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
48 views

Solving for trivial solutions of a matrix

My friend and I came across this problem while looking through some homework. Say you had a $3 \times 4$ matrix that reduced down to something like this: $$ \begin{pmatrix} 1 & 2 & 0 ...
0
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1answer
65 views

rank of a matrix series

Suppose $X$ is a $n \times n$ matrix with its rank $rank(X)=a<n$. How can we show that the rank of the matrix series $\sum_{i=1}^{\infty} X^i \cdot {(X^{T})}^{i}$ is $a$ as well?
0
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1answer
43 views

Understanding the change-of-coordinate matrix

I know that if you have a linear transformation $T:V \rightarrow W$ where $V,W$ are finite-dimensional then there is a matrix $A \in M_{m \times n}(\mathbb F)$ that represents $T$ (not sure if that's ...
0
votes
2answers
220 views

Invariant determinant in change of basis

So I've seen the proof for why the determinant of a transformation $T$ is the same under a change of basis, but I must have some basic misconception about this since I can't figure out what's wrong ...
1
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1answer
90 views

Determining whether binary matrix B is obtainable from binary matrix A via row and column permutations

Say you have two binary (i.e., (0, 1) ) m x n matrices A and B. Their row and column sums match up - i.e., for each attained row (column) sum k in A, there are the same number of rows (columns) with ...
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1answer
44 views

$A,B\in\mathbb R^{n\times n}$ then $ \|A^{-1}-B^{-1}\|\leq\|A^{-1}\|\|B^{-1}\|\|A-B\|$

I am reworking my lectures and in one proof our prof used the following: Let $A,B\in\mathbb R^{n\times n}$ invertible. Then $ \|A^{-1}-B^{-1}\|\leq\|A^{-1}\|\|B^{-1}\|\|A-B\|$ Unfortunately I have ...
1
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1answer
87 views

Gauss Elimination for Colorability Problem

Consider the following system of linear equations modulo 2: $A.X + B.Y = Z, $ where $A$ is a non-singular(modulo 2) $n$ x $n$ boolean matrix, $B $ is $n$ x $m$ boolean matrix, $X$ is n-dimensional ...
1
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1answer
195 views

What are Riemann Matrix?

I got this information regarding Riemann matrix: RIEMANN is a N-by-N matrix associated with the Riemann hypothesis, which is true if and only if: ...
1
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2answers
255 views

How to prove the existence of solution of a non linear system of equations

Writing the ortogonality condition for any element of O(n), I've arrived to: If we take n=2, we know that $\Lambda\Lambda^{T}=\mathbb{I}$, so: $$\begin{pmatrix} x & y \\ z & t \end{pmatrix} ...
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0answers
33 views

What are Riemann Matrix? [duplicate]

I got this information regarding Riemann matrix: RIEMANN is a N-by-N matrix associated with the Riemann hypothesis, which is true if and only if: ...
3
votes
1answer
72 views

A question about minimal polynomials

$$ H=\pmatrix{A&0&\cdots&\cdots&0\\ I&A&0&\cdots&0\\ 0&I&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ ...
0
votes
3answers
343 views

Eigenvalue of all matrix with all duplicate rows

I've been asked to determine an eigenvalue given the following matrix without any calculations: $\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{bmatrix}$ My ...
0
votes
1answer
111 views

Count the number of rational canonical form&find similarity classess

For a finite field $F=F_q$ having $q=p^d$ elements ($p$ a prime integer), compute the number of similarity classes in the vector space $M_n(F)$ of $n\times n$ matrices over $F$. (Maybe count the ...
1
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2answers
54 views

Complex Matrix Limit

If $A$ is an $n \times n$ complex matrix, show that if $\lim_{k\rightarrow\infty}||A^kv||=0$ for every vector $v \in \Bbb C^n$, then $|\lambda|\leq1$ for every eigenvalue $\lambda$ of $A$.
1
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1answer
95 views

Repeated Iteration of a 2x2 matrix

Suppose I am given a $2$x$2$ matrix $A=$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} And an initial vector $x_n$ = \begin{pmatrix} x_0 \\ y_0\end{pmatrix}. Under repeated iteration $x_{n+1} ...
2
votes
1answer
82 views

Coordinate System Rotation and Cross Term

If I have a conic equation $$ 5x^2 - 4xy + 8y^2 = 36 $$ and $ \left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right] $ in matrix form, whose eigenvalues are 4 and 9, how would I rotate ...
1
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1answer
32 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
1
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1answer
46 views

Decide if the vector (1,1,1) is in the row space of the matrix

Decide if the vector $(1,1,1)$ is in the row space of the matrix $$ \begin{bmatrix} 1& 1& 3\\-1&0&1\\-1&2&7 \end{bmatrix}$$ Yes. To see if there are $c_1$ and $c_2$ such ...
2
votes
2answers
172 views

Relation between linear maps and matrices

I've been reading Axler's "Linear Algebra Done Right", and have learned more about linear operators/ maps, but I'd like to make sure that I understand how to properly relate this information to ...
0
votes
2answers
108 views

Calculating the matrix corresponding to linear map

How does one go about converting a linear map in functional form to a matrix; for instance: For a fixed unit vector $\hat{n} \in \mathbb{R}^{3}$, define the map $f:\mathbb{R}^{3}\to\mathbb{R}^{3}$ ...
1
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1answer
86 views

Finding Eigensystem of Hermitian Matrix

I am trying to find the system of eigenvalues and corresponding normalized eigenvectors for the following Hermitian matrix: $$\mathbf{H}=\begin{pmatrix}10 & 3i \\ -3i & 2\end{pmatrix}$$ I ...
2
votes
1answer
77 views

Jordan Form Superdiagonal

How do you know how many of the super diagonal entries in the Jordan Form are zeros and how many are ones, and where they are placed? Thanks.
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2answers
94 views

Idempotency and Jordan cells

Which Jordan cells $J(\lambda,k)$ are idempotent? And how can I use that to determine the Jordan canonical form of any square idempotent matrix?
0
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2answers
53 views

Calculating a matrix

I have tried searching but was not able to find a question similar to mine. The question is as: Consider the matrix $$ A= \begin{bmatrix} 3 & 1 \\ 3 & 5\end{bmatrix}$$ Let $$ D= ...
1
vote
3answers
185 views

What is the *correct* (matrix) square-root of $A_2=\begin{bmatrix} 0&-1 \\ 1& 2 \end{bmatrix} $?

In studying the problem of some trivial(?) generalization of the NSW-numbers [ OEIS,wikipedia ] (see my other related question) there came up one detail where I think I have the correct answer but ...
0
votes
1answer
46 views

Equality of bilinear forms

The Problem: Given two fixed vectors $\mathbf{u}$ and $\mathbf{v}$, which conditions should the matrices $\mathbf{A}$ and $\mathbf{B}$ fullfill, for the following matrix equation to hold: ...
1
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3answers
87 views

Differentiating a non-linear functional with respect to a vector

I have the functional: $$F=v^T\times A \times v$$ Where $A$ is a function of $v$. The non-linear system of equations necessary to find $v$ is obtained doing: $$\frac{\partial F}{\partial v}=0$$ ...
1
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1answer
142 views

Expected number of random binary vectors to make matrix of order n

I have the following problem: I pick random vectors from $\mathrm{F}_2^n$. The chance that position $i$ is $1$ equals $p_i$, $0$ otherwise (each position is picked independently). Let $X$ be a random ...
0
votes
1answer
76 views

Conic matrix and diagonalization

If I have the conic $C$: $$ 5x^2 - 4xy + 8y^2 = 36 $$ How would I express it as a matrix of the form: $$ \begin{bmatrix}x&y\end{bmatrix} \begin{bmatrix}a&b/2\\b/2&c\end{bmatrix} ...
0
votes
1answer
37 views

Hermitian matrices, non-zero

If I have a hermitian matrix whose eigen values are non negative, and the trace=0, must the matrix=0? I gather that the eigen values must all be 0, but I could not find an example of a hermitian ...
1
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1answer
403 views

How to create a 2d geometric transformation matrix to stretch an image along a given direction

I would like to define a matrix to stretch an image along a given direction. For exemple, I have an angle alpha and a scaling ratio r. How can I construct the transformation matrix in order to apply ...
1
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0answers
125 views

Second-Order Tikhonov Regularization

In the second-order Tikhonov regularization approach $\min\left\|Gm - d \right\|_2^2 + \alpha\ ^2 \left\|\Gamma x\right\|_2^2$ (1) given that $\Gamma\ $ contains second order derivatives, ...
5
votes
2answers
176 views

Matrices such that $A^2=A$ and $B^2=B$

Let $A,B$ be two matrices of $M(n,\mathbb{R})$ such that $$A^2=A\quad\text{and}\quad B^2=B$$ Then $A$ and $B$ are similar if and only if $\operatorname{rk}A = \operatorname{rk}B$. The first ...
25
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9answers
2k views

Are most matrices invertible? [duplicate]

I asked myself this question, to which I think the answer is "yes". One reason would be that an invertible matrix has infinitely many options for its determinant (except $0$), whereas a non-invertible ...
2
votes
2answers
121 views

About linear transformations

Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$ $$T(A)(B)=AB-BA$$ I have to prove that this is a linear ...
7
votes
2answers
287 views

How find this matrix $A=(\sqrt{i^2+j^2})$ eigenvalue

let the matrix $$A=(a_{ij})_{n\times n}$$ where $$a_{ij}=\sqrt{i^2+j^2}$$ Question: Find the difference $sign{(A)}$ can see this define:http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia My ...
1
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2answers
195 views

Problem regarding filling squares inside a $n\times n$ grid.

Assuming a $n\times n$ square grid, what is the most number of squares that can be filled in such that there are no completed rows, columns, or diagonals? Is there a formula to calculate this? ...
0
votes
1answer
79 views

equivalent in cauchy integral for matrices

I don't know why $(zI-A)^{-1} = \frac{1}{z} \sum_{k=0}^\infty \frac{A^k}{z^k}$ in a link!
1
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0answers
405 views

Finding error and proving Romberg Integration Method

Let $f$ be a function, which its integral has to be approximated by using romberg method.The $n\cdot m$th cell in romberg matrix (a lower triangular matrix) is given by ...
6
votes
4answers
150 views

Simple examples of $3 \times 3$ rotation matrices

I'd like to have some numerically simple examples of $3 \times 3$ rotation matrices that are easy to handle in hand calculations (using only your brain and a pencil). Matrices that contain too many ...
2
votes
0answers
65 views

How many rotation matrices with simple rational entries

This is a follow-on from this earlier question, which asked for examples of simple rotation matrices. I'm interested in rotation matrices whose entries are simple rational numbers, because these are ...
4
votes
3answers
202 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
0
votes
2answers
102 views

Regular matrix and field extension.

Let $F$ be a field. We say a matrix $A\in M_n(F)$ is regular over $F$ if the minimal polynomial of $A$ over $F$ equals the characteristic polynomial of $A$. Suppose $L$ is a field extension of ...
0
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1answer
55 views

matrix representation of $f$ with respect to the union of two ordered bases for $\ker f$ and $\ker (f-id_V)$

Can you help me with this problem? Let $f:V\longrightarrow V$ be a linear tansformation such that $f\circ f = f$. Let $B'$ be an ordered basis for $\ker f$ and $B''$ be an ordered basis for $\ker ...
3
votes
2answers
686 views

If N is elementary nilpotent matrix, show that N Transpose is similar to N

If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ ...
6
votes
2answers
128 views

show that the characteristic polynomial of this matrix has negative coefficients

Let $n\geq 2$, $A$ be the $n\times n$ matrix $A=(a_{ij})$ where $a_{ij}=\max(i,j)$. Can anybody show that the characteristic polynomial $P(x)=\det(xI-A)$ has all its coefficients negative except the ...
2
votes
2answers
136 views

How to express a matrix as a product of two symmetric matrices?

Let $A$ be a matrix and $J$ its Jordan canonical form. How can one express $A$ as a product of two symmetric matrices? I expressed $J$ as a product of two symmetric matrices: block by block in the ...
2
votes
1answer
45 views

Would this be a free variable?

Let's say I have the RREF matrix $$ A= \begin{bmatrix} 1 & 3 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$$ If i'm looking for the solution of this ...
0
votes
0answers
756 views

Linear algebra Pruning X, linear combinations and Spans

Consider the following subset of the vector space $\mathbb{P}_4(\mathbb{R})$ (real polynomial functions of degree at most 4): $X := \{f_1,f_2,f_3,f_4,f_5 \}$ with $f_1(x) = 1 + x^3 + x^4$, ...
0
votes
1answer
37 views

Is the matrix defined by $\bar{K}_{ij}=f(x_i)f(x_j)$ for a real valued function $f$ semi-positive-definite?

Let $f:\mathbb{R}^m \rightarrow\mathbb{R}$ be a real valued function and define $K: \mathbb{R}^m \times \mathbb{R}^m \rightarrow \mathbb{R}$ by $K(x,y)=f(x)f(y)$. For any vectors $x_1,x_2,...,x_n ...