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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8 views

Firstly what does relative tolerance mean?

Apply the Jacobi iteration method to the system Ax=b with $$ A= \begin{pmatrix} 3 & -1 & 1 \\ 3 & 6 & 2 \\ 3 & 3 & 7 \\ ...
10
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1answer
91 views

Could one make a ring of matrices of uncountable size?

I've seen several kinds of matrices. You could see a real square matrix as a mapping: $$ A \quad : \quad \{1, 2,\cdots, n \}^2 \ \longrightarrow \ \mathbb{R} $$ I've seen that there were also infinite ...
0
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0answers
16 views

Normalize matrix so that its projection equals identity - which to use?

I wish to normalize a given matrix M, n by k, so that its projection matrix equals the identity. This is to speed up computations. The transformation/decomposition would look like this: Generate M by ...
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0answers
21 views

Non positive definite covariance matrix

Please help understanding which of the following would lead to a non positive definite covariance matrix and, most importantly, why? A. Changing all the correlations to be unity B. Changing all the ...
2
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2answers
39 views

Existence of $(I-A)^{-1}$ implies convergence of $\sum A^k$? [duplicate]

Suppose $A$ is a square matrix (over $\mathbb{R}$ or $\mathbb{C}$, take your pick) such that $(I-A)^{-1}$ exists. Then is it necessarily true that $$I + A + A^2 + \dots + A^n + \dots = (I-A)^{-1}$$ ? ...
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1answer
42 views

Matrix multiplication verification

So I have this ought to be simple example in my lecture notes but I just can't wrap my head around this guys solution. I understand how there are $n$ multiplications but how are there $n$ additions ...
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0answers
10 views

What it means by “asymptotic normality” properties of a random matrix?

I know that for the case of a random variable and a random vector, one can using (multivariate) density of normal distribution and concepts of convergence to define an asymptotic normality of a random ...
4
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2answers
73 views

Does a positive definite matrix have positive determinant

Let $A$ be a positive-definite real matrix - in other words, $x^T A x > 0$ for every real vector $x$. I don't require $A$ to be symmetric. Does it follow that $\mathrm{det}(A) > 0$?
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0answers
20 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
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0answers
23 views

Gaussian Elimination method with respect to maximum XOR subset problem?

Can anyone explain me Gaussian Elimination method with respect to maximum XOR subset problem? I am not able to figure out the various posts posted on Internet of the above solution. So I am ...
1
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0answers
27 views

Techniques to find matrix inverses of general classes of matrices?

Suppose you're given some general description of an $n\times n$ matrix, and asked to find its inverse. By "general description" I mean that the matrix can be described in one or more sentences, and ...
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0answers
12 views

how would I represent supply/demand with augmented matrices?

I don't want to post the exact problem because then someone will just solve it and it's a practice problem but if I'm being too vague let me know. Theres 3 industries each with outputs expressed as a ...
1
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1answer
36 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
1
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1answer
57 views

If $AB+BA=0$ and $B=AX+XB$, then $B$ is nilpotent.

Suppose $A,B,X \in M_n(\mathbb{R})$ and that $AB+BA=0$ and $B=AX+XA$. Prove that $B$ is a nilpotent matrix.
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4answers
118 views

Order of matrix multiplication intuition

Let $A$, $B$ be linear transformations $\mathbb{R}^2 \to \mathbb{R}^2$, for example let $A$ be a rotation and $B$ a scale. Let $\mathbf{v}$ be a 2 component column vector that I want to transform by ...
2
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0answers
37 views

Finding a solution basis of differential equation

Find a solution basis of $$y'=\left[ \begin{matrix}3&-4&-2\\2&-3&-2\\0&0&1\\ \end{matrix} \right]y \,\text{ and find the solution } \Phi \text{ with } \Phi(0) = (1,1,1).$$ I'm ...
1
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1answer
77 views

Proof of a matrix is positive semi-definite

For $\ i = 0, 1, \cdots m$, $f_{i}(x): R^n \rightarrow R$ is defined to be $$ f_i(x) = x^TQ_ix + 2p_i^Tx + r_i $$ , where $Q_0 \cdots Q_m$ are real symmetric matrices, $p_0 \cdots p_m \in R^n$, and ...
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1answer
32 views

Is adjoint of singular matrix singular? What would be its rank?

Let $A$ be square and singular matrix of order $ n $. Is $ adj (A) $ necessarily singular? What would be rank of $ adj (A) $.?
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votes
1answer
35 views

What is the least integer $k$ such that any $n$ by $n$ integer matrix is a $\mathbb Z$-linear combination of $k$ indempotents?

Let $n$ be an integer $\ge2$. (a) What is the least integer $k$ such that any $n$ by $n$ integer matrix is a $\mathbb Z$-linear combination of $k$ indempotents? (The idempotents are also ...
1
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0answers
22 views

an inequality on matrix sign function : $A \leq \operatorname{sign} (A) \times A $

Let $A$ be an $n\times n$ real symmetric matrix, and $\operatorname{sign}(A)$ is the matrix sign function, which is defined as $\operatorname{sign}(A) := Z \pmatrix{-I_p & 0\cr 0 & ...
4
votes
1answer
148 views

What is the connection between $\rho$ and $\sigma$ if $\rho\rho^T=\sigma\sigma^T$?

I want to prove that there exists a Borel function $R(\rho,\sigma)$ with values in $M^{d\times d}$ defined on $D=\lbrace(\rho,\sigma)\in M^{d\times d}\times M^{d\times d}\,: ...
3
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1answer
52 views

If diagonalizable matrices commute does it neccesarily mean that they can be simultaneously diagonalized?

If matrices $M_1$ and $M_2$ can be simultaneously diagonalized, than they commute, which can be easily shown: \begin{align} M_1M_2&=P^{-1}D_1PP^{-1}D_2P \\ &=P^{-1}D_1D_2P \\ ...
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0answers
16 views

Traces and linear combinations of idempotents

Let $R$ be a commutative ring and $S$ a subring, and consider the following conditions on an $n$ by $n$ matrix $A$ with entries in $R$: (a) $A$ is an $S$-linear combination of idempotents of ...
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2answers
124 views

Prove or disprove that AB=AC $\implies$ B=C

I proved it as follows but I'm not so sure about it. A, B and C are square matrices of the same order. Assume $ B \neq C $ $$ AB \neq AC$$ $$ B \neq C \implies AB \neq AC$$ $$ \neg ( AB \neq AC) ...
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1answer
25 views

Positive principal minors in non-symmetric matrix

Can anything be said about a (not necessarily symmetric) matrix $A$, all of whose principal minors (upper-left squares) have positive determinant? Do these matrices have a name? I would like to know ...
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2answers
102 views

How do you quickly find the eigenvalues of this matrix?

I have a final exam tomorrow, am sure a 3x3 eigen value problem like the one below is there. But I find it very hard to find eigen values without zeros in the matrix Show me how you do it quickly so ...
3
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0answers
17 views

Gerschgorin's Disks of a 3x3 complex matrix (2)

Take the following matrix as an example, $$\left( \begin{array}{ccc} -1 & -i & 0 \\ \frac{i}{2} & 0 & 0 \\ 0 & i & 1 \\ \end{array} \right)$$ The eigenvalues are ${-1.37, ...
0
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0answers
53 views

show that the determinants are equal

Prove that the determinants are equal $$ \begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \\ \end{vmatrix}= ...
2
votes
1answer
57 views

Recurrence - using power series

Could you help me in solving this recursion( a closed form ) using power series $\mu(n)=\mu(n−1)k_0+(n−1)\mu(n−2) k_1 \tag 1$, where $k_0,k_1$ are constants $\mu(0)=3,\mu(1)=5$ HINT: We can think ...
0
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3answers
25 views

Positive definiteness of MM'

If $M$ is a $k\times l$ matrix of rank $l$. Can we say $MM'$ is positive definite, as $x'MM'x = (M'x)' (I) (M'x)$ and $I$ is positive definite?
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3answers
25 views

Matrix commutation of $\boldsymbol{M}$ and $[\boldsymbol{I} - \boldsymbol{M}]^{-1}$

We consider a matrix $\boldsymbol{M}$. We suppose it is diagonalizable, with eigenvalues $\lambda_{i}$. We always assume that $\forall i \, , \,\lambda_{i} \neq 1$. As a consequence, the matrix $[ ...
1
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0answers
17 views

Gerschgorin's Disks of a 3x3 complex matrix

How to estimate the radii of Gerschgorin's Disks from a 3x3 square complex matrix? Is the disks just simply center at the diagonal elements with radius equal to the sum of the absolute values of other ...
0
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0answers
29 views

Rotation plane on the sphere (quarternion)

I asked similar question on stackoverflow but still no answers.http://stackoverflow.com/questions/25185329/image-rotation-with-the-gyro-data-math I assume it is more math than programming problem. ...
2
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2answers
66 views

Generalization of a formula for 2x2-matrices

It is well known that $$|det(v_1,...,v_n)|\le ||v_1||_2...||v_n||_2$$ with equality if and only if the vectors are pairwise orthogonal. For n = 2, the following formula holds : $$det(\pmatrix ...
0
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0answers
15 views

How to find orthogonal complement of space of complex matrices

Let $V=\mathbb{C}^{n\times n}$. How we can find the orthogonal complement of $V$? It's maybe? Is it the set of all matrices with zero trace?
2
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0answers
10 views

Showing Hat matrix equal specific values

Consider a one way layout model $y_{ij}$ = $\mu_i + e_{ij}$ (1 $\leq$ i $\leq$ a, 1 $\leq$ j $\leq$ $n_i$) where a = 3 and $n_1$ = 2, $n_2$ = 3, $n_3$ = 4. Show that the hat matrix for this design ...
1
vote
1answer
90 views

find a matrix transform

Given a vector $v={(v_1,v_2,...,v_n)}^T$, I would like to find some matrix operations on $v$ to create an $n \times n$ matrix $X$ such that its entry $X_{i,j} $ satisfy (1), (2), (3), (4), ...
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0answers
24 views

Is there a name for this operation: $x_k = \sum_{ij} y_{ij} a_{ik} b_{jk}$?

Is there a name for this operation: $x_k = \sum_{ij} y_{ij} a_{ik} b_{jk}$ ?
0
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0answers
13 views

the rank propertity for the column combination of a matrix

There is a matrix, $M*N$; and suppose that the rank is $M$. Can I assume that any combination of $M$ columns will consists of a full rank matrix?
1
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1answer
26 views

Stuck at Smith normal form

Can somebody help me with the Smith normal form of this matrix? I know what it should be, but I get stuck at some point. Can you show how to take it from the point I'm stuck? This is the matrix: ...
0
votes
1answer
33 views

covariance matrix is not positive definite

I have a feature vector(FV1) of size 1*n. Now I subtract mean of all feature vectors from the feature vector FV1 Now I take transpose of that(FV1_Transpose) which is n*1. Now I add do matrix ...
5
votes
0answers
129 views

Inequality involving traces and matrix inversions

The following question kept me wondering for some weeks: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n\times n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
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0answers
29 views

What can be the possible rank of adjoint of matrix of order n?

Let $ A $ be matrix of order $ n $. What may the possible ranks of $\mathop{\rm adj} (A) $? I think the possible answers are $0$, $1$, and $n$.
2
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2answers
79 views

Prove that the determinants are equal

$$ Let\ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
2
votes
0answers
72 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
0
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0answers
2 views

How can I apply a median filter directly to a time-varying rotation matrix?

I need MatLab script which would take a series of rotation matrices (referring to an actual physical object's orientation) and apply median filter to it to eliminate speckle noise from it. The way ...
2
votes
1answer
36 views

Is the trace of an idempotent matrix a sum of idempotents?

Let $R$ be a commutative ring, $n$ a positive integer, and $A$ an idempotent $n$ by $n$ matrix with entries in $R$. Is the trace of $A$ necessarily a sum of idempotents of $R$?
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2answers
42 views

What does Determinant of Covariance Matrix give?

I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. ...
3
votes
1answer
114 views

Dominant eigenvector by looking at rows of matrix raised to a power

I'm not strong in linear algebra. I encountered this thing and being curious I want to know a bit more about it. I'm playing with 3x3 real valued matrices in some graphics application, I'm developing. ...
1
vote
2answers
68 views

When is this matrix positive semi-definite?

I have a symmetric $n \times n$ matrix as follows. I want to find the eigenvalues of this Hessian matrix to state that it is not Positive Semi-Definite (i.e. some eigenvalues are negative while the ...