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
37 views

problems on define set with polynomials

I'm trying to say set A is the set of nonnegative integers that not of this two forms $3x^2 + (6y-4)x - y\ $ and $\ 3x^2 + (6y-2)x + (y - 1)$, for example: $4=3 \cdot1^2+(6 \cdot1-4) \cdot 1-1\ $ is ...
3
votes
1answer
111 views

Using matrices to calculate fibonacci?

I have been told a couple of times it possible to calculate the fibonacci sequence much quicker using matrices but I never understood/they never elaborated. Would somebody be able to show how this ...
2
votes
3answers
69 views

A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same?

Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form? Positive defined quadratic form is A(x,y), that ...
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2answers
34 views

How to prove: a quadratic form with a matrix $ B = CC^T $ is positive defined?

Let a matrix $ C \in \Bbb K^{n \mathtt x n} : det(c) \ne 0 $ (K is any field - C or R) $ \Rightarrow $ a quadratic form with a matrix $ B = CC^T $ is positive defined one. How to prove it?
2
votes
4answers
124 views

How to prove the inequality $\det (AA^T) \ge 0$?

How to proof for any matrix $A \in \Bbb R^{n \mathbf x n}$, that the next inequality $\det(AA^T) \ge 0$ is true?
1
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2answers
425 views

What does the inverted V represent in math

I know that A V B represents Logical disjunction which means A OR B and the result of it is false only when both A and B are false . But I still didn't understand what an inverted V means as shown in ...
5
votes
2answers
306 views

Prove: matrix A is diagonalizable iff exp(A) is diagonalizble

I need to prove: matrix A is diagonalizable iff $\exp(A)$ is diagonalizble. exp means exponent function. I know to prove that if $A$ is diagonalizable so $\exp(A)$ is diagonalizable, but have a ...
1
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1answer
967 views

block matrix multiplication

If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then $$AB = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot \left[ \begin{array}{cc} b_{11} ...
2
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1answer
78 views

What is $\frac{\partial }{{\partial A}}tr({A^T}BA)$

What is $\frac{\partial }{{\partial A}}tr({A^T}BA)$? My thoughts: $\frac{\partial }{{\partial A}}tr(AB{A^T}) = A(B + {B^T})$, hence $\frac{\partial }{{\partial {A^T}}}tr({A^T}B{A^{}}) = ...
-1
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3answers
287 views

Symmetric matrix 10 x 10

Can You give me the example of symmetric matrix 10 x 10. Or if You know online symmetric matrix generator then give me the links. I tried to make this with Wolfram Mathematica but I did not find a ...
1
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1answer
30 views

Is there any definition for homogeneous rotations?

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin. ...
1
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0answers
56 views

How to complete one vector to create a basis of a certain subspace?

I have a subspace of $R^3$ that its basis is the vector $(0,1,1)$. I want to add vectors to it, such that they will create a basis of a subspace, that is equal to the subspace that its basis is ...
4
votes
0answers
109 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
0
votes
1answer
89 views

Element-wise derivative of the inverse of a matrix

I would appreciate if you could help me to obtain the element-wise derivative of $Z = (-A-BX)^{(-1)}$ where all of elements of $A$, $B$ and $X$ are positive. I conjecture that if I increase any of ...
2
votes
1answer
65 views

Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
4
votes
1answer
393 views

Condition for degenerate eigenvalues for a matrix

Given a diagonalizable matrix $M$ (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ...
0
votes
2answers
44 views

Square of a matrix problem

A real 2 × 2 matrix M such that $M^2=\begin{pmatrix}-1 &0 \\ 0 &-1-\epsilon \end{pmatrix}$ (A) exists for all $\epsilon$ > 0 (B) does not exist for any $\epsilon$ > 0 (C) exists for some ...
1
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1answer
34 views

Vectors to Matrices in algebraic equations

This question is based off of Dave Eberly's 3D Game Engine Design, 2nd Edition. I am reading it slowly to gain a larger algebraic grasp of 3D graphics, which this book seems to offer. When finding a ...
1
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1answer
80 views

Existence of isomorphism between groups of upper triangular matrices.

Is there an isomorphism between this group of matrices $$ \begin{pmatrix} 1 & k \\ 0 & 1 \\ \end{pmatrix},~~k\in\mathbb Z $$ and this one $$ \begin{pmatrix} 1 & k_1 & k_2 \\ 0 & 1 ...
1
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1answer
54 views

Proof of why multiplication between a matrix and its eigenvector equals the eigenvalue times its eigenvector

Why is the following true? $$\lambda \vec{x} = P \vec{x} \: \: \: \lambda \:\: \text{is the eigenvalue of P} $$ If the column vectors are the input space, and the row vectors are the output space, ...
0
votes
0answers
37 views

Calculate B^2013 where B is P^-1AP

Using A and P, denote B=P^-1AP and calculate B^2013 I'm unsure of where to go with this question. I know it involves diagonalization, but I'm not sure how that even works. A is ...
1
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1answer
27 views

minimal polynomial of a matrix with no real roots

Is it possible to have a matrix in $M_3(\mathbb{R})$ with a quadratic minimal polynomial $m(x)$ in $\mathbb{R}[x]$ possessing no real roots? Does this correspond to no real eigenvalues, or can the ...
2
votes
1answer
65 views

Completely multiplicative matrix norm for certain semigroups of matrices.

I am currently working on some properties of matrix products and their norms for $\mathbb{R}^{n \times n}$ matrices and i was wondering if there exists a completely multiplicative matrix norm, i.e. ...
2
votes
3answers
42 views

Finding the characteristic polynomial in a square matrix

The example in the textbook had a square matrix \begin{pmatrix} 0&1&0\\0&0&1\\4&-17&8 \end{pmatrix} Then proceed to say $ \ (\lambda \cdot I - A) \ $ is \begin{pmatrix} ...
3
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0answers
46 views

If $A^2=A$, $B^2=B$ and $I-(A+B)$ is invertible, then $R(A) = R(B)$ [duplicate]

If $I-(A+B)$ is invertible and $A$ and $B$ are idempotent matrices, then how do I show that $A$ and $B$ have the same rank?
0
votes
2answers
149 views

How to find center and radius of hand-drawn circle? [duplicate]

You are given a set of points {(X1,Y1), (X2,Y2),...} which represent a hand-drawn circle, so it's not perfect. You are asked to find the center and radius of this circle. My intuition tells me this ...
0
votes
1answer
35 views

Function over matrices, continuous and differentiable?

How can I prove that a function which takes an nxn matrix and returns that matrix cubed, is a continuous function? Also, how can I analyze if the function is differenciable or not? About the ...
1
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1answer
116 views

3x3 real matrices with same minimal polynomial but different jordan normal form [duplicate]

This is to show that the minimal polynomial does not determine the jordan normal form. I've tried a few but haven't got anywhere. Why is this marked as duplicate? The other post does not give me an ...
3
votes
1answer
77 views

Formula for position in an upper triangular matrix

I'm currently working on the Travelling Salesman's Problem in a computer science module. I have implemented some linear programming techniques using the software lp_solve. I've ended up with an upper ...
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0answers
28 views

Notation for appending 2 submatrices

I have a matrix $M$ with $i$ rows and $(e+n)$ columns: $M_{i,e+n}$ I would like to express that $M_{i,e+n}$ is the result of appending $M_{i,e}$ and $M_{i,n}$ What is the algebraic notation to ...
2
votes
1answer
92 views

fast multiplication for a matrix and its transpose.

I know Strassen and other methods can achieve better than $O(n^3)$ for general square matrix multiplication. I am curious of the spacial case where the multiplication is between a $n*m$ matrix $A$ ...
0
votes
2answers
50 views

What could be the rank of a matrix multiplied by its transpose ?

Let $A$ be a full rank $m×n$ matrix $(m<n)$, i.e. $\operatorname{rank}(A)=m$. Can the rank of $A'A$ be $n$? Under what condition would this hold? Thanks!
3
votes
1answer
39 views

Simple SVD and Polar Decomposition Question

Given an $n \times n$ matrix $A$. (1) Show $AA^{*}$ is similar to $A^{*}A$ by singular value decomposition. (2) Consider polar forms that $A=UP=QV^{*}$ in which $U$ is $m \times n$; $P$ is $n \times ...
0
votes
1answer
46 views

Matrix Inequality: $A^\top B A \preccurlyeq B$

Consider $A, B \in \mathbb{R}^{n \times n}$, $A$ invertible, $B \succ 0$. Say if the following holds: $$ A^{- \top} B A^{-1} \preccurlyeq B \ \Longleftrightarrow \ B \preccurlyeq A^\top B A. $$ I do ...
1
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1answer
177 views

Decomposing the set of $2 \times 2$ complex matrices into orbits under left multiplication

I have some issues with a problem which is asking me to decompose the set of $2 \times 2$ complex matrices $\mathbb{C}^{2 \times 2}$ in orbits under the left multiplication operation on the group ...
1
vote
2answers
4k views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
1
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0answers
48 views

Matrix multiplication in game theory doesn't add up? Min y^T*Ax

I'm studying game theory and something seems weird to me. My book says y is the probability of the row player and x is the probability of column player, both x and y are vectors. A = [a$_i$$_j$] is ...
3
votes
1answer
67 views

Spectral norm of symmetric matrices only with the diagonal, the first column, and the first row non-zero

Consider a real symmetric matrix $$\mathbf{M}=\left[\array{a_0&a_1&a_2&\cdots&a_n\\ a_1&b_1&0&\cdots&0\\ a_2&0& b_2&\cdots&0\\ \vdots &\vdots ...
0
votes
1answer
20 views

Characterising a relation between two matrices

Say we have a $3\times 3$ matrix $\mathbf{M}$ and vectors in $\mathbb{R}^3$ such that: $$\mathbf{Mu}_i=\mathbf{v}_i,\quad i=1,2,3$$ where $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ are linearly ...
2
votes
0answers
60 views

Drawing phase portrait

This is the question in my textbook. I am a bit lost for 3 hours now. Could anyone please point me to the right direction? Let the $2 \times 2$ matrix $A$ have real, distinct eigenvalues $\lambda$ ...
1
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0answers
26 views

Constructing Matrix with Normal Distribution

I have a vector given whereby each element of the vector is assumed to be the average of one of a matrix' rows. Now I want to construct the matrix belonging to this vector, whereby the elements of the ...
1
vote
1answer
44 views

Inverse Identity + Constant Matrix

I need to invert a square symmetric matrix $$ C = c\, I+cs\, B $$ Where: (1) $B$ is a constant matrix of 1 for each entry. (2) $c$ and cs are just positive real numbers. (3) $I$ is the identity. ...
1
vote
1answer
38 views

Prove $||A||_2 = max_{x \neq 0, y \neq 0}\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}\}$ for $A\in \mathbb{K}^{n\times m}$

I'd like to prove that the spectral norm of a matrix that is not necessarily square can be written as the following subordinate norm $||A||_2 = max\{\frac{|\langle y,Ax \rangle|}{||x||\cdot ||y||}, y ...
0
votes
0answers
30 views

Computing PageRank

I'm following the book here to compute PageRank scores: http://nlp.stanford.edu/IR-book/html/htmledition/markov-chains-1.html I built the transition matrix P as indicated in the book. Then I computed ...
1
vote
1answer
16 views

'Splitting' Identity matrix around others?

Say I have some arbitrary matrix, $A$. $$A=IA=AI$$ where $I$ is the Identity matrix. If I have another matrix, $B$, which I know to be unitary and invertible, then: ...
2
votes
1answer
55 views

Positive semi-definite Matrix and its eigenvalues (Please help checking/ improving my presentation)

Let $A$ and $B$ are two $n \times n$ Hermitian matrices . Suppose $A-B$ is positive semidefinite. (a) Show that $\lambda_k(A) \geq \lambda_k(B)$ for $k=1,2,\dots ,n,$ where $\lambda_i(A)$ and ...
1
vote
1answer
51 views

Linear Algebra, matrix representation

Let $V$ be a vector space of all real differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with basis $(e^{3t}, te^{3t},t^2e^{3t})$. Let $D: V \rightarrow V$ be the derivation operator on ...
1
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1answer
46 views

The product of two diagonizable matrices is another diagonizable matrix?

I guess this question has been asked before but I have not found it. So I will re-ask. I have a diagonal matrix and a symmetric one, both are diagonizable. If I multiply them, would they be always ...
8
votes
2answers
2k views

How to prove Fibonacci sequence with matrices?

How do you prove that: $$ \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n\\ F_{n} & F_{n-1} \end{pmatrix}$$
7
votes
5answers
439 views

Find the determinant of the following;

Find the determinant of the following matrix, and for which value of $x$ is it invertible; $$\begin{pmatrix} x & 1 & 0 & 0 & 0 & \ldots & 0 & 0 \\ 0 & x & ...