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)

4
votes
2answers
41 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
29 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
12 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 ...
1
vote
1answer
33 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
20 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
51 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
136 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
31 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
22 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
25 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
26 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
17 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
13 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
13 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
80 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
15 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
20 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
243 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
0answers
36 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
38 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
42 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
68 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
99 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 ...
-1
votes
0answers
28 views

What is ${\sigma _{\varepsilon ,W}}(P)$?

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 ...
1
vote
2answers
36 views

Matrix Equation

Imagine the question: If $K$ and $L$ are $2\times 2$ matrices (knowing all of their components) and $KM=L$, solve for the matrix $M$. One simple solution is to set the components of $M$ as $x,y,z,w$ ...
0
votes
2answers
28 views

Equivalence of two different versions of “change of basis matrix”?

I have a question regarding basis change and the matrix that represents it. I understand the concept, though I've noticed a different formula/proof in different math books and I don't understand how ...
1
vote
1answer
29 views

Different formulas for matrix transformations

I am a bit confused about how to get a matrix in a new basis. On the one hand, we always use the multiplication by transformation matrix when we want to receive a matrix in a new basis: $A' = CA$, ...
1
vote
0answers
37 views

How to solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ in Matlab when nullitity$(A)\neq 0$

Say, $A= \begin{pmatrix} 1 & 0 &1 \\ 0 & 1 &1 \\ 0 &0 &0 \end{pmatrix}$ and $\overrightarrow{b}= \begin{pmatrix} 8 \\ -5 \\ 0 \end{pmatrix}$ and I want to solve ...
5
votes
5answers
551 views

Why the determinant of a matrix with the sum of each row's elements equal 0 is 0?

I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and ...
0
votes
0answers
47 views

Finding permutation matrix

Let $P_{\pi}$ denote a permutation matrix associated to the permutation $\pi:\{1,...,n\}\rightarrow \{1,...,n\}$ and $\sigma$ denote the cyclic permutation $(1 2 ...n)$. If T is the $n\times n$ lower ...
2
votes
2answers
62 views

Conditions for an orthogonal matrix equation

Let $B_1$ and $B_2$ be given $n \times n$ real non-singular matrices and consider the system of equations $$\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\begin{bmatrix}B_1 ...
1
vote
1answer
33 views

How to solve this vector equation for optical flow

I am unable to solve for $\textbf{h}$ in the following equation $\sum\limits_{\textbf{x}=1}^n2\partial F(\textbf{x})/\partial\textbf{x}(F(\textbf{x}) + \textbf{h}^{T}\partial F(\textbf{x})/\partial ...
1
vote
1answer
24 views

Replacing pinv with inv in MATLAB

Let $\mathbf{y} = \mathbf{Ax}$ represent a system of equations where $A\in\mathbb{R}^{m\times m}, x\in\mathbb{R}^{m\times 1}$. However rank of $\mathbf{A}$ is $m-1$. I add another equation ...
-1
votes
0answers
23 views

What is subordinate matrix norm?

What is 'subordinate matrix norm' in this question? .
0
votes
0answers
40 views

Wondering how to rotate a normal vector in 4 dimensions?

Saw another post that suggested a answer but need help with the answer and the other post is inactive. I know how to rotate in 3-space using matrix transforms for each axis no problem. Have a very ...
-1
votes
0answers
16 views

Affine Transform Matrix (Rotation) [on hold]

Can someone help me with this question? I know how rotation matrix looks like after rotating by y axis but don't know how. Show how to derive the Rotation Matrix about the Y-Axis.
0
votes
0answers
13 views

SVD of partitioned matrix where all cells except one are zero

Let $A$ be real valued matrix of size $n \times n$. Let the SVD of $A$ be $$A= UDV^T.$$ I am interested in $$Q=VU^T.$$ Now assume we expand $A$ with zero rows and columns to get the block matrix ...
1
vote
1answer
31 views

Non square Matrix multiplication

Assuming we have the following matrix multiplication problem $$ {\bf A x} = {\bf b}$$ and that the dimensions of ${\bf A,x,b}$ are the following $3\times2$, $2\times 1$ and $3\times 1$ How can one ...
1
vote
0answers
26 views

Cayley Hamilton Theorem using LU decompostion

I am trying to find the characterisitic equation of n*n matrix by Cayley Hamilton Theorem using LU Decompostion. Below is my algorithm to find U matrix. ...
1
vote
3answers
70 views

Matrix Exponential and Logarithm

Consider the following matrix $A$: $A = \begin{bmatrix} \cos^2(1) & -\sin(2) & \sin^2(1) \\ \cos(1)\sin(1) & \cos(2) & -\cos(1)\sin(1) \\ \sin^2(1) & \sin(2) & ...
0
votes
1answer
32 views

Permutation and signature matrices “almost commute”

Let $\mathcal{P}$ be the set of all permutation matrices of order $n$ and $\mathcal{S}$ the set of all signature matrices of order $n$. Furthermore, let $$\mathcal{P}\mathcal{S} = \{PS \mid ...