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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20 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
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1answer
38 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
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3answers
30 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
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1answer
29 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
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2answers
36 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
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0answers
67 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
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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
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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, ...
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2answers
39 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 ...
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0answers
24 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 ...
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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 ...
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1answer
18 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
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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 ...
0
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1answer
35 views

On the decomposition of 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
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2answers
30 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
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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
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0answers
27 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 $M$?
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0answers
61 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
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3answers
28 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
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1answer
72 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
67 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! :)
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1answer
51 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 \\ ...
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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. ...
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0answers
40 views

Suppose A is a $3\times5$ matrix, B is a $5\times s$ matrix and C is a $5\times5$ matrix. if $A(BC)^T$ is defined, then what is the size of B ?? [closed]

Suppose A is a $3\times5$ matrix, B is a $5\times s$ matrix and C is a $5\times5$ matrix. if $A(BC)^T$ is defined, then what is the size of B ?? $A. 5\times5$ $ $ $B. 5\times s$ $ $ $C. ...
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1answer
11 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 ...
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0answers
20 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 ...
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0answers
27 views

Applications of matrices – CAT scans [closed]

Question 1 From the diagrams below, explain why one or two x-rays are insufficient in finding the location or number of tumours accurately, remembering that the x-ray source moves from left to right ...
7
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1answer
72 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 ...
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0answers
20 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 ...
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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 ...
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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
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0answers
38 views

Intersection of lines in 3D space [closed]

Given two or more pairs of points in 3D space, I should calculate the intersection of the lines passing through each of these pairs of points: for each pair of points, I get the linear system of two ...
0
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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}\\ ...
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1answer
18 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
58 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
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0answers
17 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
34 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 ...
2
votes
2answers
51 views

Can units in $M_n(\mathbb{Z})$ be moved to the other side?

Let $M, U_1 \in M_n(\mathbb{Z})$ with $U_1$ a unit (i.e. $\lvert \det(U_1) \rvert=1$). Can I always find another unit $U_2\in M_n(\mathbb{Z})$ such that $U_1 M = M U_2$?
1
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1answer
29 views

Solution to two equations with three unknowns

So I'm a student studying through correspondence and I need some help. This is an assignment question, and I have tried everything I know how, to answer it which has lead me to the conclusion that ...
0
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0answers
42 views

What conclusions we can draw from $|A|$=0? if $A$ is a positive-semidefinite matrix [closed]

What conclusions we can draw from $|A|$=0? (the determinant is 0) $A$ is a positive-semidefinite matrix. Many thanks
0
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0answers
24 views

Tightest upper bound for $\sum_j g_{ij}$ of an adjacency matrix of a graph

If I have an adjacency matrix of a graph $G$ (i.e. $g_{ij}=1$ if $i$ and $j$ are connected and $g_{ij}=0$ if not. $g_{ii}=0$), is there any tighter upper bound on $\sum_{j} g_{ij}$ than just $n-1$ ...
1
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2answers
223 views

Matrix Equation- solution

Sir, We have given $A= \begin{bmatrix}q_1 & q_2&q_3 \\ q_4 & q_5&q_6\\ q_7 & q_8&q_9 \end{bmatrix} \tag 1$. A is a matrix with determinant 1,orthogonal , invertible and ...
1
vote
1answer
68 views

Do these two rearranged matrices have the same singular values (or the same rank)?

This is the origin of my problem: I have a set of data which expresses which user ($U$ set) applies what tag ($T$ set) to which item ($I$ set). So it is actually a $U×I×T$ tensor $A$ (or 3-dimensional ...
4
votes
1answer
49 views

Solving a set of non-linear matrix equations

Consider the following set of equations $$\begin{cases}PAQ^{-1}&=T \\ QBR^{-1}&=T\\ RCP^{-1}&=T, \end{cases} $$ where A,B,C and T are known real-valued $3\times3$ matrices and P, Q, R are ...
0
votes
2answers
16 views

Verifying eigenvalues

How would you check whether eigenvalues $\lambda_1=8$, $\lambda_2=3$, $\lambda_3=-1$ belong to a matrix? $$ \begin{matrix} 7 & 1 & 1\\ 3 & 1 & 2 \\ 1 ...
1
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0answers
47 views

Fast checking Matrix multiplication in mod 10

I recently faced this problem in a programming contest: Given 3 square matrices N x N of size N up to 1000. All elements in 3 matrices are from 0 to 9. Check if matrix A x B equals to C, mod 10. In ...
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0answers
49 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
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0answers
29 views

Similar eigenvalues imply in similar rows? [closed]

My question is simple: If a matrix has rows that are similar in their elements this implies that all eigenvalues of the matrix are also similar to each other? Similarity here is defined as: B= ...
0
votes
1answer
17 views

Algebraic multiplicity and similarity between rows

I know that if two rows in a square matrix are identical, one eigenvalue will have multiplicity of at least 2. I was wondering if two rows are very similar in their elements but not identical it would ...
2
votes
4answers
25 views

Independence of the columns of triangular matrices

Let $M$ be a square upper triangular matrix with nonzero diagonal entries. Prove that the columns of $M$ are linear independent. I understand that this proof can be done with some sort of ...