0
votes
1answer
515 views

Smith Normal Form

Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$ $$\begin{pmatrix}   (x+a)(x+b) & 0 & 0 &0 \\  0 & (x+c)(x+d) & 0 & 0 \\   0 ...
3
votes
0answers
145 views

Dimension of cells in the Bruhat decomposition of $GL_3$

I'm trying to understand the Bruhat decomposition of $GL_n$ so I've been doing a simple example with $GL_3$. We have the Weyl group $W$, which is isomorphic to $S_3$, and a simple root system namely ...
3
votes
1answer
796 views

2-norm and Frobenius norm of a matrix

Im prooving the inequality: $\|AB\|_F \leq \|A\|_2 \|B\|_F$. To prove this I need to know, if the following is true: Lets $B_{n \times r}~=~(\mathbf{b_1}, \ldots, \mathbf{b_r})$ is a matrix, ...
0
votes
1answer
239 views

Eigenvector corresponding to zero eigenvalue / identical eigenvalues, not-identical eigenvectors

Assume symmetricmatrix $B\in\mathbb{R}^{n\times n}$ is given, and a transformation $$A=JBJ,$$ where $J=I - \frac{1}{n}1_n1_n^T$ and $I$ denoting the identity matrix, hence centering its rows and ...
3
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1answer
272 views

Matrices over a ring

How might I find $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a,b,c,d \in \mathbb Z[x]$ such that there does not exist $B, C \in M_2(\mathbb Z[x])$ such that $B^{-1}AC$ is ...
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2answers
3k views

Determining the eigenvectors from a reduced-row echelon matrix

I am given the following matrix: $$A = P\left(\begin{matrix} -5 & -6 & 3\\3 & 4 & -3\\0 & 0 & -2\end{matrix}\right)$$ After finding the following eigenvalues by finding the ...
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3answers
100 views

Triadiagonal Symmetric Positive Definite matrix show that $B_N = C_N C_N^T$

I am given a matrix $N$ × $N$ symmetric positive definite matrix $B_N$ $$B_N = \begin{bmatrix} 2 & -1 & 0 & 0 & \cdots & \cdots\\ -1 & 2 & -1 & 0 & \cdots & ...
0
votes
1answer
367 views

Uniqueness of Cholesky factorization

I am given $A$ a symmetric positive definite matrix, and $U$ which the Cholesky factor of $A$. I am also told that if $V$ is an upper triangular matrix such that $A$ = $V^TV$. I have to show that ...
1
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1answer
245 views

Retrieving angles from a rotation matrix

I'm working with rotations in n dimensions. I represent these rotations as a sequence of $(n^2 - n)/2$ angles, one for each pair of axes, in a fixed order. I can easily compute a rotation matrix from ...
7
votes
3answers
1k views

Significance of eigenvalue

When I represent a graph with a matrix and calculate its eigenvalues what does it signify? I mean, what will spectral analysis of a graph tell me?
0
votes
0answers
146 views

Binary matrix as a product of two matrices

Is it possible to represent a binary( 0 and 1) matrix $A$ of size $ m \times n$ as a product of two martices $B$ of size $m \times k$ and $C$ of size $k \times n$. Various cases can also be considered ...
0
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1answer
131 views

strange matrix -> quaternion conversion problem

I find two different rotation matrices are mapped to a single quaternion. $$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ 0 & ...
1
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2answers
386 views

I have to show that the matrix $M^TM$ is SPD if and only if the columns of the matrix M are linearly independent

I have been given M be an $m\times n$ matrix. I have to show that the matrix $M^TM$ is symmetric positive definite if and only if the columns of the matrix $M$ are linearly independent My thoughts ...
1
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1answer
654 views

Matrix-vector multiplication before evaluating the matrix inverse

Suppose I need to evaluate the expression $$\begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} s+2 & -1 & 0 & 0\\ 0 & s+3 & 0 & 0\\ -1 & -2 & s ...
3
votes
2answers
156 views

Show that $\det(A+A^t)\neq p$ where $p$ is a prime number and $A \in M_{p \times p}(\mathbb{Z})$

I've stumbled across the above question. Usually, I don't spend my time on such problems, though it's a nice question in my opinion. My idea: Case $p=2$ is easy, follows from $x^2 \equiv 2 \;\;\; ...
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votes
0answers
97 views

How do I compute the eigenvectors of a general matrix using the QR decomposition (PLAPACK)?

I am working on an eigensolver in PLAPACK, which has built-in functionality to find the QR decomposition of a general matrix of double precision floating point numbers. I am familiar with the concept ...
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votes
1answer
120 views

What is this unary matrix operation called?

I have the folowing problem! $B \cdot \overline{X} = C$ where $B$ is $(12 \times 12)$ matrix and $C$ is $(12 \times 4)$. $X$ is unknown. What does $\overline X$ mean? Also I need to solve this in ...
6
votes
2answers
2k views

Derivative of a trace with respect to a matrix

Could someone explain this equation? $$ \frac{d \operatorname{tr}(AXB)}{d X} = BA $$ I understand that $$ d\operatorname{tr}(AXB) = \operatorname{tr}(BA \; dX) $$ but I don't quite understand ...
5
votes
3answers
614 views

0 as the unique eigenvalue of a matrix

I want to prove the following statement : Let $B$ be a matrix, such that $B$ has the eigenvalue $0$ and no other eigenvalue. Then $B^2=0$. In the context of the statement, $B$ is of size $2$. Is ...
4
votes
2answers
212 views

Eigenvalues of a matrix $A$ and $e^{A}$

If I know the eigenvalues of $e^{A}$, what can I say about the eigenvalues of $A$ itself?
7
votes
1answer
326 views

Holomorphic function of a matrix

A statement is made below. The questions are: (a) Is the statement true? (b) If it is, does it appear in the literature? Here is the statement. For any matrix $A$ in $M_n(\mathbb C)$, write ...
0
votes
1answer
152 views

How do I solve these simultaneous equations with restrictions?

Let's say I have $an_1+bn_2+cn_3=n_T$ $ap_1+bp_2+cp_3=p_T$ $ak_1+bk_2+ck_3=k_T$ where $a,b,c \geqslant 0$ What's the best way to find solutions for a, b and c so that the results of the sums are ...
2
votes
1answer
341 views

Matrix equation involving commutator

Given two matrices $N\times N$ $A,B$, is there some method to solve the matrix equation: $$e^{[A,B]}=A$$ where the symbol $[A,B]$ means the commutator of the two matrices: $[A,B]=AB-BA$? Thanks in ...
1
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1answer
343 views

Jacobian matrix normalization

I have a problem with normalization of the Jacobian matrix. There seems to be no clear method for doing it: in some literature, it has been normalized by using some characteristic length, which is ...
-1
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1answer
165 views

calculating characteristic row-vector

I am fairly new to matrices, especially stochastic matrices. In an effort to become more comfortable with them I am doing working out some problems. One of them that is giving me a hard time is to ...