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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3
votes
2answers
30 views

How may I use a 3x3 matrix to simulate a larger square matrix?

I am using a game engine where the library only provides 3x3 matrices with the multiplication and inverse operation. I could build my own matrix library to provide larger matrices, but it would be ...
1
vote
2answers
30 views

If A is positive definite (but not necessarily symmetric) can you decompose it?

If A is a $2 \times 2$ matrix that is positive definite but may or may not be symmetric, does there exist another matrix B such that $A=B^TB$?
1
vote
1answer
12 views

is the trace of inverse of positive, positive definite matrix decreasing?

Let $A, B$ be non-negative, and symmetric positive definite matrices. If $A\le B$, is it true that $\mbox{trace}(A^{-1}) \ge \mbox{trace}(B^{-1})$?
0
votes
0answers
14 views

A non-strict inequality on skew symmetric matrices

As we know that skew-symmetricity means $A=-A^\top$ where $A\in\mathbb{R}^{n\times n }$. But recently I came across an inequality that states, $A+A^\top\preceq0$ can also be considered as an ...
-1
votes
0answers
23 views

MAthematical notation for sorting submatrix and replacing it back

I need help in expressing the following paragraph in mathematical form as much as possible. I have a matrix $A$ which is $N\times M$. For each element of $A$, $A(i,j)$, I consider a submatrix of $A$ ...
2
votes
1answer
41 views

Black are berries and maroon are cherries. Place 8 more cherries removing berries 1 from each row and each column. No of ways?

I tried to see it as a matrix where for a position (i,j) , i+j = 8, 9, 16 means you can't change that position. Any help?
0
votes
0answers
24 views

Null spaces and their dimensions. how to decide the dimension of the null space

Here is a quote from a textbook. Within four dimensional space ofa ll possible vectors x the solutions to $$Ax=0$$ form a two dimensional subspace - the nullspace of A In this specific A we ...
0
votes
0answers
20 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What can we learn about \begin{align} \text{Tr} (UX) \end{align} In particular Is there a closed form expression? When does $\text{Tr} (UX)$ vanish?...
0
votes
0answers
12 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
votes
1answer
38 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 ...
2
votes
3answers
37 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 $f_A=xA^t$, ...
0
votes
0answers
4 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 ...
0
votes
3answers
28 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&\...
0
votes
0answers
10 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
votes
2answers
33 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)^...
1
vote
4answers
90 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)...
0
votes
2answers
20 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
1answer
21 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
vote
2answers
20 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
53 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
38 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
votes
1answer
30 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
3answers
328 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?
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 ...
2
votes
4answers
52 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
votes
2answers
75 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
28 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
38 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
20 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-...
-1
votes
2answers
37 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 $p$.
7
votes
2answers
480 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
votes
1answer
56 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
788 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 ...
1
vote
1answer
22 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
votes
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 ...
1
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
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 ...
1
vote
1answer
16 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\...
0
votes
0answers
29 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^{...
0
votes
0answers
22 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 ...
1
vote
2answers
54 views

How to calculate matrix rotation

Given the following rotation matrix $$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{matrix} \right]$$ ...
1
vote
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
vote
3answers
54 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
votes
0answers
31 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 ...
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, \...
0
votes
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 ...