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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1
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3answers
39 views

How to prove $I-BA$ is invertible

Show that $I-BA$ is invertible if $I-AB$ is invertible. And also, we have to prove that eigen values are same for $AB$ and $BA$ Till now, i used the equation $(I-AB)(I-AB)^{-1}=I$ which gives $(I-AB)...
2
votes
1answer
55 views

Problem about linear algebra [duplicate]

Suppose we have two $n \times n$ square matrices A and B such that $AB=BA$. It is known that A, B and AB all have n distinct eigenvectors that is a basis of $\mathbb{C}^n$. Can we then show that there ...
1
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2answers
32 views

how can I create a random matrix with specific entries

I would like to create/generate a random square $n \times n$ matrix with the following specifications: the first and the last row of the matrix are nonzeros (i.e all the elements in the first and ...
0
votes
2answers
17 views

Finding inverse by elimination

Find the inverse of the matrix $A$ below by elimination on [A I] By expanding the matrix into an alternating matrix. $$ A= \begin{bmatrix} 1 & -1 & 1 & -1 \\ 0 & 1 & -1 & 1 \\ ...
0
votes
0answers
19 views

Deciding positive definite function

Is there a characterization other than using Bochners Theorem (computing its Fourier transform) to decide whether the function is positive definite function or not? Any suggestion would be ...
0
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0answers
13 views

is I both a lower triang enad upper triangle ( Also proving L1=L2 )

First part of the question is $$ A= L_1D_1U_1\\ A = L_2D_2U_2\\ Prove\\ L_1= L_2\\ D_1 = D_2 \\ U_1 = U_2 \\ $$ My attempt seems correct but not quire sure whether it's mathematically constructed. $$...
1
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2answers
18 views

Choosing independent entries in a symmetric matrix

So, the question is how many entries can be chosen indepently in a symmetric matrix of order n? 2) How many entries can be chosen indepently in a skew-symmetric matrix $$ K^T=-K $$ of order n. The ...
0
votes
3answers
46 views

Showing A is not invertible

$$ A= \begin{bmatrix} 2 & 1 & 4 & 6 \\ 0 & 3 & 8 & 5 \\ 0 & 0 & 0 & 7 \\ 0 & 0 & 0 & 9 \\ \end{bmatrix} $$ We are asked to show A is not invertible ...
3
votes
1answer
34 views

Is there an $\alpha\in\mathbb{R}^m$, such that $\alpha_i > 0$ and $A\alpha\in S$?

$A$ is a real $n\times m$ matrix and set $S\subseteq \mathbb{R}^n$ is defined as $$S = \{(x_1,\dots, x_n)\in \mathbb{R}^n\mid \forall(i,j)\in I.\; x_i< x_j\}\text{,}$$ where $I$ is a possibly empty ...
1
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0answers
5 views

upper bound on the difference between two Perron-Frobenious eigen values

Let $\lambda, \mu$ be the Perron-Frobenious eigen value of the non-negative matrices $A,B$ respectively. I am interested in knowing whether there are any results available on the upper bound of $|\...
11
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2answers
15k views

Understanding rotation matrices

How does $ {\sqrt 2 \over 2} = \cos (45^\circ)$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? ...
0
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1answer
1k views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 = ...
6
votes
2answers
269 views

Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements

I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is not diagonal or triangular. Is there a term for such matrices, and have they been researched?
2
votes
1answer
28 views

If both of $A,A^{-1}$ have entries from non negative integers then can we say $A$ is a permutation matrix?

I've shown if both of $A,A^{-1}$ (assuming $A$ to be invertible) are $n\times n$ matrices with entries from natural numbers then both of them have to be permutation matrices. Now my question is if ...
8
votes
2answers
773 views

Why is some power of a permutation matrix always the identity?

If you take powers of a permutation, why is some $$ P^k = I $$ Find a 5 by 5 permutation $$ P $$ so that the smallest power to equal I is $$ P^6 = I $$ (This is a challenge question, Combine a 2 ...
0
votes
1answer
52 views
+50

Rank and null space of a particular block matrix.

Let $D_1, D_2 \in \mathbb{R}^{N \times N}$ be diagonal matrices with diagonals that are linearly independent vectors. Let $A, B \in \mathbb{R}^{N \times N}$ be rank-deficient matries. Define $S = \...
1
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1answer
16 views

Eigenvalues of product of p.d. Matrix with upper-triangular Matrix

Let $A$ be a positive definite matrix (positive eigenvalues). Let $B$ be an upper triangular matrix, with ones in its main diagonal (i.e. all its eigenvalues are 1). Is there anything I can say about ...
2
votes
0answers
28 views

Creating a tight frame of $\mathbb{R}^{n}$ when already knowing some of its vectors.

I'm wondering whether or not it's possible to start with a matrix $S\in\mathbb{R}^{m\times mn}$, $m<n$, and add rows to it so that the columns of the resulting matrix form an orthogonal system of ...
-1
votes
2answers
35 views

If $A$ is diagonizable then $p(A)$ is diagonalizable

Show that if a matrix $A$ of size $n \times n$ is diagonalizable, then $p(A)$ is diagonalizable for each polynomial $p$.
0
votes
0answers
26 views

First steps in derivation of matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
0
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1answer
1k views

Difference between rotation and pure rotation

Hi i am trying to understand my teacher's assignment. I have 2 write 2 Matlab functions ...
0
votes
1answer
20 views

Gradient of a matrix of only constants

I am confused about calculating the gradient of a matrix when the matrix is composed of only constant values. I'm doing an online interactive course in C++ that requires me to find this. I can't even ...
1
vote
1answer
100 views

Transformation Matrix for cube in 2D

My task is to transform the cube from the left corner to the big cube in the middle: What I did was: First i scale the cube: $$ \begin{pmatrix} 4 & 0 & 0 \\ 0 & ...
3
votes
1answer
2k views

Java Tetris - Using rotation matrix math to rotate piece

I'm working on building tetris now in Java and am at the point of rotations... I originally hardcoded all of the rotations, but found that linear algebra (matrix rotations) was the better way to go. ...
3
votes
2answers
69 views

Prove that the determinant is $(a-b)(b-c)(c-a)(a^2 + b^2 + c^2 )$

Prove that $$ \begin{vmatrix} 1 & a^2 + bc & a^3 \\ 1 & b^2 + ac & b^3 \\ 1 & c^2 + ab & c^3 \\ \end{vmatrix} =(a-b)(b-c)(c-a)(a^2 + ...
2
votes
4answers
51 views

How to calculate the negative half power of a matrix

I have a square matrix called A. How can I find $A ^ {-1/2}$. Should I compute $a_{ij} ^ {-1/2}$ for all of its elements? Thanks
0
votes
1answer
12 views

The map $f$ is degenerate or non-degenerate?

Let denote by $M_{3,2}(\mathbb C) $ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
3
votes
1answer
27 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...
3
votes
1answer
36 views

$p(x) \in \mathbb R[x]$ be non-constant polynomial , $n>1$ , the function $A \to p(A)$ is surjective on $M(n, \mathbb C)$?

Let $p(x) \in \mathbb R[x]$ be a non-constant polynomial and $n>1$ , then is it true that the function $f:M(n,\mathbb C) \to M(n, \mathbb C)$ defined as $f(A)=p(A) , \forall A \in M(n, \mathbb C)...
0
votes
1answer
22 views

Zeros in pivot position

When zero appears in a pivot position, $$ A = LU $$ is not possible. What do we have to do here to make A=LU possible then? Do we have to find a specific P (permutation matrix) for A and continue ...
7
votes
2answers
464 views

Additive rotation matrices

Let's assume that we want to find a rotation matrix which added to a given rotation matrix gives also a rotation matrix. I would name such matrix a rotation additive matrix for a given rotation ...
0
votes
1answer
21 views

Elimination and exchanging rows

Solve by elimination, exchanging rows when necessary $$ v + w = 0\\ u + v = 0\\ u + v + w = 1\\ $$ Which permutation matrix is required? answer is $$ P= \begin{bmatrix} 0 & 1 & 0 \\ 1 ...
0
votes
1answer
438 views

How do you calculate the dimensions of the null space and column space of the following matrix?

I understand you are supposed to get the reduced row echelon form, which I did, and this is what I came up with: 1 -2 0 19 -6 0 -37 0 0 1 -6 2 0 6 0 0 0 0 0 1 3 0 0 ...
0
votes
1answer
63 views

Why can't we sum two $n\times m$ and $u \times v$ matricies for all positive integer $n,m,u,v$? [on hold]

Why does the sum$$\left[\begin{matrix}1&2\\0&-1\\2 &3\end{matrix}\right]+\left[\begin{matrix}1&2&3&4\\0&-1 &1 &7\end{matrix}\right]$$ undefined? Let's expand these ...
1
vote
1answer
15 views

Inverse of a quasipositive matrix with negative spectral bound

A square matrix is quasipositive if all off-diagonal elements are nonnegative. The spectral bound of a square matrix is defined as $$s(A) = \max\{\Re (\lambda) : \lambda \mbox{ is an eigenvalue of } A\...
1
vote
2answers
645 views

Lower bound for the trace of product of two symmetric matrices

i am stuck on finding a lower bound of $tr(XY)$ of two symmetric matrices in $M_{n}(\mathbb{R})$. I know that it holds $tr(XY)=tr(YX)$ and thus $tr(XY-YX)=0$ and i can remember, that XY-YX is also ...
3
votes
1answer
11k views

Determine a basis for the solution set of the homogeneous system

Determine a basis for the solution set of the homogeneous system: $$\begin{align*} x_1 +x_2 +x_3 &=0\\ 3x_1+3x_2+x_3 &=0\\ 4x_1+4x_2+2x_3&=0 \end{align*}$$ Then the augmented ...
5
votes
0answers
68 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
1
vote
1answer
16 views

Improper rotation matrx in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
1
vote
1answer
19 views

Converting Fractional Coordinates to Cartesian

I'm confused about what I am reading online - different sites tell me different answers. Lets say I have a point pair in fractional coordinates, [xf,yf,zf]. I know that to convert them to their ...
1
vote
2answers
70 views

Solving this matrix equation.

Given the following matrix equation, $$\begin{bmatrix}a && b \\ c&& d\end{bmatrix}^n\begin{bmatrix}\alpha\\\beta\end{bmatrix}=\begin{bmatrix}\gamma \\ \kappa\end{bmatrix}$$ $\alpha, \...
1
vote
1answer
21 views

Primitive = Non-negative + Irreducible + 1 Positive element on main diagonal

Can anyone provide me with the proof for the sufficient condition for a matrix to be primitive as described by the definition from planetmath.org? (http://planetmath.org/primitivematrix)
0
votes
1answer
24 views

distance between two eigen vectors corresponding to two different matrices in a normed space

Let $A$ and $B$ are two $n\times n$ matrices. Let 1) $Ax = \lambda x$ and 2) $By=\mu y$ for $x,y$ in a normed space. $\lambda, \mu$ are scalar. Also, for $x,y$ are unique eigen vectors (upto a ...
0
votes
1answer
454 views

condition number with respect to spectral norm

I would like to show that "the condition number for inversion of $A$, with respect to the spectral norm is $k(A)=\rho(A)\rho(A^{-1})$" for $A\in M_n$ as nonsingular and normal matrix . Can anyone ...
22
votes
7answers
2k views

Is there such a thing as a matrix of functions?

Do we ever put functions as entries of a matrix? If so, are these matrices used in linear algebra or do they have some other special use? There have been minor not neccessarily conflicts per se, but ...
0
votes
0answers
26 views

Is the determinant of the following class of matrices non-zero?

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$. Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is $M[i,j]={(...
0
votes
1answer
24 views

Proving a fact about non-nilpotent matrices

Let $A$ be a square matrix such that all its eigenvalues are less than or equals 1 in absolute value. If A is not nilpotent, then prove that $$ \text{There exists an } i_0 \text{ such that rank }(A^{...
1
vote
2answers
51 views

How to calculate matrix rotation

Given the following rotation matrix $$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{matrix} \right]$$ ...
0
votes
0answers
20 views

The kernel of the transpose of the differentiation operator - Solution check

I tried to solve the following problem and I'd like some feedback on my solution: Let $n$ be a positive integer and let $V$ be $P_n(\Bbb R)$the space of all polynomials functions over the field of ...