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

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
0
votes
1answer
11 views

Derivative of vector and vector transpose product

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$. I am finding difficult to understand the part in red. What rule is that ? If I apply multiplication rule, ...
0
votes
0answers
10 views

Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix?

Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we ...
0
votes
1answer
12 views

Characterising Adjugate(adjoint) of a matrix

If $A$ is an $n\times n$ matrix over a field, then adj$(A)$ is an $n\times n$ matrix (obtained from $A$) such that $$\mathrm{adj}(A)\,A=A\,\mathrm{adj}(A)=\mathrm{det}(A)I_n.$$ Question: If $B$ is ...
0
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1answer
43 views

Linear Algebra: Question about determinants

The following matrices are $4 \times 4$ matrices. $$A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1& 1 &0\\ 1 &1 &0 &0 \end{bmatrix}\\ B= ...
0
votes
1answer
13 views

Sylvester's law of inertia for generic matrices.

By Sylvester's law of inertia, the positive and negative indices of a symmetric matrix $A$ are also the number of positive and negative eigenvalues of $A$. I was wondering if a similar result is known ...
0
votes
2answers
32 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$.

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
0
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3answers
30 views

Inverse of partitioned matrices

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
1
vote
1answer
27 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
1
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0answers
8 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
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2answers
40 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
2
votes
3answers
46 views

Producing lower bounds for $\text{trace}(A^2)$ for a positive semidefinite, symmetric matrix $A$

Are there any lower bounds on $\DeclareMathOperator{trace}{trace}$ \begin{align*} \trace(A^2), \end{align*} where $A$ is positive semi-definite and symmetric? I am aware of the inequality $$ ...
-1
votes
0answers
19 views

Find the solution to the following LPP by solving its dual. [on hold]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}
0
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0answers
15 views

How to multiply the elements within a vector using matrix operations (e.g., dot product)?

Suppose a vector $\vec{v}^T=(v_1, v_2, \ldots, v_n)^T$. To sum the elements within the vector, I can use the dot product with a column vector of ones, $\sum_i v_i = \vec{v}^T \cdot \vec{1}$. My ...
1
vote
0answers
20 views

how to understand a matrix with order $O(n^{-1})$

I am reading a paper in which an assumption is that a matrix (for example $A_{n\times n}$) is $O(n^{-1})$. I have difficulty to understand that assumption. Does that mean the norm of the matrix is ...
0
votes
1answer
40 views

Mean value for $\tiny\left( \begin{array}{cc} X & X \\ -X & 1-X \\ \end{array} \right)$

We have a matrix $$\left( \begin{array}{cc} X & X \\ -X & 1-X \\ \end{array} \right)$$ where $X$ is a random variable between $0$ and $1$. I heard about "random matrices". Is it an ...
2
votes
1answer
18 views

properties of the solution to a non-homogeneous matrix equation with a non-singular M-matrix

I have a matrix equation $Ax=b$, where $A$ is a $4\times4$ non-singular M-matrix ($A$ has negative off-diagonal and positive diagonal entries) and $b$ is a strictly positive vector. Let $x=(x_1, x_2, ...
1
vote
0answers
31 views

Finding the Jordan form and basis failing

Let $$A = \left(\begin{array}{cccc} 3&4&-1\\0&-2&0\\1&-4&1 \end{array}\right)$$ Find the Jordan form $J$ and $P$ such that $P^{-1}AP = J$. So here's what I did: $f_A(x) = ...
4
votes
3answers
53 views

$A$ is a symmetric postivie definite matrix. Prove that $A^k$ is also a positive deinite

Let $A\in M_n(\mathbb{R})$, a symmetric positive-definite matrix. Prove that for every $k\in\mathbb{N}$, $A^k$ is also positive definite. So since $A\in M_n(\mathbb{R})$ is symmetric and positive ...
2
votes
1answer
34 views

$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$

Suppose $A \in {C^{n \times n}}$ $\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm. Is this true that $\left\| {\mathop ...
2
votes
1answer
14 views

Commutative Monoid - matrix set

Let $M$={$\begin{bmatrix} a & b & c \\ c & a & b \\ b & c & a \end{bmatrix}|a,b,c\in \mathbb{R}, a+b+c=0$}. The matrices in $M$ are a special kind of Toeplitz matrices ...
2
votes
1answer
37 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
0
votes
1answer
23 views

Analytical result for element-wise vector division?

I have two vectors $$a=[a_1,a_2,...,a_n], b=[b_1,b_2,...,b_n]$$ Is it possible to express the result $$c=[a_1/b_1,a_2/b_2,...,a_n/b_n]$$ by some standard matrix operations such as matrix ...
0
votes
1answer
53 views

Which matrices diagonalizes a diagonal matrix? [on hold]

I think the answer is the set of all diagonal matrices but I am not sure. Can anyone give the answer with a proof?
2
votes
4answers
144 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
1
vote
2answers
36 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
0
votes
1answer
12 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
3
votes
2answers
34 views

For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
0
votes
1answer
25 views

how to calculate the projection of a vector onto a closed convex set? [duplicate]

suppose we have a vector $x \in \mathbb{R}^{n}$, and a closed convex set $C \in \mathbb{R}^{n}$. $C =\{x|Ax=b\}$ how to calculate the vector $y \in \mathbb{R}^{n}$, which is the projection of $x$ ...
1
vote
1answer
27 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
1
vote
1answer
28 views

Matrix Properties Problem

If $A\in M(n\times n;R)$ and $K= \dfrac {A+A^T}{2} $ and $L= \dfrac{A-A^T}{2}$. Prove: i) that $K$ and $-L$ are symmetric ii) that $K+L=A$ iii) that $K$ and $L$ are unique matrices with the properties ...
0
votes
0answers
18 views

Notation sumation confusion

I am reading paper about additive schwarz preconditioner, where following notation is used in order to obtain matrix C $$C_i = \sum_k (I^k B^k (P^k u_i)R^k)$$ . My question is, what's dimension of ...
3
votes
2answers
35 views

Eigenvalues of Matrix Product.

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their product? What about the special case when one of these matrices is a diagonal (positive) matrix? I ...
1
vote
0answers
16 views

reoder basis vectors to get 'more diagonal' representation of NxM matrix

I am trying to reorder the basis vectors of an NxM matrix in a way that leads to a representation with 'as much weight close to the diagonal as possible'. I would be happy, if instead of a matrix ...
1
vote
0answers
14 views

Inner product inequalities with a diagonal matrix defining the inner product

This question came about from analyzing symmetric positive definite bilinear form decompositions and trying to understand what conditions would ensure certain inequalities hold. Suppose we have 3 ...
0
votes
2answers
82 views

Possible to express the diagonal matrix $D=\tiny\left(\begin{matrix}1&0&0\\0&2&0\\0&0&3\end{matrix}\right)$ as function of $3\times3$ identity matrix?

Is possible to express the diagonal matrix $$D=\left(\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}\right)$$ as function of $3\times3$ identity matrix? For ...
2
votes
0answers
16 views

derivative of quadratic form with regard to inverse of lower-triangular matrix

I have a quadratic form of the form $Q(\Sigma; x, \mu) = (x-\mu)'\Sigma^{-1}(x-\mu)$ where $\Sigma$ is a positive-definite non-singular matrix with (modified) Cholesky decomposition $\Sigma = LDL'$ ...
-2
votes
0answers
22 views

A qustion in matrix polynomial [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
3
votes
2answers
247 views

If two matrices have the same trace and determinant, do their powers have the same trace?

Let $A,B$ be two $2 \times 2$ matrices over some finite field $\mathbb{F}_q$, such that they have the same trace and determinant. Does this imply that tr $A^k$ = tr $B^k$ for any integer $k$? I've ...
0
votes
1answer
59 views

Calculation of $trace(L^THL)$, L is lower triangular, H is symmetric.

I am working on a problem where I had to find the following expression: $$ l = Tr({P'HP})$$ I already modified my model formulation using cholesky decomposition for PSD matrices and came up with ...
0
votes
0answers
21 views

SVD proof to If $A$ is of full rank, then $A^{*}A$ is of full rank

Provided $A$ is a full rank matrix $\in\mathbb{C^{m\times n}}$, then $A^{*}A$ is of full rank. Suppose $m\gt n$. There is a solution to this problem: solution link, and the top solution makes ...
1
vote
1answer
23 views

How to denote a tensor in terms of matrices product?

How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where $T_{i,j,k} = \sum_{m=1}^{b} ...
1
vote
1answer
20 views

Ranks of matrices after multiplication by a nonsingular matrix

Consider an $n \times n_1$ matrix $A_1$ and an $n \times n_2$ matrix $A_2$ with the following properties: $\mathrm{Rank} (A_1)=n_1$, $\mathrm{Rank} (A_2)=n_2$, $\mathrm{Rank} (A_1 : A_2)=n_1+n_2$ ...
2
votes
1answer
41 views

Comparing $\text{tr}(A^{-1})$ and $\text{tr}(A(B+A)^{-2})$ for pd $A$ and psd $B$

Suppose that $A$ is positive definite and $B$ positive semidefinite, both with dimension $n\times n$. Is there some inequality between $$ \text{tr}(A^{-1})\quad\text{and}\quad\text{tr}(A(B+A)^{-2})? ...
0
votes
1answer
24 views

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of v

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of $v$, when $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
-5
votes
0answers
45 views

Would you please give me your opinion about solving this equation? [on hold]

Would you please give me your opinion about solving this equation? [![enter image description here][1]][1] $\sum_{i=0}^{1}\left (-X \right )^{i}*\sum_{J=0}^{7}\sum_{i=0}^{j}\, \gamma _{j}*X^{i}=-T$ ...
3
votes
2answers
71 views

Eigenvalues of the sum of two matrices: one diagonal and the other not.

I'm starting by a simple remark: if $A$ is a $n\times n$ matrix and $\{\lambda_1,\ldots,\lambda_k\}$ are its eigenvalues, then the eigenvalues of matrix $I+A$ (where $I$ is the identity matrix) are ...
-3
votes
2answers
39 views

If $\det(A)=0$, must the null space of $A$ be zero? [on hold]

Came along this question: If $\det(A)=0$ for an $N\times N$-dimensional matrix $A$, the null space of $A$ is equal to zero. True or false? Why? Thank you already!
0
votes
7answers
100 views

Is there a matrix with real entries such that $A \ne I_2$ but $A^3 = I_2$.

Is there a matrix with real entries such that $A \ne I_2$ but $A^3 = I_2$. I've actually encountered with this post: $A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$ ...
-1
votes
1answer
13 views

Augmented matrix [on hold]

Animals in an experiment are to be kept under a strict diet. Each animal should receive $20$ grams of protein and $6$ grams of fat. The laboratory technician is able to purchase two food mixes: Mix ...