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), ...

learn more… | top users | synonyms (2)

0
votes
0answers
5 views

Basis of linear map

I've got a problem with this task. Can anyone help me? We know that $B,C,D$ are basis of $R^3$ and $F$ is linear map from $R^3$ to $R^3$. Also we know that $B = \left\{(1,1,1)^T, (0,1,1)^T, ...
1
vote
0answers
8 views

Matrix space, with <A,B>=tr(AB*) isn't Hilbert space, how can i find a counter example?

Generally, I'm having quite troubles thinking about counter examples. So I would love if someone could guide me into finding the example for the following question by myself (and not just giving it ...
1
vote
1answer
11 views

What do symbols “\” and “/” mean in the context of matrix computation?

Does it indicate this? $L=U$ \ $C=U^TC$ $Q=U$ / $C=UC^{-1}$ Can you please provide any references for these two symbols for matrix computation? [Update] according to ...
0
votes
0answers
10 views

Solution of Matrix ODE

Specifications It is given that $ \psi'(s)=(A+Bs)\psi(s)\tag 1$ where A,B are constant $3 \times 3$ skew symmetric matrices with determinant $0$ $\psi(s)$ has determinant $1$ , orthogonal and has ...
0
votes
0answers
25 views

Prove that this expression about determinants is true

Let D be a square matrix where every entry is an integer, and the determinant of D is 1 or -1. Prove that every entry in D^-1 is also an integer.
0
votes
2answers
40 views

Prove or disprove: $\det(A+B) = \det A + \det B$, $\det(A^{T}A)\geq0$ [on hold]

I have two statements I need to prove or disprove with a counterexample: $\det(A+B)=\det(A)+\det(B)$ $\det(A^TA)≥0$ Please explain how I could go about doing this.
0
votes
2answers
32 views

What is the special name of vectors like <0,1,0,0,0> or <1,0,0>?

Is there a special name for vectors whose elements are all 0's except one 1? Thanks.
2
votes
2answers
34 views

Find the values of $k$ so that the matrix is not invertible

So, the question is indeed asking for what you just read. I have the following matrix for which I have to find the values of $k$ in order to make it not invertible. I have understood that the inverse ...
1
vote
1answer
14 views

Trouble understanding the proof for linear independency of a basis for a linear transformation

I am reading Matrix Analysis and Applied Linear Algebra by Meyer and the following statement and proof are given What I am having trouble understanding is how the author is showing that ...
2
votes
1answer
21 views

Given a matrix, how can you tell if it is an exponential matrix?

Of course if you can find a matrix $B$ such that $A=\exp(B)$ then you know that $A$ is an exponential matrix. But finding such $B$ is not always trivial. Are there criteria to know if a matrix is an ...
1
vote
1answer
23 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
13 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
11 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
votes
0answers
11 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 ...
1
vote
0answers
13 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
1answer
29 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
votes
0answers
18 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
votes
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
votes
2answers
33 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
20 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
11 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
44 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
23 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
36 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
11 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
16 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
17 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
32 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
35 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
vote
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
29 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 ...
1
vote
3answers
72 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
49 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
28 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
62 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
1answer
32 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}, ...