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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2
votes
1answer
36 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 A\...
2
votes
1answer
20 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 (...
3
votes
2answers
1k views

rotating a rectangle via a rotation matrix

I want to rotate a 2D rectangle using a rotation matrix. After the rotation, I want the points (x, y) of the rectangle to be: ...
0
votes
3answers
73 views

Given $A^2-4A+I=0$, show that $ A^3=15A-4I$

If have a question like this , can we using equation method or deduction method to answer the question?? Or we need to answer the question by substituting the matrix??
2
votes
1answer
40 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 theorem?...
0
votes
1answer
29 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
103 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 $x$ ...
2
votes
1answer
155 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 ...
3
votes
2answers
38 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
19 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 x=...
1
vote
1answer
35 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 ...
3
votes
2answers
157 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 $\{...
9
votes
2answers
147 views

Can you determine from the minors if the presented module is free?

Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function $$\varphi(x)=\...
6
votes
2answers
263 views

Determinant of a Certain Block Structured Positive Definite Matrix

PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED VERSION....
3
votes
2answers
58 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 ...
3
votes
1answer
80 views

A matrix as a point in $\mathbb{R}^{nm}$

I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said "In a ...
1
vote
0answers
33 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
23 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
91 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 ...
0
votes
1answer
25 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 $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
1
vote
1answer
36 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: $A' = CA$, ...
0
votes
2answers
48 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 ...
3
votes
2answers
389 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 ...
2
votes
0answers
48 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 $\Sigma = LDL'$ ...
1
vote
1answer
38 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} F_{...
1
vote
1answer
110 views

How do the rows of a change of basis matrix form a basis for expressing columns?

I am reading this article on Principal Component Analysis (PCA) and in section III-B (page 3) it has strange definition I don't understand. In the toy example $\mathbf{X}$ is an $m \times n$ ...
1
vote
1answer
405 views

Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $v$

Let $V$ be the plane with equation $x_1 + 4x_2 + 2x_3 = 0$ in $\mathbb{R}^3$. Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $v = \begin{bmatrix} -12 \\ 4 \\ ...
3
votes
3answers
1k views

Are there non-affine matrices?

Matrices are useful for proving statements like The ratio between the areas of a parallelogram and the quadrilateral formed by joining their midpoints is $2$. The ratio between the volumes ...
0
votes
0answers
36 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 sense ...
1
vote
1answer
56 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: $\mathrm{Rank} (A_1)=n_1$, $\mathrm{Rank} (A_2)=n_2$, $\mathrm{Rank} (A_1 : A_2)=n_1+n_2$ ...
1
vote
1answer
41 views

Connection between $S$ and $A$ when $S^{-1} A S$ is a diagonal matrix

In diagonalizing a matrix $A$, we use a matrix $S$, which consists of eigenvectors of $A$. To compute $S$, we simply take each eigenvector of $A$ and write it as a linear combination of the standard ...
2
votes
2answers
63 views

Positive semi-definiteness of a matrix whose diagonal elements slightly differ from the sum of the absolute values of other elements in the row

I have a matrix which has the following form: $$ A= \begin{bmatrix} a+b-\varepsilon_1 & -a & -b \\ -a & a+c-\varepsilon_2 & -c \\ -b & -c & b+c-\varepsilon_3 \end{bmatrix} $$ ...
3
votes
1answer
47 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
2answers
105 views

Is “basis times square matrix” a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of $B$. Thus $BS$ should be a new ...
1
vote
3answers
200 views

The rows of an orthogonal matrix form an orthonormal basis

A matrix $A \in \operatorname{Mat}(n \times n, \Bbb R)$ is said to be orthogonal if its columns are orthonormal relative to the dot product on $\Bbb R^n$. By considering $A^TA$, show that $A$ ...
0
votes
0answers
27 views

Reading a basis for $\operatorname{Null}(A)$ and $\operatorname{Null}(A^T)$ from the reduced row-echelon form of $A$

I know that it is possible to read the basis for $\operatorname{Null}(A)$ and $\operatorname{Null}(A^T)$ by simply looking at the reduced row-echelon form (RREF) of the matrix $A$. I have only an ...
1
vote
1answer
55 views

Prove an $n\times n$ matrix is negative definite

I wonder is there any way to prove the $n\times n$ matrix with elements below is negative definite: $$ \sigma_{ij} = \frac{a_ia_j}{\sum_k s_ka_k} \space; i \neq j \text{ (off diagonal terms)}$$ $$\...
2
votes
6answers
187 views

$A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$

Let $A$ a $n \times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$. a) Give an example that satisfies this conditions. b) what are the eigenvalues ​​of $A$? Well for $a)$ i ...
0
votes
7answers
122 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
vote
1answer
59 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 \...
-4
votes
2answers
49 views

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

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!
6
votes
5answers
2k 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 it'...
1
vote
1answer
63 views

markov chain computation

I consider a 2 state Markov chain: $X = \{1,2\}$, transitions are $M(i,j)$ and the matrix has a unique stationary distribution $\pi$: $$ \pi(1) = \frac{M(2,1)}{2-M(1,1) - M(2,2)} \\ \pi(2) ...
3
votes
3answers
351 views

Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
1
vote
2answers
50 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 $x,y,z,w$ ...
17
votes
2answers
1k views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $ A_{ij}=\frac{1}{i + j - 1}$. I need to show that $A$ is invertible and the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
2
votes
2answers
77 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
0answers
47 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 ...
9
votes
0answers
123 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
0
votes
0answers
73 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 ...