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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0
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0answers
14 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
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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
2answers
33 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
27 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
19 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
17 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
31 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
10 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
4 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
24 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
13 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
13 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
13 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
27 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
25 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
30 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
60 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
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0answers
27 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
24 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
54 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
47 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
23 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
11 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^* = ...
0
votes
0answers
20 views

What is the constraint matrix in the assignment problem formulation?

The linear assignment problem may be posed as $$ \min_{X\in \mathbb{R}^{n \times n}} <X, C>_F, \;\; \text{subject to}\;\; \sum_jx_{ij}=1, \forall i \; \sum_i x_{ij}=1, \forall j \;\; x_{ij} ...
2
votes
2answers
25 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
16 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
37 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
26 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
11 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
25 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 ...
1
vote
2answers
31 views

Characterize stochastic matrices such that max singular value is less or equal one.

By a stochastic matrix, I mean any non-negative square real matrix with rows summing to one. It is well-known that singular values of stochastic matrices can be more than one. Is there a ...
1
vote
0answers
15 views

prove the identity of the kronecker product of four matrix

Suppose that $H_i$($i=1,2,3,4$) are four arbitrary matrices, the Kronecker product relation below $$ (H_1\otimes H_2)\otimes(H_3\otimes H_4)=H_1\otimes (H_2\otimes H_3)\otimes H_4 $$ holds due to the ...
3
votes
1answer
22 views

When is Block-Partitioned Matrix Invertible?

Suppose I have a block partitioned matrix \begin{equation} \begin{bmatrix} \mathbf{X}_1^{\top}\mathbf{X}_1 & \mathbf{X}_1^{\top}\mathbf{X}_2 \\ \mathbf{X}_2^{\top}\mathbf{X}_1 & ...
0
votes
1answer
22 views

Inverse Matrix Methods to find Nash Equilibriums

The board of directors of two companies determines the salary of its CEO according to the following reaction functions: S1 = 100,000 + (1/2) S2 S2 = 70,000 + (2/3) S1 Where Si is salary of company ...
1
vote
0answers
21 views

Determine whether the function is a linear transformation between vector spaces

Problem: $T:\Bbb R\rightarrow \Bbb R$, $T(x) = -2x$ I approached this problem trying to find counter examples to show that one of the properties of linear transformations fails to hold. I was unable ...
0
votes
1answer
57 views

How can I use cramers rule to solve this problem?

I have deduced and found three equations in which I need to solve this problem (Stated Below) How can I format this to fit into matrices to be solved with Cramers rule. The formulas I have are: ...
-1
votes
0answers
31 views

Need help to find diagonal dominant matrix [on hold]

I need to solve system of linear equations using iterative method. So one of the steps is to make the matrix be diagonally dominant Please help me to find diagonal dominant matrix from: $$ ...
3
votes
2answers
44 views

All the $k\times k$ minors determines the matrix?

Suppose two $n\times n$ matrices $A$ and $B$ with the same $k\times k$ minors, that is, for each $1\leq i_1<\cdots<i_k\leq n$, $$\det\left(\begin{matrix} a_{i_1i_1}&\cdots&a_{i_1i_k}\\ ...
1
vote
1answer
20 views

Proving the Determinant of a Tridiagonal Matrix

Let $A_n$ denonte an $n \times n$ tridiagonal matrix. $$A_n=\begin{pmatrix}2 & -1 & & & 0 \\ -1 & 2 & -1 & & \\ & \ddots & \ddots & \ddots & \\ & ...