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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4
votes
1answer
81 views

Can a elementary row operation change the size of a matrix?

My question is very simple - Can an elementary row operation change the size (eg: $2\times2$ or $3\times 2$) of a matrix? I think the answer should be no, but while reading Linear Algebra by Hoffman ...
0
votes
3answers
56 views

Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
1
vote
0answers
27 views

Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
0
votes
0answers
19 views

$A^{\pi}$ property

Can someone give an example for the matrix mentioned in the following definition. Definition is taken from "Iterative Solution Methods" of Owe Axelsson. Definition: The matrix $A$ said to have ...
1
vote
1answer
24 views

Algorithm to check if number x exists in matrix

I have the task to develop an algorithm which checks if a specific number x exists in an int-array[][]. Further the 2-dim array entries have following terms: $$array[i][j] \leq array[i][j+1] \space ...
2
votes
1answer
65 views

Prove that $R(+,.)$ is a division ring but I disproved it

QUESTION: Let $R=\left[\begin{matrix}\alpha & \beta \\ \bar\beta & \bar\alpha\end{matrix}\right]\in \mathbf{M_2(\mathbb{C})} $ where $\bar\alpha,\bar\beta$ denote the conjugates ...
9
votes
1answer
89 views

Prove the n-th power of a matrix is the null matrix

Let $A,B$ squared matrixes with complex elements, $dim(A)=dim(B)=n, AB=BA, \det(B)\ne0$, having the following property: $|\det(A+zB)|=1, \forall z \in \mathbb{C}, |z|=1$. Prove ...
1
vote
2answers
31 views

Eigen-values of a matrix $P^{-1}AP$

QUESTION: If A and P be $2$ non-singular $n\times n$ matrices and $\lambda$ is the eigen-value of $A$, then show that $\lambda$ is also the eigen-values of a matrix $P^{-1}AP$. I could simply ...
0
votes
1answer
25 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and ...
-1
votes
0answers
16 views

Find the center of Group in 2 cross 2 matrices in group theory [closed]

Find the center of Group in 2. 2 matrices
1
vote
0answers
16 views

Proving that an element of a ring annihilates a module [duplicate]

Let $R$ be a commutative ring with $1$, $M$ be a finitely generated $R$-module, $\mathfrak{i}$ an ideal of $R$, and $\phi$ an $R$-homomorphism such that: $$1.\;\phi(M)\subseteq \mathfrak{i}M=M$$ i. ...
1
vote
1answer
37 views

Existence of Unimodular Congruence Transformation for Symmetric, Integer matrices

Two symmetric, integer valued matrices, $K_1$ and $K_2$, are congruent if there exists a unimodular integer matrix, $X$, such that $$X^T K_1 X = K_2$$ What are the conditions on the existence of such ...
0
votes
0answers
23 views

Rank of a symmetric matrix after removing a column and row.

If I have a $n\times n$ symmetric matrix $M$ with real entries, zeros on the diagonal, and two of the column vectors are identical and I remove one of these columns, and the corresponding row, then ...
1
vote
1answer
36 views

Effect of simple linear transformation

Consider the linear transformation given by $$T\left\{\begin{bmatrix}x \\y\\z\end{bmatrix}\right\} =\begin{bmatrix}-x\\y\\z\end{bmatrix}$$ Find a matrix $A$ such that $T(x) = Ax$, where x = $[x, y, ...
3
votes
1answer
26 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
0
votes
0answers
48 views

Does $B^2 \leq A^2$ imply $\| A^{-1} B\| \leq 1$ for the operator norm?

Assume we have two $n \times n$ real symmetric matrices $ A^2 $ and $B^2$, such that it holds for some $0\leq\rho<1$ $$ 0 < (1-\rho)B^2 \leq A^2 \leq (1+\rho)B^2, $$ where "$\leq$" means ...
0
votes
2answers
38 views

Generate random variables without cholesky decomposition

How would you generate n standard normal random variables with $n\times n$ correlation matrix $R = (r_{ij} )$ where $r_{ij}$ is $1$ if $i = j$ $\rho$ if $i \neq j$, with $\rho \geq 0$ without ...
5
votes
2answers
154 views
+100

Given a symmetric positive-definite matrix $M$, find all $A$ such that $A^\top M A=M$

Given $M$ a real symmetric positive-definite matrix, I would like to characterise all matrices $A$ such that $A^\top M A=M$. Note that the question of finding $A$ solutions to $A^\top M A=M$ for all ...
1
vote
1answer
34 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So ...
-1
votes
1answer
28 views

Generating normal random variables with mean and variance [closed]

I wish to generate normals $X$, $Y$, and $Z$ with the correlation matrix $R$ but with means $0$, $1$, and $2$, as well as variances $4$, $16$, and $25$, respectively. How would you do this?
0
votes
1answer
31 views

Find vector x such that matrix multiplication Sx = 0

I have the following matrix $$S= \begin{bmatrix} -1 & 1 & 0\\ -1 & 1 & 1\\ 1 &-1& -1\\ 0 &0 & 1 \end{bmatrix} $$ I wish to find a non-negative, ...
1
vote
4answers
39 views

Is the L in LU factorization unique?

I was doing an LU factorization problem \begin{bmatrix} 2 & 3 & 2 \\ 4 & 13 & 9 \\ -6 & 5 &4 \end{bmatrix} and I was going to multiply the second row by ...
0
votes
1answer
37 views

Cholesky decomposition of a $4 \times 4$ matrix

I want to decompose the following matrix using Cholesky. I know that $R=LL^T$ where L is the lower-triangular, but I do not know how to find the lower-triangular or if that formula suffices for this ...
1
vote
1answer
24 views

Lower bound for norm of matrix

I have the following problem: $A$ is a positive definite, symmetric matrix. Firstly I was required to find a matrix $B$ such that $B^n = A$. I believe this to be $C(D^{\frac1n}) C'$ where C is the ...
0
votes
0answers
27 views

If some columns of $XA, A$ are equal, does it mean $XA=A$?

I'm working on a problem related to the row space $R(A)$ of a matrix $A \in K^{k \times n}$, where $k < n$. This space is invariant under a left-action of $GL(k, K)$ on the matrix $A$. Say I have ...
1
vote
1answer
47 views

Does $A^2 \geq B^2 > 0$ imply $ACA \geq BCB$ for square positive definite matrices?

Assume we have two $n \times n$ real nondegenerate matrices $ A^2 $ and $B^2$, such that $$ A^2 \geq B^2 > 0, $$ where "$\geq$" means positive semidefinite (Loewner) ordering. Does the following ...
0
votes
1answer
39 views

Significance of Rank of Matrix

Why we determine the Rank of Matrix ? Instead of this just asking for my info : What is the easiest way to find Rank of Matrix ?
1
vote
1answer
29 views

Eigen values of a matrix depending on k

If $A = \begin{bmatrix} 2 & k \\ 0 & 1 \end{bmatrix}$. Find all values of $k$ for which A has eigenvalues 3 and -1. A has no real eigenvalues. (David Poole, Linear Algebra). The ...
-2
votes
1answer
36 views

Square Matrices and Determinant question [closed]

For invertible matrices $A$ and $B$ of order $n$, show the following $(AB)^{-1} = B^{-1} A^{-1}$ Thank you!
0
votes
0answers
14 views

Sum of principal minors of order $n-1$ equals the sum of products of eigen values taken $n-1$ at a time

Let $A$ be a symmetric matrix of order $n$ .Prove that Sum of principal minors of $A$ of order $n-1$ equals the sum of products of eigen values taken $n-1$ at a time . Now if I consider the ...
1
vote
0answers
28 views

Projection from high dimension to lower, for visualization

I want to project high dimensional data points onto 2D screen coordinates, for visualization purposes. I want to be able to control the angles of projection manually (eg, with the mouse). I have ...
3
votes
3answers
93 views

Is a diagonalization of a matrix unique?

I was solving problems of diagonalization of matrices and I wanted to know if a diagonalization of a matrix is always unique? but there's nothing about it in the books nor the net. I was trying to ...
1
vote
1answer
45 views

If the determinant of a matrix goes to infinity, does it means it has no inverse?

Context I have a linear time-invariant (single-input, single-output) system in state space representation (https://en.wikipedia.org/wiki/State-space_representation#Linear_systems): $$ \mathbf{x'}(t) ...
0
votes
2answers
49 views

Matrix or vector product

This is probably a simple question A factory produce a good (1) that requires 3 labor-hours in the assembly department and 1 labor-hour in the finishing department. Assembly personnel receive 19 per ...
0
votes
0answers
57 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
0
votes
0answers
25 views

Bounding the off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
2
votes
2answers
32 views

How do I build this band matrix in MATLAB?

I need to build a pentadiagonal matrix in MATLAB like this: $\begin{pmatrix} 1+2\lambda & -\lambda_1 & 0 & -\lambda_1 & 0 & \cdots & 0\\ -\lambda_1 & 1+4\lambda_1 & ...
0
votes
4answers
27 views

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$?

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$? I tried with $\det(A - aI) = (\cos\phi - a)^2 + \sin^2 \phi = 0$ and I got somehow to $2\cos\phi = a$, and I believe ...
0
votes
0answers
11 views

Symmetry properties charge conjugation matrices in even dimension.

While reading a paper on supersymmetry (by Peter West) i faced the following problem. Its about the symmetry property of charge conjugation matrix in different space time dimension. The charge ...
0
votes
0answers
6 views

Properties of $\nabla T_k(x)\cdot \nabla T_i(x)$ for a diffeomorphism $T$

Let $T:A \subset \mathbb{R}^n \to B \subset \mathbb{R}^n$ be a smooth diffeomorphism between $A$ and $B$. Is there anything I can say about the quantity $$\nabla T_k(x)\cdot \nabla T_i(x)$$ where ...
1
vote
3answers
45 views

Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
0
votes
1answer
21 views

Rule of thumb on number of zero entries for invertibility of a $4\times 4 $ matrix?

I have to determine whether a $4\times 4$ matrix $A$ is invertible. Suppose that there are no zero columns or zero rows. Is there any rule of thumb saying how many zero entries can be at most in $A$, ...
0
votes
2answers
12 views

what is the difference between multi-linear coefficient and multiple linear regression

what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression. I am confused at this point.
0
votes
2answers
43 views

Find the eigenvector and eigenvalues for the following 3 x 3 Matrix?

$$ \pmatrix{5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 &-4 & -11} $$ I already got the eigenvalues that is $\lambda = 1$ and $-3$. And I managed to solve the eigenvector corresponding to ...
1
vote
1answer
35 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
0
votes
0answers
17 views

Prove two matrices are similar?

Let $G_1=[I(k), \mathcal G_1]$, $G_2=[I(k), \mathcal G_2]$, $H_1=[\mathcal H_1,I(m)]$ and $H_2=[\mathcal H_2,I(m)]$, where $\mathcal H_1, \mathcal H_2$ are transpose of $\mathcal G_1,\mathcal G_2$, ...
0
votes
0answers
8 views

Aligning matrices, normalization. Calculating coefficients.

So as a pre-task for my upcoming exam this is one of the rehearsal assignments. I can't wrap my head around this one at all, haven't seen anything like it earlier, and I can't seem to find any ...
0
votes
1answer
18 views

Can you multiply a matrix out of another one?

This is actually from a computer graphics problem. I calculate a transformation matrix by multiplying a few other ones. ...
1
vote
1answer
59 views

Prove that this $10 \times 10$ matrix is diagonalizable. [closed]

Suppose that $A$ is an non-invertible $10\times10$ real matrix, and that $\mathrm{rank}(A-3I)=7$ , $\mathrm{rank}(A-I)=4$. How do I prove that $A$ is diagonalizable?
0
votes
1answer
48 views

Does $AA^T = A^TA$ imply that A is normal?

A is $n\times n$ matrix over complex numbers. Does $AA^T = A^TA$ imply that A is normal? If not what will be a counterexample?