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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0answers
7 views

Distance between all rows in 2 matrices expressed as a matrix equation

I have two matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. All real numbers. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element ...
0
votes
0answers
11 views

Eigenvalues of (restrictions of) the standard representation of $S_n$

Let the permutation group on $n$ elements $S_n$ act on a set $S$ of size $k < n$ via permutations. Fix some ordering on the elements of $S$ to make this sensible. Is there any way to understand ...
-2
votes
0answers
20 views

Linearly independent columns [on hold]

Let $M$ be a square upper triangular matrix with non-zero diagonal entries. Prove that the column vectors of $M$ form a linearly independent subset of $\mathbb{R}^n$. Please somebody answer and soon.
2
votes
2answers
25 views

Skew symmetric 4x4 matrix of full-rank

I have come across the fact that a 4x4 skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 & ...
-3
votes
0answers
7 views

HOW TO PLOT DAG (DIRECTED ACYCLIC GRAPH) in BNT toolbox for matlab.

I have used markov chain monte carlo (MCMC) in BNT toolbox for matlab, from which i have got one output "sampled_graphs " which is cell array. Now how to plot DAG (Directed acyclic graph ) from ...
0
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0answers
29 views

explaining the pattern

I have been given the following math puzzle: you are given a matrix that is filled by the following rule: every cell i,j is evaluated by taking the lowest non-negative number that is not present in ...
1
vote
1answer
13 views

Divide matrix using left division

In matlab, I defined a=[1;2;3] b=[4;5;6] both a and b are not square matrix. and execute a\b will return ...
0
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0answers
19 views

Is there a relation to determine condition (positive or negative definite) of C, if C = A+B and A, B are positive and negative definite?

I have a question: Matrix A and B are positive and negative definite, respectively. Is there a relation to determine whether C is positive or negative definite, if C = A+B?
0
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1answer
31 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...
-1
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0answers
13 views

Transforming roll along a path

I'm working on a script to transform a bunch of planes in 3d space. The source data is a bunch of points in 3d space that first form an L shape and then the movement path. Point 2 is the starting ...
0
votes
1answer
9 views

Rank of block matrix

Given a $q\times n$ matrix $E$ whose rank is $n$. Imagine that every element $[e_{ij}]$ of $E$ is replaced by a $m\times p$ matrix $F_{ij}$, whose rank is $p$. And in general, each $F_{ij}$ is ...
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0answers
10 views

Proof for Determinants using Laplace and induction.

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ I need to prove that det(A)=det(B). I thought induction might be one solution, but I don't know how to apply the Laplace ...
0
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0answers
20 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
0
votes
1answer
19 views

Given a square matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues?

Given a matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues? If instead the matrix had its nonzero entry component at ...
1
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0answers
21 views

How to determine if a set represents a line, plane or hyperplane?

How do you approach a question that gives you a set and asked to determine if it represents a line, plane or hyperplane? The Question: https://www.dropbox.com/s/0gscqur18kqg3ma/SpanningQuestion.PNG ...
2
votes
0answers
20 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
1
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1answer
23 views

Solve the following matrix equation $X'X=A$

I have square matrices $X$,$A$ and $X'X-A=0$. $A$ is given and is positive definite and I need to get matrix $X$. I know $X$ is not unique since $TX$ such that $T'T=I$ will satisfy. My problem is ...
-1
votes
1answer
32 views

Prove the equality of two determinants. [on hold]

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ Proof that $det(A)=det(B)$? Thanks in advance.
1
vote
1answer
19 views

Triangularization of a matrix.

so I need to find an invertible matrix $P$ such that $P^{-1}AP$ is upper-triangular, where $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$ So I found that the eigenvalue is $1$ which ...
0
votes
1answer
16 views

Linear algebra - projection matrix - inverse matrix

I am not sure how to prove this one: Let $A$ be a projection matrix so that $A^2=A$ and $A$ is not equal to zero. Find the inverse matrix of $I+cA$. Thanks.
1
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2answers
53 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
0
votes
1answer
22 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
0
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1answer
35 views

$\operatorname{rank}(A) = $max number of rows of submatrix $B$; Proof

I don't understand how to proof the following: The rank of a matrix $A \in M$ ($m \times n$, Field) equals the maximum number of rows of a square submatrix $B$ of $A$ with $\det (B) \neq 0$. The ...
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0answers
17 views

Operator norm of a matrix less than or equal to one

Do all matrices of operator norm $\leq 1$ have the sum of the absolute values of their rows $\leq 1$?
1
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3answers
172 views

To show two matrices are conjugate to each other

Given two matrices A and B $$ A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 ...
3
votes
2answers
24 views

Eigenvalues of a transpose multiplication

Say I have a matrix $\mathbf B \in \mathbb R^{m\times n}$. Is it correct to say that the eigenvalues of $\mathbf B^T\cdot\mathbf B$ are always positive?
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0answers
22 views

What does matrix decomposition really mean?

Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & ...
2
votes
0answers
38 views

Is $2^{xy}$ a positive definite kernel?

Is $2^{xy}$ a positive definite kernel on $\mathbb{N}$? i.e. for all $a_1, ..., a_n \in \mathbb{R}$, for all $x_1, ..., x_n \in \mathbb{N}$, $\sum_{i,j} a_i a_j 2^{x_ix_j}\geqslant 0$
-1
votes
0answers
27 views

does a closed form solution exist for this equation?

I have a cost function $J$, which depends on a projection matrix $W$, which is unknown. When I get the partial derivative $\frac{\partial J}{\partial W}$ the equation is: $\frac{\partial J}{\partial ...
1
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0answers
20 views

Gauss-Jordan elimination in the form of (A|I)

So Gauss-Jordan elimination can be performed through the form of $(A|I)$ where $I$ is the identity matrix. We carry out row elementary operations as usual until the matrix becomes the form $(I|B)$, ...
1
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1answer
18 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
3
votes
3answers
144 views

“Orthogonal” Rectangular Matrix

Is it possible to have a matrix $\mathbf B \in \mathbb R^{m\times n}$ such that it satisfies: $$\mathbf B^T\cdot\mathbf B = \mathbf I_n$$ Where $\mathbf I_n$ is the $n\times n$ identity matrix. Or ...
2
votes
1answer
28 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.
0
votes
2answers
35 views

Change of Basis for $2\times2$ matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
-1
votes
1answer
36 views

how to creat a matlab program? [on hold]

How can I creat a tridiagonal matrix in matlab if the elements of matrix is again a matrix instead of a scalar.n how to solve them using crouts methods.I am new to matlab.
1
vote
1answer
29 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
0
votes
1answer
16 views

Find the standard matrix of T given T is a linear transformation

$T:\mathbb{R}^2\to \mathbb{R}^2$ first performs a horizontal sheer that transforms $e_2$ into $e_2 + 2e_1$ (leaving $e_1$ unchanged) and then reflects points through the line $x_2 = -x_1.$ I am ...
1
vote
1answer
22 views

Quickest way to calculate $(I-M)^{-1}(I-M^{n+1})$

What is the quickest way, as computationally efficient, to calculate $(I-M)^{-1}(I-M^{n+1})$, where $M$ is a given $n \times n$ singular matrix of $0$'s and $1$'s? My experiments show that the matrix ...
0
votes
2answers
45 views

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, Prove that $|(A+A^T)(B+B^T)|=0$

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, prove that $\det[(A+A^T)(B+B^T)]=0\ \ $ with $ \ k\in \mathbb{N}$ I don't have ideas for this problem. Thanks !
1
vote
2answers
17 views

Implications on structure of $B$ when $rank(A-B) = s$ for a fixed $A$

Consider the case where $A \in \mathbb{R}^{n \times K}$ where $n > K$ and $\text{rank}(A) = K$. Suppose we know $$ \text{rank}(A - B) = s$$ for some $s < K$. What does this say about the ...
0
votes
0answers
18 views

How to prove that a matrix with specific property is invertible?

If we have a square matrix $$ M = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & ...
2
votes
1answer
53 views

Eigenspaces and jordan normal form

I have a question here regarding the jordan normal form of two matrices where the eigenspace is one is contained in the other. Let $A,B$ be two $nxn$ matrices s.t $AB=BA$. I firstly proved that the ...
0
votes
0answers
29 views

Sparse Matrices and Tridiagonalization.

Assume that we are given a sparse matrix,let it be 90*90(1000*1000), would you say that a vector with lots of zeros(let it be 90*1(1*1000),and 65(500) zeros are there),is a smart option to initialize ...
2
votes
0answers
31 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...
1
vote
2answers
26 views

Show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$ for a normal matrix $A$ with an eigenvalue $c$

Suppose $A \in M_{n\times n}(\mathbb C)$ is a normal matrix and $c$ is an eigenvalue of $A$. I'm trying to show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$. I know that if we were ...
1
vote
1answer
43 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
3
votes
2answers
117 views

How to show that $A^2=AB+BA$ implies $\det(AB-BA)=0$ for $3\times3$ matrices?

Let $A$ and $B$ be two $3\times 3$ matrices with complex entries,such that $A^2=AB+BA$. Prove that $\det(AB-BA)=0$. (Is the above result true for matrices with real entries?)
1
vote
1answer
20 views

Clarification of some doubts on the definition of submatrix

I don't fully understand how I can choose a submatrix in a matrix. Judging from this definition and picture (http://mathworld.wolfram.com/Submatrix.html), I would assume that you can't pick as a ...
2
votes
1answer
50 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
3
votes
1answer
42 views

Reversed Cayley transformation for any unitary matrix

It is well known that if $Q$ is a complex unitary matrix such that $I+Q$ is invertible (where $I$ is the identity matrix), that is, $-1$ is not an eigenvalue of $Q$, then $$ A:=(I-Q)(I+Q)^{-1} $$ is ...