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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5
votes
1answer
69 views

Mutually commuting matrices

Let $A_{1},..., A_{m}$ be $n \times n$ matrices with entries in a field $K$ such that $A_{i}A_{j} = A_{j}A_{i}$ for all $ 1 \leq i, j \leq n$ and the product $A_{1}A_{2} ... A_{m} = 0$ is the zero ...
0
votes
1answer
39 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
1
vote
1answer
16 views

Finding variables, basis and dimension given linear transformation with representing matrice

Let $T:R^4 \to R^4$ be linear transformation such that $T(x)=Ax$, when $A = ...
0
votes
1answer
44 views

Is there a solution to this system for the diagonal matrix?

I'm trying to find a solution to a system of equations, but its quite different from anything I've come across before. I believe there is a solution, but I could be wrong. $\mathbf{A} = ...
6
votes
4answers
377 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
0
votes
1answer
24 views

Find a basis $B$ such that the matrix has the desired shape.

Let $p=(x-(a+bi))(x-(a-bi))$ be the characteristic polynomial of linear operator $T$ ($T$ in $\Bbb C$) and its basis of eigenvector is $A=\{u+iv,u-iv\}$. Find a basis $B$ in $\Bbb R^2$ such that ...
1
vote
1answer
30 views

Lower bounds on eigenvalues of a symmetric matrix based on the diagonals

A symmetric matrix $A$ always has real eigenvalues. If I know the elements on the diagonals, is it possible to have a lower bound on the smallest eigenvalue? How sharp would this bound be? For now I ...
0
votes
1answer
28 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
0
votes
0answers
37 views

Is a set of some $m \times n$ matrices a relation?

A relation between sets $A_i, i = 1, \dots, n$ is defined as a subset of $\prod_i A_i$. Given $m, n \in \mathbb N$, is a set of (some or all) $m \times n$ matrices over $\mathbb R$ considered a ...
0
votes
0answers
31 views

traingular matrix

I am not much in to the matrices. so I am sorry if I can not put forward the question properly. Assume $G_{n\times n}$ is a rank $n$ matrix (Indeed $G$ is the generator matrix of a lattice). I need ...
3
votes
1answer
63 views

how to find $(I + uv^T)^{-1}$

Let $u, v \in \mathbb{R}^N, v^Tu \neq -1$. Then I know that $I +uv^T \in \mathbb{R}^{N \times N}$ is invertible and I can verify that $$(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}.$$ But I am not able ...
0
votes
1answer
26 views

How do get eigenvalues of a matrix B if add a row/column pair of a matrix A?

I have a matrix of size N×N of the form: where and A is N-1 x N-1 matrix, a=0. I known the eigenvalues of A. Any possible for getting eigenvalues of B from eigenvalues of A?
0
votes
0answers
17 views

$LU$ decomposition of a matrix

My question is: Is it okay to switch rows to find the $LU$ decomposition (Lower, Upper Triangle) of let's say matrix $A$. Let's say we found matrix $U$ easily without having to switch rows. Now, can ...
1
vote
1answer
44 views

Jordan-Chevalley decomposition of $T$ acting on $k[T]/(\pi(T)^e)$

Given an algebraically closed field $K$, a f.d. vector space $V$ over $K$ and $A\in{\rm GL}(V)$, we can view the space $V$ as a $K[T]$-module, where $T$ acts by $A$. Using the fundamental theorem of ...
0
votes
1answer
23 views

Non-Unitarily Diagonalizable Matrices

When searching for matrices that are similar to a diagonal matrix but not in a unitary way then a first hint would be to exclude the normal ones. But apart from that is there a general form for such ...
0
votes
1answer
49 views

Null space of a matrix mcq

Let $M$ be the set of all $m\times n$ matrices with real entries. Which of the following statement is correct? There exists $A$ of order $2\times 5$ belonging to $M$ such that the dimension of the ...
0
votes
0answers
21 views

How to get the transformation matrix for Linear Discriminant Analysis?

I am trying to implement Linear Discriminant Analysis. Is the eigen vectors of the product of within scatter matrix and between scatter matrix inverse (Sw*Sbinverse), the transformation matrix? Could ...
0
votes
1answer
39 views

Understanding a matrix notation

I am trying to understand A Fast Random Sampling Algorithm for Sparsifying Matrices (Arora, Hazan, Kale). I don’t understand the meaning of the notation: $$\Vert A \Vert_2 = \max_{\Vert x \Vert_2 = ...
1
vote
3answers
33 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
0
votes
1answer
30 views

Power of matrix expansion

We know about the expansion $(a+b)^n\tag 1$,for scalar variables. What will be the equivalent when we want to find $(A+B)^n \tag 2 $, when A and B are square matrices? Can we treat it as same as ...
-1
votes
2answers
37 views

How to get A,B and C given XYZ?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{a+b+c}{3}$$ in other words How do i get $a$, $b$, and $c$ on the left ...
3
votes
0answers
72 views

Matrix partwise multiplication

I am working on an artificial intelligence application that (among other things) combines "opinions" of several "experts" who each have access to different aspects of a "situation". I can build this ...
0
votes
2answers
37 views

How does a matrix change the magnitude of a vector?

I have the following problem: $z=Ax$, in which $z$ and $x$ are $N\times 1$ vectors and $A$ is a $N\times N$ matrix. I am interested in how the magnitude of $x$ changes after applies $A$ on it. Is ...
2
votes
0answers
38 views

Show that $(Au,Bv)=(u,A^tBv)$

Let $ A, B $ be matrices of order $ n $, and $ \vec{u}, \vec{v} $ vectors from euclidean space $ \mathbb{R}^n $, then $ (Au,Bv) = (u,A^tBv) $ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
0
votes
2answers
42 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
1
vote
0answers
25 views

Is finding a matrix out of some set with a given determinant a hard problem?

Given $n\ge 2\ \ ,\ u,v,k\ $ integers. Decision problem : Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist? In which complexity class is this problem ? Is it ...
1
vote
1answer
22 views

Existence of 3 Matrices with given restrictions

Would it be possible to have 3 square matrices (preferably 2x2 or 3x3) $A$, $B$ and $C$ such that: $A\neq B \neq C$; The product $A\cdot B\cdot C$ equals the Identity Matrix; All 3 matrices are ...
1
vote
1answer
19 views

Making it positive semdefinite

What conditions( on $a$ and $b$) I need to impose on the following matrix to make it positive semidefinite? $$A=\begin{pmatrix}a&b\\b&0\end{pmatrix}.$$ Thanks in advance.
3
votes
1answer
68 views

Is det(A) maximal, if det(A+E) is maximal?

Let A be a binary matrix of size n x n and E be the matrix of the same size with all entries $1$. Proof or disproof : If det(A+E) has the maximal possible value, then det(A) also has the maximal ...
1
vote
1answer
51 views

On Stochastic Matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
1
vote
2answers
33 views

Is factoring a semiprime easier than matrix multiplication?

I'm currently dealing with complexity estimates of various algorithms and the connected mathematical problems. Up until now, I had in mind that problems such as integer factorization and the discrete ...
0
votes
1answer
59 views

Matrix-valued differential equation $A'(t)=A(t)B(t)$

How to solve matrix-valued differential equations of type $$A'(t)=A(t)B(t) \tag 1$$ All the given functions are square matrices of dimension $3$ and only $A(t)$ is invertible (not $B(t)$ or ...
0
votes
0answers
28 views

The derivative of matrix vector product with respect to matrix

Given function $$ f(M) = Mv$$ where $M$ has dimension $n \times n$, and $v$ is a vector with dimension $n \times 1$. What's the derivative of $f(M)$ with respect to $M$?
1
vote
0answers
62 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
0
votes
3answers
33 views

The i's and j's in a Matrix

I know that i means row and j means column, what i don't understand is what are they meaning when they say that the row is greater than or = to 1? And the column is less than or equal to 3? I don't ...
1
vote
1answer
79 views

Finding eigenvalues of a block matrix

I have a block matrix of size $2N \times 2N$ of the form $$B = \begin{bmatrix} A_N & C_N \\ C_N & A_N \end{bmatrix}$$ where $A_N$ and $C_N$ are both $N \times N$ matrices. Specifically, $$A_N ...
0
votes
1answer
69 views

Can an Elementary Matrix's Inverse's Determinant = 0?

Can someone explain to me why an elementary matrix's inverse determinant cannot equal 0? Or can it? Is there some theorem to elementary matrix inverses? THANKS FOR YOUR INSIGHT! :)
1
vote
1answer
52 views

Inverse of LU decomposition

The LU decomposition is $A=LU$, where $L$ is lower and $U$ is upper triangular. For the example of a 3*3 matrix $$A= \begin{pmatrix} 1 & 0 & 0 \\ l21 & 1 & 0 \\ ...
0
votes
0answers
13 views

Given the position and number of columns find the coordinates of the array

Given the following matrix. If the position in the array can be found using the formula pos = y * number_of_columns + x; given x, y and number_of_columns. ...
1
vote
1answer
16 views

Transpose : Matrices and Orders of

Here is my Answers and Reasoning, all I ask is that you check it and direct me if I have gone wrong, Thank you! Number 2 - because (BC)^T equals C^T x B^T which is a 5x2 matrix. This multiplied by ...
0
votes
0answers
21 views

Positive definiteness and eigenvalue inequalities of two symmetric matrices

Let $A = A^T \in \mathbb{R}^{n \times n}$ and $B = B^T \in \mathbb{R}^{n \times n}$ be any symmetric matrices. Then, I know that if $A$ and $B$ are positive definite, and $A-B$ is positive ...
7
votes
1answer
73 views

Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

How many diagonalizable matrices are there in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$ ? Where $p$ is a prime number. Attempt : By definition a matrix is called diagonalizable if there exists an ...
1
vote
0answers
21 views

Similarity and Jordan forms

Let $A \in \mathrm{Mat_{3}}(\mathbb{C})$ with A invertible such that $A$ is conjugate to $A^{2}$. Find the Jordan form of $A$. Suppose $B \in \mathrm{Mat_{3}}(\mathbb{C})$ such that $B$ is conjugate ...
1
vote
1answer
38 views

Need help understanding the proof: if v is a left singular vector of A then v is a unit eigenvector of $AA^{T}$

This is the proof in my textbook: What I don't understand it why " $AA^{T}u = 0u $ means that u is an eigenvector. Is this a theorem that I don't know? That if you multiply a matrix by a vector ...
1
vote
2answers
32 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
0
votes
0answers
11 views

Extract translation vector from two homogenous transformation matrices

Given two homogenous transformation matrices $$ A = \begin{pmatrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ ...
1
vote
1answer
22 views

Verification Matrices & Linear Equations Part 2

...Continued Question 3 A - True because if it equals 4 then there will be infinite solutions B - True because any gradient except for one that is equal (4) will intersect giving a unique ...
0
votes
2answers
60 views

Multiplication and derivation of 3D matrix

I have $A(q)=\begin{bmatrix}q_1 &q_2 & q_3\\ 2q_1 &3q_2 & 4q_3\\ 2q_1 &3q_1 & 10\\ \end{bmatrix}\tag 1$ $ q= {\left(\begin{array}{c}q_1\\q_2\\q_3\\q_4\\q_5\\q_6 ...
1
vote
0answers
19 views

Verification - Matrices and Linear Equations Part 1

Would just like to verify my Answers and bounce off my ideas and thinking with someone as I feel quite alone in this course. I am usually great at maths and enjoy it but these matrices and linear ...
2
votes
0answers
35 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...