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)

1
vote
2answers
120 views

How do I find the adjacency matrix for the nodes of an n-dimensional finite grid?

I have an orthotopic grid, in n-dimensions (usually small ~<3), where each node is connected to it's orthogonal neighbours. The grid may be any number of nodes long, but is finite (and usually ...
0
votes
1answer
103 views

Proof: $ker(g) \subset ker(f)$ ..

Let $V = \mathbb{R}_{\le 3} [x]$ with basis $ B = (1, x, x^2, x^3)$. And $f: V \to \mathbb{R}, p \to \int_{-1}^1 p(x) dx$ and $g: V \to \mathbb{R}^3, p \to ^t( p(-1), p(0), p(1) )$. (1) I had to ...
2
votes
4answers
234 views

if $A=AB-BA$ then $A^n=0$?

Let matrix $A_{n\times n}$, be such that there exists a matrix $B$ for which $$AB-BA=A$$ Prove or disprove there exsit $n\in N^{+}$such $$A^n=0,$$ I know $$tr(A)=tr(AB)-tr(BA)=0$$ then I ...
0
votes
1answer
35 views

Representing a bijection as a matrix

Is it possible to represent a bijection using matrices? For example, if I have a bijection from a set of vectors back onto itself. A vector in the set could be v = [10 2 11], and the mapping would be ...
1
vote
2answers
80 views

Given a matrix, how to build an equation or inequality using variables that gives unique solution?

I want to solve a system of equations of the form $$ Ax = b $$ where $A$ is a $6\times 15$ matrix, $x$ is $15\times 1$, and $b$ is $6\times 1$. In the absence of other constraints, this system of ...
0
votes
0answers
109 views

Square Matrix, Row Echelon Form

I'm reading Artin's Algebra. There is the following theorem: A square (reduced) row echelon matrix $M$ is either the identity matrix $I$, or else its bottom row is zero. I am able to prove this ...
0
votes
2answers
48 views

Algebraic proof with matrices

I need to proof the following: Given $A$ is a $n\times n$ matrix so that $A^2 - 3A + I = 0$ Prove that $A^{-1} = 3I - A$ So I laid out a matrix: $$ A =\begin{pmatrix} a & b \\ c & d ...
0
votes
1answer
652 views

Shortest distance matrix given an adjacency matrix?

If I have an adjacency matrix, how can I find a matrix that has the shortest distance between each pair of nodes? (distance matrix, but the nodes are not in a euclidean space) I'm trying to implement ...
7
votes
1answer
398 views

Number of binary $M\times N$ matrices with even row sums, even col sums and $K$ ones, $K$ even

A combinatorial problem arising with certain checksums: When sending messages, the user data are protected by adding a parity bit for bit positions $1\dots8$ and a parity bit for each byte. So, the ...
0
votes
1answer
24 views

form the equivalent matrix equation and augmented matrix

what does it mean when it tells you form the equivalent matrix equation and augmented matrix lets say I have something like this ...
1
vote
5answers
95 views

What does it mean for $AA^T$ to be symmetric?

What does it mean for $AA^T$ to be symmetric? A question in my book says to show that $AA^T$ is symmetric so I took a very simple matrix to try and understand this: $A=\begin{bmatrix} 2 \\ 8 \\ ...
0
votes
1answer
57 views

Pseudo Inverse matrix

I have a matrix from which I have to find the pseudo inverse, but none of the methodes that I found gave me the correct answer (one that wolfram alpha gave me) The matrix is: $ \pmatrix{0 & 1 ...
1
vote
1answer
312 views

Eigenvalues less than or equal to 1

What proprieties does a square $n\times n$ real matrix $\mathbf M$ need to have in order to have all it's eigenvalues be less than or equal to one in absolute value? I'm looking for proprieties such ...
1
vote
1answer
62 views

Operator Norm = 1

Let there be a linear map $T$ such that $T: \mathbb R^n\to\mathbb R^m$. The operator norm $\lVert \cdot\lVert_{op}$ of $T$ is then defined as the largest value of $c$ for which $\lVert T(\vec v)\lVert ...
1
vote
2answers
501 views

The determinants of upper triangular matrices (For any 2x2 and 3x3 matrix)

I am trying hard to figure out what am I supposed to do, if I am supposed to go on write a conjecture about the particular question. How can I go on about to prove it?
0
votes
4answers
70 views

Decomposing a given $2\times2$ matrix as a product of two non-identity matrices

Let $X = \left( \begin{matrix} -8 & 7 \\ 2 & 0 \end{matrix} \right) $. Give an example of two $2 \times 2$ matrices $A$ and $B$, neither of which is the identity matrix $I$, such that $AB = ...
2
votes
2answers
150 views

connection between PCA and linear regression

Is there a formal link between linear regression and PCA? The goal of PCA is to decompose a matrix into a linear combination of variables that contain most of the information in the matrix. Suppose ...
0
votes
1answer
4k views

Finding matrix inverse by Gaussian Elimination With Partial Pivoting

Hello guys I am writing program to compute determinant(this part i already did) and Inverse matrix with GEPP. Here problem arises since i have completely no idea how to inverse Matrix using GEPP, i ...
1
vote
1answer
36 views

Represent derivation as a standard matrix (Linear mapping)?

Given a matrix $a$ of coefficients $\left( \begin{array}{cc} a_0 \\ a_1 \\ .. \\a_n\end{array} \right)$representing $a_0 + a_1 x + a_2 x^2 + ... a_n x^n$, how can I find a standard matrix D such that ...
2
votes
6answers
317 views

Definition of matrix exponential

Is there an alternative definition of a matrix exponential so I can use it to prove that $$e^{A}=\sum_{m=0}^{\infty} \frac{1}{m!}(A)^m \;?$$ Thanks a lot in advance!
1
vote
2answers
49 views

Showing that two Matrices are not similar over GL$_n(\mathbb{F}_2)$

Problem: Show that $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ Are not similar over GL$_n ( \mathbb{F}_2)$ Note: We write ...
1
vote
1answer
41 views

Degree $4$ polynomial $p$ such that $p(A) = 0$ for all $A \in M_{2 \times 2}(\mathbb{R})$?

Someone asked me how to prove that there exist real numbers $a_0, \ldots, a_4$ not all zero such that $$ a_0 I_2 + a_1 A + a_2 A^2 + a_3 A^3 + a_4 A^4 = 0 \quad \forall A \in M_{2 \times ...
0
votes
2answers
125 views

Find all $2\times 2$ orthogonal matrices $A$ such that $2A^3 = B$

I need to find all possible $2 \times 2$ orthogonal matrices $A$ such that $$ 2A^3 = \begin{pmatrix} 1 & -\sqrt{3}\\ \sqrt{3} & 1 \end{pmatrix} $$ I have thought of the given matrix, divided ...
1
vote
1answer
59 views

Linear Algebra, power of matrix

[Linear Algebra]Please help. Without using the method of Diagonalization. I have tried changing the matrix into [sin(pi/6), -tan(pi/6)] [-tan(pi/6), sin(pi/6)] and try to figure out what is the ...
1
vote
0answers
73 views

Index notation for unitary matrices

I was wondering if someone could confirm this for me. I'm attempting to re appropriate a paper into matrix notation but i keep getting confused. I first have a unitary matrix that makes a ...
2
votes
1answer
216 views

Number of binary n x m matrices, with at most k consecutive number of 1 in each column

I am trying to compute the number of $n x m$ binary matrices with at most $k$ consecutive values of $1$ in each column. I've figured out that I it will be enough to find the vectors with $1$ column ...
1
vote
1answer
83 views

If $S$ is an non-empty subset of $\mathbb{R}^n$, then $S^\perp$ is a subspace of $\mathbb{R}^n$

How do I prove the following statement: If $S$ is an non-empty subset of $\mathbb{R}^n$, then $S^\perp$ is a subspace of $\mathbb{R}^n$.
1
vote
1answer
31 views

polynomial with real coeficients for a matrix

let $A$ belongs to $M_{n \times n}$ then we have to show that there exist a polynomial $f(x)$ with real coefficients such that $f(A)=0$..we know that this is true for characteristic polynomial i.e for ...
1
vote
2answers
608 views

Consider the trace map $M_n (\mathbb{R}) \to \mathbb{R}$. What is its kernel?

The map is the trace map. I.e, it takes any $n$ by $n$ matrix and associates to that matrix, a number of the form $\mathrm{Tr}(A) = \sum_{i=1}^n a_{ii}$, where $A \in M_n (\mathbb{R})$. I need to ...
0
votes
1answer
101 views

Fitting 2nd Order multivariate quadratic with matrices

Hopefully you at least entertain this question as it took forever to construct the below matrix using TeX. Any ways, so I have a list of data points ($X_1$,$X_2$,Y), with the X's being independent ...
0
votes
5answers
144 views

Do non-square infinite matrices exist?

Sorry, I tried to wrap my head more around this, but I failed. Given non-square matrix $A$ that has dimension $kn \times n$. Now let $n$ goto infinity. Is the matrix finally square?
0
votes
1answer
161 views

Givens method for sparse matrix

I have to solve a linear equation system, Ax=b, where A is a (very) large sparse matrix (compressed sparse column form). From what I've read I think I'm supposed to use Givens method, but I don't ...
2
votes
2answers
43 views

Proof that matrix $B^{-1}$ = matrix $A^{-1}$ with 2 columns swapped given that B = A with 2 rows swapped.

I'm trying to prove the following. Given that $A$ is a nonsingular $n \times n$ matrix, and $B$ is the nonsingular matrix obtained by interchanging rows $i$ and $j$ of $A$, where $i \neq j$, show ...
0
votes
1answer
62 views

Constructing a similarity matrix between points

I have two images with two sets of corresponding points. In order to align the images I'm trying to compute the similarity matrix that describes the relationship between the corresponding points. I ...
2
votes
2answers
264 views

Row swap changing sign of determinant

I was wondering if someone could help me clarify something regarding the effect of swapping two rows on the sign of the determinant. I know that if $A$ is an $n\times n$ matrix and $B$ is an $n\times ...
0
votes
2answers
51 views

Proof that Jordan form with eigenvalues of 0 is nilpotent

I can see that if I have a nxn matrix Of Jordan form with eigenvalues of zero $\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}$ How do I prove that $A^n = 0$ ...
8
votes
4answers
379 views

What does it mean if $\det(A)$ equals $1$?

What does it mean if $\det(A)$ equals $1$? Does it mean that the identity matrix can be obtained from $A$ by only adding multiples of rows onto others?
1
vote
1answer
60 views

When do i use row vectors over column vectors when showing the basis of null space of a matrix?

Let $$A = \begin{bmatrix}1&-5&4&-2&2\\1&-6&5&-3&2\\-2&11&-8&5&-2 \end{bmatrix} $$ and define a linear transformation, $T: \mathbb{R}^5 \to \mathbb{R}^3 ...
3
votes
1answer
173 views

How do I find upper triangular form of a given 3 by 3 matrix??

We are asked to find an invertible matrix $P$ and an upper triangular matrix $U$ such that: $P^{-1}\begin{pmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{pmatrix}P=U$ I'm ...
7
votes
2answers
444 views

Maximal dimension of subspace of matrices whose products have zero trace

In the space of all real matrices with dimension $n$, find the maximal possible dimension of a subspace $V$ such that $\forall X,Y\in V,\, \operatorname{tr}(XY)=0$.
7
votes
0answers
163 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
1
vote
1answer
743 views

find a change of basis matrix

given basis $\beta$ = {(1, 1, 0), (1, 0, -1), (2, 1, 0)} and matrix $$ A = \begin{matrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ ...
0
votes
1answer
78 views

paper about linear independence in altered Vandermonde and Cauchy Matrices

Both Vandermonde and Cauchy matrices with $n$ rows and $k$ ($n \geq k$) columns have the property that any $k$ rows are linearly independent (assuming the coefficient are independent). It seems to me ...
3
votes
2answers
136 views

A formula in 'The Matrix Cookbook'

In section 9.4 (Idempotent Matrices), the book says that : if $A$ is idempotent, which means that $AA = A$, then $f(sI + tA) = (I-A)f(s) + Af(s+t)$ but I don't understand the meaning of this ...
0
votes
1answer
37 views

Restrictions on a Matrix-Vector product

Suppose I have a $m\times n$ matrix $\mathbf M$, and a unit vector $\hat v$, of dimension $n$. What restrictions do I need to apply to $\mathbf M$ so that $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ ...
1
vote
1answer
91 views

Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn't include the full proofs. That's fine for the question as it ...
-2
votes
2answers
60 views

Why can this product of rectangular matrices not be the identity matrix?

Let $m > n$ be positive integers. Show that there do not exist matrices $A ∈ R_{m×n}$ and $B ∈ R_{n×m}$ such that $AB = I_m$, where $I_m$ is the $m × m$ identity matrix If someone could explain it ...
1
vote
3answers
65 views

Very confused about linear Algebra

My confusion is a result from the methodology used to solve some problems. As an example, in order to find the kernel I set $Ax=0$ and the way I find a basis I also set $Ax=0$ and to find something ...
3
votes
5answers
536 views

How to find Determinant of a matrix

I could not understand the concept while googling. can anybody provide help? what will be the determinant of the following matrix? $$ \left[\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 ...
1
vote
0answers
87 views

Whats the name of this sort of matrix

What the name of such a matrix \begin{pmatrix} 1 & 2 & 5 & 10 \\ 3 & 4 & 7 & 12 \\ 6 & 8 & 9 & 14 \\ 11 & 13 & 15 & 16\\ \end{pmatrix} Its properties ...