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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0
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1answer
17 views

Proofing that an matrix is idempotent

My task was to show that certain matrices are idempotent, that is, ${AA} = {A}$. I struggled a with the proof for one case and when I allok at the solution, I have problems understanding onse step. ...
0
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0answers
12 views

Application of Frobenius inequality [on hold]

What are the most interesting applications of the Frobenius inequality?
0
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1answer
23 views

For what values of $k$ will these equations have no solution/infinite solutions/unique solution

Here are the 3 linear equations: $$x+y-z=-1$$ $$2x-4y-6z=-1$$ $$x-y+(k^2-1)z=k$$ I understand a $4\times3$ matrix must be set up in order to solve this particular problem.The part which I get ...
4
votes
8answers
229 views
+100

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
3
votes
0answers
20 views

Easy elementary proof of Farkas Lemma?

Is there any elementary proof of Farkas lemma which does not use convex analysis and hyperplane separation theorem? What about special case below: If the Matrix $A$ is invertible, then there is ...
2
votes
1answer
77 views

How prove this matrix $B^{-1}-A^{-1}$ is positive-semidefinite matrix,if $A-B$ is positive matrix

Question: Let $A,B$ be positive $n\times n$ matrices, and assume that $A-B$ is also a positive definite matrix. Show that $$B^{-1}-A^{-1}$$ is a positive definite matrix too. My idea: ...
0
votes
1answer
16 views

Optimising using Hessian matrix

I am bit perplexed in optimisation problem if the principal minor is zero. If the principal minor is zero does it mean that the Hessian matrix is always indefinite and the point of extremum will refer ...
0
votes
0answers
5 views

How to calculate discrete cosine transform for a matrix

I have a 8x8 matrix and I want to calculate its discrete cosine transform (DCT-II). I have this formula but I don't know to use it with a matrix. In the French Wikipedia they gave an example for ...
3
votes
1answer
101 views

When is there a vector $D$ with positive coordinates such that $e^{Ct}D$ has a negative coordinate?

Let $C$ be a $2 \times 2$ asymmetric matrix with real entries. Assume that $C$ has strictly negative, real eigenvalues. Fix $D\in\mathbb{R}^2$, where $D > 0$ (i.e., both coordinates are strictly ...
1
vote
0answers
9 views

Identities for the Hilbert–Schmidt norm of products of projections.

I've been studying different metrics on the Grassmannian $Gr(k,n)$ of k-dimensional linear subspaces of $\mathbb{R}^n$ and found myself needing some identities for the norm of a product of orthogonal ...
0
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1answer
37 views

how to write formal proofs involving nxn matrices

i have problems like these: Prove that if A is a nxn matrix, then tr(A-A^T)=0 Prove that if A and B are nxn matrices then tr(A+B) = tr(A) + tr(B). I can clearly understand why these hold true and ...
0
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0answers
18 views

Are there any sure-fire methods for correctly arranging matricies for Gaussian Elimination?

I am attempting to make a Gaussian Elimination solver for systems of linear equations that contain less than 100 equations. I have roughed out a method for creating and filling in the diagonal of a ...
0
votes
0answers
5 views

Open Leontief input output model

An economy is divided into three sectors: Manufacturing, Agriculture and Services. For each unit of output, Manufacturing needs 0.1 units of Manufacturing, 0.3 units of Agriculture, 0.3 units of ...
0
votes
0answers
16 views

Changing a negative definite matrix to a positive definite matrix

Consider a negative definite matrix $X$,then $(I-e^X)$ is a positive definite matrix. What condition should matrix $X$ satisfy?
0
votes
0answers
22 views

Finding a parity check matrix of a binary code

I'm supposed to find a parity check matrix of a binary [6,3,3] code. Given a generator matrix G I can find a parity check matrix by row reducing until I get the identity matrix, then take $-A^{\top} ...
0
votes
4answers
35 views

commutative matrix multiplication of nxn matrices?

If there are two matrices A and B that are both nxn matrices, will AB = BA always? Is there a way to have those two matrices so that AB = 0 but BA ≠ 0?
4
votes
1answer
101 views

General formula for $\det(A+I)$ where I is identity. Worked it out for $2 \times 2$ and $3 \times 3$.

Does anybody know a general formula for $|A+I|$ where $A$ is a $\textbf{symmetric}$, real (square) matrix? For a $2\times2$ system I worked out: $|A+I| = |A|+\text{tr}(A)+1$. This is very friendly. ...
0
votes
1answer
232 views

What are the $\succ$ and $\prec$ operators for when used with matrices?

I understand that $A\succ0$ means that "A is a positive definite matrix" (i.e.; all of the eigenvalues of A are positive). But what does it mean when the right hand side is a different value than ...
0
votes
4answers
43 views

Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$ A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix} $$ I have found the eigenvalues to be 1 ...
1
vote
0answers
23 views

Closed-form expression for this matrix equation?

I have the following matrices $P \in \mathbb{R}^{N \times N}$, $q(k) = \begin{bmatrix} q_1(k) \\ \vdots \\ q_N(k) \end{bmatrix}$. With $q_i(k) \in \mathbb{R}^n$ and thus $q(k) \in \mathbb{R}^{Nn}$. ...
14
votes
2answers
708 views

Are 10x10 matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
0
votes
1answer
42 views

If A is a matrix, what does A' mean?

If A is a matrix, what does A' mean? I have tried google this but nothing came up. My new stats course had some review problems, and these multiple choice came up. Which statement is true? (a) ...
3
votes
1answer
46 views

Matrix manipulation using trace

Suppose that $u$ is an $N\times 1$ random vector and $M$ is an $N\times N$ nonrandom positive semi-definite matrix that is also idempotent: $M\times M=M$. Claim: $E(u'Muu'Mu)=\text{Tr}\{M ...
4
votes
1answer
35 views

Are primitive row stochastic matrices diagonalizable?

Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are ...
2
votes
1answer
628 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
5
votes
3answers
200 views

What “is” a matrix in the context of a vector space?

I'm familiar with the definition of a vector space $V$ over a field $F$ I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space ...
0
votes
1answer
20 views

$A$ and $B$ conjugacy

Show that the matrices $A=\begin{pmatrix}2&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix}2&0\\1&0\end{pmatrix}$ are not $\mathbb{Z}$ conjugate (there exists no matrix ...
1
vote
1answer
18 views

Trace of vectors

Does that sound about right? Given that x is $m\times 1$ and y is $m\times 1$ vectors, show that $ tr(\mathbf{xy'})=\mathbf{x'y}$. Attempt: By using the property of ...
1
vote
1answer
24 views

If we know nullspace of matrix, how to find reduced row echelon form of that matrix?

vectors u = [4 1 0 0] and v = [1 0 2 1] form a base of nullspace of matrix $$ A\in M_{5,4}(R) $$ Find a reduced row echelon form of Matrix A. Since $ n-r = dimN(A) $ we know we got two base ...
-1
votes
3answers
43 views

Matrix Algebra $(A+B)^2$ help [on hold]

Does $(A + B)^2$ = $A^2$ + $B^2$ or $A^2$ + $B^2$ + $AB$ + $BA$ ? Where A and B are both matrices.
0
votes
2answers
35 views

Matrix equation xA=x

The equation $Ax=x$ seems trivial, but how one could solve xA=x. Assuming A is reversible, the best I can get to is $x(a^{-1}+I)=0$. However, I'm not sure what to do from here. Thank you for your ...
0
votes
2answers
48 views

Is it true: A real symmetric matrix is either positive definite or negative definite or indefinite?

I got a real symmetric matrix that is neither positive definite or negative definite, so can I just say that this matrix is indefinite? Thanks in advance:)
0
votes
0answers
22 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
0
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0answers
12 views

How to find a function from a matrix?

Suppose I have a matrix like this: H2 N2 G2 H1 0 3 8 N1 2 4 7 G1 1 5 6 How would I find a nice function $f(x, y)$ that ...
0
votes
3answers
49 views

Proofs about Matrix Rank

I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the ...
0
votes
2answers
35 views

Transformation Matrix project

My task is to find the Transformation Matrix, that projects, any point of the xy-plane, on the line $$ y = 4x$$ The solution should be: $$T=\pmatrix{0.06&0.235\\0.235&0.94}$$ But somehow i ...
3
votes
2answers
58 views

Is the set of invertible upper triangular matrices open in $GL_n(\mathbb R)$? Is it open in the set of all upper triangular matrices?

I think the answer to the second question is yes, but can't quite prove this. I've no idea about the first part. I've done a few exercises of this kind but all have used the continuity of the ...
1
vote
0answers
20 views

Prove that this sequence of fractions is returned unchanged after the divisor recurrence, the matrix inverse, and the sum over divisors.

Consider the fraction of binomial coefficients and powers of $4$: $$a(n)=\frac{\binom{2 (n-1)}{n-1}}{4^{n-1}}$$ starting: ...
0
votes
2answers
90 views

How to show a matrix can't be written as exponential?

How can I show the matrix $$A = \left( \begin{array}{c c} -2 & 0 \\ 0 & -1 \\ \end{array} \right)$$ can't be written as $A = exp(a)$? I've tried to write A like $$A = \left( ...
4
votes
1answer
277 views

Rank of matrix as a difference of ranks

If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank ...
2
votes
0answers
28 views

Does the distance between points determine the shape?

Given three pairwise distances between three unknown points in a plane, the positions of the triangle vertices are uniquely determined up to a rotation and translation. Is this true for an arbitrary ...
0
votes
0answers
15 views

Is a complex function really just an infinite dimensional matrix?

I have recently sort of come to the understanding that integrating two functions multiplied together is a sort of infinite dimensional dot product, and I only know this from taking an undergraduate ...
1
vote
1answer
39 views

Matrices over PID

Let $R$ be a PID and $A,B\in\operatorname{M}_n(R)$ are $n\times n$ matrices such that $\det(A)\sim\det(B)\neq0$,i.e., the ideals generated by $\det(A)$ and $\det(B)$ are the same, does there exist ...
2
votes
1answer
44 views

Determinant in $\mathbb Z_{5}$

I need to find $$ \det\left[ \begin{array}{cc} 2 & 4 & 0 \\ 1 & 1 & 3 \\ 3 & 2 & 1 \end{array} \right] $$ over $\mathbb Z_{5}$ What I did: $$2\det\left[ ...
0
votes
1answer
22 views

Finding a standard generator matrix given a binary code

My question is how do I find the standard generator matrix of a binary [7,6,2] code? From what I understand a generator matrix for $C$ is any $ k \times n$ matrix $ G$ with entries in $ ...
1
vote
1answer
32 views

Finding determinant of a 3x3 matrix

Assuming y is a nonzero real number, I need to find the determinant of this matrix: $$ \left[ \begin{array}{cc} 1 & y & y^2 \\ y & y^2 & y^3 \\ y^2 & y^3 & y^4 ...
4
votes
1answer
79 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
0
votes
1answer
19 views

Computing RREF on matrix with variables in coefficients

$\pmatrix{a&a&-1&1\\ 1&-1&1&a\\ -1&1&a&1\\ }$ It says to consider cases such as a=0 or a does not equal 0.
0
votes
0answers
20 views

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$ Prove for $1: e_{ii}A=Ae_{ii}$ and for $2: e_{ij}A=Ae_{ij}$ where $i\le j$ Now for 1, I understand ...
0
votes
1answer
35 views

How can I find out when a system of linear equations have a non-trivial solution?

So I have two linear equations: $(a-1)x + 2y = 0$ $2x + (a-1)y = 0$ How can I figure out for what values they have non-trivial solutions whereas $x$ and $y$ aren't 0?