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
votes
2answers
27 views

Powers of a matrix using eigen-vectors

Represent a vector as linear combination of eigen-vectors: $$u_0=c_1x_1+....c_nx_n$$ Now, $$Au_0=c_1\lambda_1x_1+....c_n\lambda_nx_n$$ where $\lambda$ is eigen-value. $Ax_i=\lambda_ix_i$ since $\...
0
votes
1answer
19 views

Is a function of a simplectic matrix still a simplectic matrix?

Given $M$, a simplectic matrix $(2n\times 2n)$, the function $f(M)=\exp(M)$ is still a simplectic matrix? More in general, what kind of properties has to have a function $f(M)$ in order to give a ...
5
votes
2answers
110 views

Compute the main diagonal of $(K + D)^{-1}$ in less than $O(n^3)$ operations

Compute the main diagonal of $(K + D)^{-1}$ in less than $O(n^3)$ operations given full-rank, dense and symmetric matrices $K$ and $K^{-1}$, and a diagonal matrix $D$ with positive elements on its ...
3
votes
2answers
71 views

Geometric interpretation of a hollow symmetrical 3D matrix

Any matrix $A$ can be presented as a sum of its symmetrical and skew-symmetrical part: $A=sym(A)+skew(A)$. Decomposition can go further and we can present symmetrical part as a sum of some ...
2
votes
1answer
57 views

Condition for a binary matrix to contain a permutation matrix

I would like to know if there is any condition to check whether a binary matrix contains a permutation matrix of the same size. E.g. $$A_1=\pmatrix{1&1&1&1\\ 1&0&0&1\\ 1&0&...
1
vote
1answer
39 views

What is the derivative of $Tr(X^{-\frac{1}{2}}D)$ with respect to $X$?

In the question, $X$ and $D$ are symmetric positive definite (SPD) matrices, and $Tr(\cdot)$ is the trace of the matrix. $X^{-\frac{1}{2}}X^{-\frac{1}{2}}=X^{-1}$, and $X^{-\frac{1}{2}}$ is also a ...
0
votes
1answer
26 views

Two planes in $\mathbb{R^3}$ can't be orthogonal and it's application

The floor and the wall are not orthogonal subspaces because they share a nonzero vector(along the line where they meet). Two planes in $\mathbb{R^3}$ cannot be orthogonal! Find a vector in both column ...
1
vote
1answer
28 views

which $X$ can satisfy $XSX^2=D$?

In the question, $X$, $S$, and $D$ are symmetric positive definite (SPD) matrices. Just for an example, $X=S^{-\frac{1}{2}}(S^{\frac{1}{2}}DS^{\frac{1}{2}})^{\frac{1}{2}}S^{-\frac{1}{2}}$ is the ...
1
vote
1answer
20 views

Matrices always permute a vector space basis?

Is it possible that given any matrix in $GL_n(K)$ (with $K$ a field) where the selected matrix has finite order, one can show a basis that is permutated by the transformation made by the selected ...
-1
votes
2answers
28 views

Finding basis and nullspace

The question is as follows : If P is the plane of vectors in $$R^4$$ satisfying $$x_1+x_2+x_3+x_4=0$$ write a basis for P perp. Construct a matrix that has P as its nullspace So below is my approach :...
0
votes
1answer
16 views

Projection matrices(meaning)

The question is as follows : Compute the projection matrices $$aa^T/a^Ta$$ for $$a_1=(-1,2,2) \\ a_2=(2,2,-1)$$ Multiply those projection matrices and explain their product P1P2 and what it is. So ...
0
votes
1answer
24 views

Which of the following subsets of $M_n(\mathbb C)$ are compact?

Let $A ∈ M_n(\mathbb C)$ and let $\rho (A) = \max \{| \lambda| : \lambda$ is an eigenvalue of $A\}$ denote its spectral radius. Which of the following subsets of $M_n(\mathbb C)$ are compact? $a. S ...
3
votes
1answer
44 views

$K = \{A ∈ M_n(R) \mid A = A^T , \operatorname{tr}(A) = 1, x^T Ax ≥ 0 \ \ \forall x ∈ R^n\}$. Then $K$ is compact.

Let $K ⊂ M_n(\Bbb R)$ be defined by $$K = \{A ∈ M_n(\Bbb R) \mid A = A^T , \operatorname{tr}(A) = 1, x^T Ax ≥ 0 \ \ \forall x ∈ \Bbb R^n\}$$ Then $K$ is compact. Considering the continuous map $A \...
1
vote
1answer
63 views

An example of a $2 \times 2$ matrix $A$ without real eigenvalues and s.t $A^2$ has $-1$ as an eigenvalue with algebraic and geometric multiplicity $2$

Basically I tried the matrix \begin{bmatrix} 0 & -1\\ 1 & 0\\ \end{bmatrix} but this eigenvalue of $A^2$ is $1$,and $-1$ which has algebraic multiplicity is $1$. I can not find any $2 \...
1
vote
0answers
20 views

I need help normalizing a Gaussian kernel matrix to integer values

I am trying to understand the mathematics behind Canny edge detection, and the first step is to apply a Gaussian blur to the image you are working with. To do a Gaussian blur, you must obtain a ...
3
votes
0answers
64 views

Upperbound of the ratio of column sums of an integer matrix

Suppose $X_{n \times n}$ is a positive integer matrix where $n\geq 2$. The element in the $i_{th}$ row and $j_{th}$ column of the matrix $X$ is defined as $x_{i,j}$. Now, consider $S_{j,j+1}=argmax_{...
0
votes
3answers
41 views

Find a matrix whose right nullspace $\neq$ left nullspace but rowspace $=$ colspace

Question is simple: Find a matrix whose right nullspace $\neq$ left nullspace but rowspace $=$ colspace I thought the inverse symmetric matrix would be a good example for this. $$ A^T=-A $$ ...
0
votes
0answers
20 views

Eigenvalues of cycle on 5 vertices [duplicate]

Reading a list of graph spectra I noticed that the golden ratio pops up 4 times in the eigenvalues for the spectrum of C_5, the cycle on 5 vertices. However I have no idea for C_n for odd n>5 (the ...
2
votes
1answer
54 views

Prove that the ellipsoid $x^T W x \leq 1$ is invariant under $f (x) = A x$ [closed]

Given matrix $W \succ 0 $ and a set $\mathcal{Z} := \{z \mid z^T W z \leq 1\}$, prove that if $Az \in \mathcal{Z}$ and $z \in\mathcal{Z}$, then the following inequality holds $$ A^T W A - W \...
1
vote
1answer
25 views

Representation of Matrix Calculations - Column Mean Subtract From Each Row

Suppose I have a matrix that looks like this: $$X = \begin{bmatrix}1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0\end{bmatrix}$$ The ...
3
votes
1answer
35 views

rank of an even-diagonal integer matrix

If $A$ is a $(2n+1)\times(2n+1)$ matrix, all of its entries integer, whose diagonal elements are even, and all the other elements odd, is the minimum of the rank of $A$ equal to $2n$ ? And I'm not ...
0
votes
1answer
36 views

Time Derivative of a Positive Definite Matrix

Suppose we have a positive definite symmetric matrix $\mathbf V(0) \in \mathbb S^{n}_{++}$, which changes with time according to the following equation, $\dot{\mathbf V}(t) = \mathbf A \mathbf V(t) + ...
1
vote
1answer
24 views

Why can the determinant of a transformation matrix, and the original are be used to find out the new area?

Im studying Year 11 Mathematical Methods. Within the book there are several questions which give vertices (or sometimes just the area) of a particular shap, plus a transformation matrix to be applied ...
1
vote
2answers
35 views

Derivative of an exponential mapping with respect to a scalar

This is a follow up question to one I've asked previously. How can I take the derivative of this exponential mapping / what is the solution? Let $$A (x) := \begin{bmatrix}0&0\\4x&1\...
0
votes
0answers
18 views

What doest W•[h, f] matrix notation mean?

I see the formula (Source) What doest W•[h, f] matrix notation mean? I know that it is the same as Wh + Wf, but don't know the notation meaning.
6
votes
1answer
86 views

Compute a diagonalizable matrix close in matrix exponential

It is known that for any matrix $A$, one can perturb $A$ slightly so that the resulting $A(\epsilon)$ is diagonalizable. I am wondering whether for any matrix $A$, $\epsilon>0$, there is an ...
11
votes
2answers
97 views

Let $S$ be a diagonalizable matrix and $S+5T=I$. Then prove that $T$ is also diagonalizable.

My solution: Since $S$ is diagonalizable, so we can write $S=P^{-1}DP$, where $P$ is an invertible matrix and $D$ is a diagonal matrix. Now $5T=I-S=P^{-1}P-P^{-1}DP=P^{-1}(I-D)P$. So $T=P^{-1}...
1
vote
0answers
21 views

Factorising Cyclic expression .

What are ways for factorising cyclic expressions? Note: I am not saying about specific one. Just ways of factorising cyclic expressions.
0
votes
0answers
16 views

How to proof $M(H,K)X=U([M(\lambda_{i},\mu_{j})]_{ij}\circ (U^{*}XV))V^{*}$

Let $M(x,y)$ be positive real function on $(0,\infty)\times (0,\infty)$ satisfies $M(x,y)=M(y,x)$ $M(\alpha x,\alpha y)=\alpha M(x,y)$ for all $\alpha>0$ $M(x,y)$ is non-decreasing in $x$ and $y$...
1
vote
1answer
25 views

Rank of a block lower triangular matrix

Let $A$ be a $k \times k$ block lower triangular matrix $$A = \left[\begin{matrix} A_{11} & 0 & \cdots & 0\\ A_{21} & A_{22} & \cdots & 0\\ \vdots & \vdots & \...
0
votes
1answer
27 views

sign the elements of a ugly matrix

I have a matrix defined as $A=(I-aT)^{-1}F$, where I is identity matrix, a is a positive constant smaller than 1, T is a stochastic (transition matrix) and F is a matrix with positive diagonal ...
0
votes
1answer
39 views

The set of invertible $k \times k$ matrices with complex entries is a connected subset of $\Bbb C^{k \times k}$. [duplicate]

The set of invertible $k \times k$ matrices with complex entries is a connected subset of $\Bbb C^{k \times k}$. Required Hint for this problem. I have recently proved that the set of invertible $k \...
0
votes
0answers
31 views

Finding the basis and dimension of a subspace of the vector space of 2 by 2 matrices

I am trying to find the dimension and basis for the subspace spanned by: $$ \begin{bmatrix} 1&-5\\ -4&2 \end{bmatrix}, \begin{bmatrix} 1&1\\ -1&5 \end{bmatrix}, \begin{bmatrix} 2&-...
2
votes
2answers
34 views

Existence and uniqueness of $A\in M_2(\mathbb R)$ such that $(A^{-1} - 3 I_2)^t = 2 \begin{pmatrix} -1 & -2 \\ 1 & -5 \end{pmatrix}$

I recently thought about this exercise that states: Does there exist a matrix $A \in M_2(\mathbb R)$ s.t. $$(A^{-1} - 3 I_2)^t = 2 \left( \begin{array}{ccc} -1 & -2 \\ 1 & -5 \\ \end{...
1
vote
3answers
67 views

Computing the $4 \times 4$ determinant of a matrix

Compute the determinant of $$\begin{bmatrix}1&-2&5&2\\0&0&3&0\\2&-4&-3&5\\2&0&3&5\end{bmatrix}$$ by first expanding along the first row (at ...
2
votes
1answer
45 views

find all matrices that com-mute with the given matrix

So I have this matrix and I need to find all matrices that commute with the given matrix A.\begin{bmatrix}2 & 3\\-3 & 2\end{bmatrix} I know how to get to the point where $2a+3c = 2a-3b$ $...
0
votes
1answer
34 views

Linearly Dependent Equations

Let $\mathbf{x}$ and $\mathbf{y}$ be $n \times 1$ vectors, $\mathbf{A}\neq \mathbf{0}$ and $\mathbf{B}\neq \mathbf{0}$ be $m \times n$ matrices with $m>n$, and, for some $u < m$, let $$ \...
7
votes
3answers
94 views

Trace norm of a triangular matrix with only ones above the diagonal

For $n\in\mathbb N^*$, we consider the triangular matrix $$ T_n = \begin{pmatrix} 1 & \cdots & 1 \\ & \ddots & \vdots \\ 0 & & 1 \end{pmatrix} \in M_{n,n}(\mathbb R) \,. $$ ...
1
vote
1answer
51 views

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix?

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix? Assume that $M$ is an $n \times m$ matrix with $n \leq m$. Is it possible to obtain ...
1
vote
0answers
24 views

Matrix increasing integer vector length

Given a matrix $M \in GL(n,\mathbb{Z})$, such that $M$ has an eigenvalue $\alpha$ with $|\alpha|>1$. Does there always exist a vector $x \in \mathbb{Z}^n$, $l \in \mathbb{N}$ and $\lambda > 1$, ...
2
votes
0answers
28 views

Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
1
vote
0answers
30 views

Conditions for a block matrix to be positive definite

Consider the block matrix given by $$M = \begin{bmatrix} A_{11}&A_{12}&0\\ A_{12}&A_{22}&A_{23}\\ 0&A_{23}&A_{33}\end{bmatrix}$$ What conditions should I impose on each ...
3
votes
1answer
64 views

Is this number positive?

Let $(a_{ij})$ be a collection of non-negative numbers indexed by integers $1\le i,j \le N$ where $N$ is some fixed integer. Let $(c_{ij})$ be another collection of real numbers also indexed by ...
0
votes
1answer
32 views

rank of these two matrices [on hold]

Could anyone confirm me that rank of these two matrices are $3$ or not? Thanks! $$M=\begin{pmatrix}3&0&1&0\\0&2&2&-1\\1&2&3&0\\0&-1&0&2\end{pmatrix}, \...
1
vote
1answer
29 views

Derivation of why a diagonalizable matrix can be written as a sum of outer products $\Sigma=\sum_{i=1}^n \lambda_i v_i v_i^T$

Lets say we have a symmetric positive semi-definite $n\times n$ matrix $\Sigma$, which therefore has a diagonalization $\Sigma=V\Lambda V^T$, where $V$ is an orthogonal matrix (containing the ...
2
votes
3answers
44 views

If $BNA=N$ it can happen that $B$ and $A$ identities?

Consider two matrices $A\in GL_m(K)$, $B\in GL_n(K)$ such that for any $N\in M_{n,m}(K)$ is true that $$BNA=N$$ How can I prove that $B$ and $A$ are identities?
3
votes
1answer
66 views

Is square-root of a real symmetric positive semi-definite matrix real as well?

Given a real symmetric positive semi-definite matrix $A$, will there be a root $R$ which is real symmetric positive semi-definite as well? Can you comment on it's uniqueness? It will be nice if you ...
0
votes
0answers
20 views

Solve linear system with $A_{i,j} = \langle e_i, e_j\rangle^2$, edges of a tetrahedron

I have six vectors in $e_i\in\mathbb{R}^3$ that are the edges of a tetrahedron. Let us consider now the linear equation system $Ax=b$ with $$ A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, ...
0
votes
0answers
19 views

Standard form or simple expression for matrix to select “diagonal” columns

My question isn't about understanding what's below, but if there's a standard expression for it in the literature, or a simple way to express it that lends itself to easy manipulation. I have a block ...
1
vote
1answer
41 views

If two real matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$? [duplicate]

As in the title: If two real (square) matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$?