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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14 views

Prove that these pairs of complex numbers have real part 1/2 if they are symmetric in the complex plane.

Let matrix $A$ be defined as: $\Large A(n,k)=k^{-a_k + 1/2 + ib_k}$ if $k$ divides $n$, else $A(n,k)=0$ Let matrix $B$ be defined as: $\Large B(n,k)=\mu(n) n^{a_n+1/2 -ib_n}$ if $n$ divides $k$, ...
3
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1answer
60 views

The maximal rotation matrix

Let's consider two numbers calculated for a rotation matrix which are: $s_e=$ the sum of all entries of a matrix $s_a=$ the sum of absolute values of all entries for a given matrix. It ...
3
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4answers
55 views

Why is a linear transformation expressed using its transpose?

If $A$ is an invertible matrix with entries from $\mathbb{R}$, what is the reasoning behind defining an invertible linear transformation $f_A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ as $f_A=xA^t$, ...
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0answers
7 views

MIMO static decoupling matrix

I am trying to implement a static decoupling matrix of a MIMO system in MATLAB Simulink. This static decoupling matrix is of size 2x2 with some complex numbers, and the control scheme used for ...
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3answers
33 views

Inverse of matrix with very structured submatrix

Does this matrix admit an easy analytic expression for its inverse? $$\begin{bmatrix} a_1 & 0 & 0 & 0 & 0 &0&\dots&0 \\ a_2 & 1 & -b & 0 & 0&0&\...
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0answers
13 views

What is the optimized Time complexity of Cholesky decomposition

Is there any algorithm for Cholesky decomposition that has complexity O(n^a) where a < 3? I know there are some algorithms to be better than n^3 for matrix multiplication, not sure about Cholesky, ...
0
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2answers
36 views

How come associative law of matrix multiplication won't work when permutation matrices come in. Which is the case for some

if $$x=y$$ explain why $$Px=Py$$ I believe this part is very since when we do $$P^{-1}Px = P^{-1}Py$$ from here $$x=y$$ But the other part of the question seems much more confusing then $$(Px)^...
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4answers
112 views

How to prove $I-BA$ is invertible [duplicate]

Show that $I-BA$ is invertible if $I-AB$ is invertible. And also, we have to prove that eigenvalues are same for $AB$ and $BA$ Till now, I used the equation $(I-AB)(I-AB)^{-1}=I$ which gives $(I-AB)...
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2answers
27 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 \\ ...
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1answer
28 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. $$...
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2answers
21 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 ...
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3answers
55 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
50 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 ...
2
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1answer
32 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 ...
9
votes
3answers
367 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 neither diagonal nor triangular. Is there a term for such matrices, and have they been researched?
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1answer
25 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 ...
2
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4answers
55 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
3
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2answers
87 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 + ...
3
votes
1answer
30 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
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1answer
40 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)...
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1answer
28 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-...
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2answers
43 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 $p$.
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2answers
493 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
64 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 ...
2
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1answer
58 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 ...
9
votes
2answers
800 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 ...
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1answer
23 views

Improper rotation matrix 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 ...
0
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1answer
22 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 ...
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vote
1answer
20 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 ...
0
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1answer
23 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 ...
1
vote
1answer
17 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\...
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0answers
31 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
25 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^{...
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0answers
23 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 ...
0
votes
1answer
38 views

A confusion about the definition of the “trace” norm

Given a $n \times m$ real matrix $A$ of rank $r$ one can define its SVD as $A = UD V^T$ with $D$ being a $r \times r$ diagonal matrix and $U^TU = V^TV= I$. Here clearly the diagonal entries of $D$ are ...
2
votes
2answers
53 views

Eigenvectors are unique up to a scalar

If $ A $ is a matrix with eigenvector $ v $ corresponding to the eigenvalue $ \lambda, $ can we prove that $ v $ is unique up to $ \lambda, $ that is if $ v $ and $ v' $ are eigenvectors corresponding ...
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2answers
55 views

How to calculate matrix rotation

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

Is there a faster way to determine partial orderings of basic finite sets?

For example, consider the set $S = \{ 0, 1, 2, 3 \}$, and the following relation on $S$: $$ R = \{(0,0), (1,1), (1,2), (1,3), (2,0), (2,2), (2,3), (3,0), (3,3) \}. $$ Obviously, I can go through ...
1
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3answers
56 views

Every subspace is the kernel of a linear map

I know that every kernel of a linear map from $\mathbb R^n$ to $\mathbb R^m$ is a subspace of $\mathbb R^n$. I am wondering if the converse is true, i.e. every subspace of $\mathbb R^n$ is the kernel ...
2
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0answers
32 views

Efficient way to check if a large matrix is positive definite.

Suppose I have a large $n\times{}n$ matrix with $n>1000$ say. I would like to find the quickest way to check if it is positive definite. My matrices are sparse so at the moment I am using sparse ...
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2answers
72 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, \...
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0answers
8 views

involutory matrices and their applications.

In a previous question, I was meaning this question but asked about idempotent matrices. Now the original question. The motivation for this question is that an involutory matrix is the inverse of ...
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3answers
46 views

linear algebra:prove or disprove

Does there exist a matrix of order $n\times n$ above $F$, say $A$, so that for every $\underline{b}\in F^n$ there are infinite solutions to A$\underline{x}$ = $\underline{b}$?
2
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2answers
66 views

A question in simillarity of matrices

I have a 4x4 matrix $$ A = \left[ \begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \end{matrix} \right] $$ and i want ...
0
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2answers
26 views

Transformation of a plane

I have the $(x,y)$-plane $$\left\{(x,y,z)\in \mathbb{R}^3 | x,y\in \mathbb{R}, z = 0 \right\}.$$ I need a transformation matrix to transform this to the plane $$ \left\{ (x,y,z) \in \mathbb{R}^3 | x+...
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1answer
15 views

idempotent matrices and transformations

What is the significance of idempotent matrices? Are there any idempotent transformations in signal processing that are widely used? Are there any practical applications of idempotent ...
5
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1answer
81 views

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

I'm wondering whether or not there's an optimal way for adding rows to a given matrix $S\in\mathbb{R}^{m\times mn}$, $m\leq n$, so that the columns of the resulting matrix form an orthogonal system of ...
4
votes
1answer
65 views

Nonsingular matrices with bounded coefficients

I can show that there exists $n^2$ positive integers $a_1,\ldots ,a_{n^2}$, such that each $n\times n$ matrix with coefficients $a_i$ (used once and only once) is nonsingular. Two questions: Could ...
1
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3answers
90 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 likewise nonzero at the main diagonal ...
1
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0answers
26 views

Quadratic form with one value changed.

I came across the following problem when trying to run a Metropolis algorithm. (It is related to computing a multivariate normal density.) Let us have an $n\times n$ matrix $A$ of a special kind: ...