Tagged Questions

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
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0answers
19 views

Checking wether the vectors are linearly independent.

I just finished an assignment and I would like someone that knows this material to basically check if I'm doing it right. I was given 3 vectors <1,2,3>, <1,0,1>, <2,1,0> in R^3 and need to ...
2
votes
1answer
11 views

Determinant value of a square matrix whose each entry is the g.c.d. of row and column position

Let $A=(a_{ij})$ be a $n \times n$ matrix with $a_{ij}=\gcd(i,j) , \forall i,j=1,2, \cdots, n$ , then how do we prove $\det A=\prod_{i=1}^n \phi(i)$ ? , where $\phi$ is the Euler's phi function
0
votes
1answer
10 views

(Partial) symmetry order for matrices

Does there exists commonly used partial orderings which would rank matrices as a function of their "degree of symmetry"? I am thinking one could for instance have $\succeq_{SYM}$ defined as : ...
0
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0answers
10 views

A property of weighted pseudoinverse matrix

Let me assume that $\mathbf{J} \in \mathbb{R}^{m \times n},~m<n$ is a full row rank matrix, $\mathbf{A} \in \mathbb{R}^{n \times n}$ is a symmetric positive definite matrix, and $\mathbf{J}^{-}$ is ...
0
votes
0answers
11 views

Minimum of the trace of a Cholesky factorizable matrix

Given a matrix $P$ positive semidefinite Cholesky factorizable: $$P=SS^T$$ does this equality hold? $$\arg\min_{\theta}(\mathrm{tr}(P(\theta))=\arg\min_{\theta}(\mathrm{tr}(S(\theta)))?$$ if the ...
0
votes
0answers
19 views

Find an orthogonal matrix $P$ such that $P^{T}AP$ is diagonal.

I began by finding the eigenvalues and eigenvectors of $A$ where $A=\begin{pmatrix} 4 & 0 & -2 \\ 0 & 2 & -2 \\ -2 & -2 & 3 \end{pmatrix}$. This gave $\lambda_1=0, ...
0
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0answers
13 views

Why is the state transition matrix unique if the fundamental matrix is not?

For a linear time variant system, the state transition matrix $\Phi(t,t_0)=X(t)X^{-1}(t_0)$ but you can select any linearly independent initial conditions to build the fundamental matrix $X(t)$, so ...
0
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0answers
8 views

How to transform pixel from image 2 to image 1 if camera matrices are known?

two cameras C1 and C2 take two images I1 and I2 of the same static scene, focal length and focal point are known and identical for both cameras after Bundle Adjustment: the rotation of C1 (as a ...
0
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0answers
6 views

Error propagation of non-square Matrix

I measure values of the vector $\vec{P}$ where each value $P_i$ has its own error $E_i$. I'm interested in the vector $\vec{K}$. Both values are linked by $\vec{K} = M \vec{P}$ . Since $\vec{P}$ and ...
0
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2answers
29 views

Matrix Algebra Questions

I would like you to help me with two questions I am stuck in. You can call these homework questions. It would be helpful if you can give me non-trivial hints instead of complete solution. 1) Let $A$ ...
0
votes
0answers
17 views

determinant of the covariance matrix of a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
-1
votes
0answers
10 views

question in co-matrices [on hold]

I need help to understand the definition of consistently ordered matrices and some examples about that ?
0
votes
1answer
39 views

Suppose $KA = {\bf0}$ and $K$ is idempotent. Define $G = (A-K)^{-1}$. Prove that (i) $AG = I-K$; (ii) $AGA = A$; and (iii) $AGK = {\bf0}$.

I don't know how to start this one. Should I divide these into cases where $K$ is the identity matrix, the null matrix and an idempotent matrix w/c is not null and identity? Help please. Thank you.
2
votes
2answers
30 views

Is the set of matrices with rank at most $r$ closed? [duplicate]

The question is as follows: $\DeclareMathOperator{\rank}{rank}$ Is the set $S_r = \{A \in \Bbb R^{n \times n}: \rank(A) \leq r\}$ closed in $\Bbb R^{n \times n}$ in the Euclidean topology? I ...
0
votes
0answers
20 views

Matrix inside matrix

Im stuck in a paper and hope you guys can help me I have the following defined: D(x) = transpose([b;w]) * [1;x] $w$ is the inverse covariance $b$ is $w$ multiplied with a constant $x$ is a ...
1
vote
1answer
18 views

Continuity argument to show that the derivative exists everywhere.

I have shown that, for $f(X) = \det(X)$, $$\mathrm d f_A(H) = \mathrm{tr} (\mathrm{adj}(A) H)$$ But I have only show this for invertible $A$. I wish to use a continuity argument to show that this ...
3
votes
2answers
34 views

Determinants of 'block' matrices

I am trying to simplify the determinant of \begin{pmatrix}C&A\\B&0\end{pmatrix} where $A$ and $B$ are square $m\times m$ and $n\times n$ matrices, and $C$ is some $m\times n$ matrix, $0$ is ...
0
votes
0answers
10 views

Integrate determinant of product of two matrices

Let $V\left(i,j\right) = \alpha_j^{i-1}$ be the $\left(i,j\right)^{th}$ element of the matrix $V\in\mathbb{R}^{n\times n}$. Such matrices are called Vandermonde matrices. Let $X = \left|V\times ...
0
votes
2answers
12 views

Finding a single vector that is a combination of two $3\times3$ transformations

The transformations $T_1$ and $T_2$ are defined by the matrices $\begin{pmatrix}4&1&1 \\ 1&2&-1\\3&1&1\end{pmatrix}$ and $\begin{pmatrix}1&1&1 \\ ...
-1
votes
0answers
11 views

Give an example of a matrix reduce to the cononcial form. Also find the non singular matrix P and Q such that PAQ is in the cononical form. [on hold]

Give an example of a matrix reduce to the cononcial form(normal form). Also find the non singular matrix P and Q such that PAQ is in the cononical form(normal form).
0
votes
1answer
11 views

finding clusters in a network from eigengaps

I have a usual Laplacian matrix, which describes a network. From the matrix I get the eigenvalues and from these I can compute a metric of modularity in my network based on the largest eigengap. Let's ...
1
vote
1answer
12 views

estimation of condition number for column equilibration

I have trouble with the following problem: Let $A$ be an invertible square matrix. Let $D$ be the diagonal matrix with entries $d_j=\dfrac{||A||_1}{\sum_i |a_{i,j}|}$. Show that $||D||^{-1}_\infty ...
3
votes
0answers
27 views

Range of vectors that turn into eigenvectors after recursive multiplication by a matrix

Suppose $\mathbf{x}$ is a vector, and $\mathbf{A}$ is a square matrix. Which $\mathbf{x}$'s will satisfy the equation $\mathbf{A}^n\mathbf{x} = \lambda\mathbf{A}^{n-1}\mathbf{x}$, where $\lambda$ is ...
1
vote
0answers
15 views

Trace norm identity (in bra-ket notation)

I came across the following identity in a paper: $$ \|\hspace{0.3em}|v\rangle\langle v| - |w\rangle\langle w|\hspace{0.3em}\|_{tr}=2\sqrt{1-|\langle v|w\rangle |^2}$$ where the norm on the left is ...
1
vote
0answers
14 views

LU Factorization algorithm always fails.

I'm trying to implement LU factorization in openCL but I'm struggling to get my sequential algorithm working properly. I implemented a sequential algorithm that works perfectly. Next I wanted to ...
0
votes
0answers
15 views

Is it possible to have all the rows distinct(unique) in a matrix having only 0 or 1 , after removing exactly one column?

I have to solve the following programming problem Given a M*N binary matrix. Detect if it is possible to delete a column in a manner that after deleting that column, the rows of the matrix will be ...
1
vote
0answers
31 views

One question on Matrix Equation

Assume $\hat{M}_1, \hat{M}_2, \hat{T}_{11}, \hat{T}_{12}, \hat{T}_{21}, \hat{T}_{22}$ are $2\times 2$ matrix. And $a, b, A, B, C, D$ are all numbers, satisfying the following relation: \begin{align} ...
1
vote
1answer
32 views

If A is a Hermitian matrix then SAS* is Hermitian

If $A$ is an $n\times n$ Hermitian matrix, and $S$ is an nxn matrix, then $SAS^*$ is also Hermitian. Why is this true? I have seen this claim made in several places but can't find a proof.
1
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0answers
7 views

Formula Needed by providing row column total rows and total column

Need a math formula to determine a result number based on provided input (Row number: a, Column number: b, Total Rows: p, Total column: q). let me explain the scenario : ...
1
vote
2answers
27 views

prove value of trace of a matrix

Suppose that $X$ is an $m \times n$ matrix and that the matrix $X^TX$ is invertible. $H = X(X^TX)^{-1}X^T$ where $X^T$ is transpose of $X$; $(X^TX)^{-1}$ is inverse of $X^TX$ we are asked to show ...
0
votes
0answers
36 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
1
vote
3answers
65 views

$I+A^*A$ is non-singular whenever $A$ is a square matrix with complex entries? [on hold]

Let $A$ be a square matrix with complex entries , then is it true that $I+A^*A$ is non-singular ? where $A^*$ denotes the conjugate transpose of $A$ http://en.wikipedia.org/wiki/Conjugate_transpose ...
0
votes
0answers
33 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
-3
votes
0answers
27 views

Calculus and Economics [on hold]

I have literally no idea how to even start on a question like this. Would somebody please help me find the direction I need to be headed. The goods market and money market of an economy are described ...
2
votes
2answers
59 views

Proving ($\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$, for all $x\in\mathbb{C}^n$) $\implies A$ is unitary

As the title states, I'm trying to prove that $\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$ for all $x\in\mathbb{C}^n\implies$ $A$ is unitary, where $A$ is a square matrix. This is ...
0
votes
2answers
20 views

Sum over all possible combinations of a Cholesky decomposition

Suppose to have a $n \times n$ positive definite matrix $\boldsymbol{\Sigma}$ and let $ \boldsymbol{\Sigma}= \mathbf{B}\mathbf{B}^T$ where $\mathbf{B}$ is obtained with the Cholesky decomposition. ...
0
votes
3answers
48 views

Can $A$ be singular

$A^2 + A + I= 0$ Can $A$ be singular? Justify your answer. I do not know where to start.
-1
votes
0answers
22 views

Where can we Matrix in computer [on hold]

I am actually studying the matrices and I want to find some motivation so i can have fun while studying . my question is what are some uses of the matrices in computer or others , i.e. other form , ...
1
vote
1answer
24 views

Relationship among $b_1$, $b_2$ and $b_3$ to have a solution

$$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ If $b= \begin{bmatrix} b_1 \\ b_2\\ ...
1
vote
1answer
13 views

Why left eigenvector complex conjugate transpose of right eigenvector?

My teacher today stated the following: For a matrix $A\in \Bbb R^{n \times n}$, any left eigenvalue $e^*$ is simply the transpose of the conjugate of a right eigenvector $e$ of $A$, so $e^* = ...
2
votes
2answers
27 views

Linear Algebra: Symmetric matrices, diagonalization (help with proof)

I need a bit of help with an IFF proof, here it is: {Let X be a symmetric n × n-matrix. Show: $$X=Y^2$$ for some symmetric matrix Y iff X has only non-negative eigenvalues. } My thinking: This ...
-3
votes
1answer
17 views

Compute all powers $A^{n}, n\in \mathbb{Z}$ and find their matrices. Is there a basis of $V$ so that the matrix of $A$ is diagonal? [on hold]

Let $V$ be a 2-dimensional real vector space with basis $\left\{e_{1},e_{2}\right\}$. Consider the linear operator on $V$ defined by $A\left( e_{1}\right)=e_{1},A\left( e_{2}\right)=ae_{1}+e_{2}, ...
2
votes
1answer
40 views

Is $g(A)$ diagonalizable?

Let $A$ be an $n \times n$ diagonalizable matrix; let $g(x)$ be a polynomial. Is $g(A)$ diagonalizable? If not, what are the minimum hypothesis one needs to make so that it works (if any?) (As ...
1
vote
3answers
30 views

Elementary matrix proof

I am supposing that $E$ is the elementary matrix obtained from $I$ (the identity matrix), by adding $\mu$ times the $m$-th row to the $l$-th row for some $\mu \in \mathbb{R}$ and $1\leq l,m\leq n$ and ...
3
votes
1answer
32 views

Is it true that $\|\text{diag($\pi$)} P\|_2 \leq 1$ for $P$ stochastic and $\pi P = \pi$.

$\| \cdot \|_2 $ is the matrix norm induced by $L_2$. $P$ is any given real square $n \times n$ non-negative matrix with rows summing to one, i.e. $P1 = 1$, where $1$ is the vector of ones. There is ...
0
votes
1answer
12 views

Incorrect elementary row operation in an augmented coefficient matrix

When solving the matrix $$\left(\begin{array}{ccc|c} 1 & 1 & 1 & 4\\ 1 & 3 & 1 & 4\\ -1 & 2 & 3 & -2\end{array}\right)$$ I somehow made an error with the ...
1
vote
1answer
12 views

When is $\| X \| _\star = \| F X \| _2$ submultiplicative?

All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix. Consider the norm $\| X \| _\star = \| F X \| _2$. What is the condition on ...
0
votes
1answer
26 views

Need help to understand a line of a proof of diagonalizability of real symmetric matrices

I was reading a proof of diagonalizability of real symmetric matrices using the concept of generalized eigenvalues and understood all except the very starting (and fundamental) line of the proof " if ...
0
votes
1answer
25 views

linear equations in a matrix form

Considering $$x_1 − x_2 + x_3 − x_4 = 2$$ $$x_1 − x_2 + x_3 + x_4 = 0$$ $$4x_1 − 4x_2 + 4x_3 = 4$$ $$−2x_1 + 2x_2 − 2x_3 + x_4 = −3$$ We have the following matrix $$ \begin{pmatrix} ...
0
votes
2answers
39 views

If A = BCD show that $C^{-1}$ = $DA^{-1}B$

I came across this question in a past paper, If A = BCD show that $C^{-1}$ = $DA^{-1}B$. All these matrices are sqaure and have inverses. I attempted a solution but I am not sure if- 1. The solution ...