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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1answer
203 views

Rank of a graph matrix

$G$ is a bipartite graph with $2m$ nodes on the left $(u_0..u_{2m-1})$, and $2^{m}$ nodes on the right $(v_0..v_{2^{m}-1})$. There is an edge (connection) between $u_i$ and $v_j$ iff $(i+1)$'th ...
5
votes
3answers
517 views

Freedoms of real orthogonal matrices

I was trying to figure out, how many degrees of freedoms a $n\times n$-orthogonal matrix posses.The easiest way to determine that seems to be the fact that the matrix exponential of an antisymmetric ...
8
votes
1answer
3k views

Relationship between eigendecomposition and singular value decomposition

Let $A \in \mathbb{R}^{n\times n}$ be a real symmetric matrix. Please help me clear up some confusion about the relationship between the singular value decomposition of $A$ and the eigen-decomposition ...
0
votes
1answer
866 views

help converting 2D rotating affine matrix to 3D rotating (matlab / octave code)

Greetings All I have a function that creates an affine matrix that works for 2D rotation around an arbitrary point but now I would like it to work for 3D rotations also. The working function example ...
2
votes
1answer
461 views

Matrix bracket notation

I am reading a section in a book that talks about normal matrices, and I see the following: A normal matrix is a matrix that commutes with its adjoint, Eh? ...
3
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1answer
182 views

Finding SVD efficiently for $AB^T$

I posted this on cs theory yesterday but did not get an answer and hence I am posting here. I have a low rank matrix given as $AB^T$ where $A,B \in \mathbb{R}^{n \times p}$ and $p \ll n$. (I know $A$ ...
6
votes
2answers
5k views

Eigenvalues of product of matrices

If $\mathbf{A}_{n\times n}$ is a positive semi-definite matrix with eigenvalues $\{\alpha_k\},\ k\in\{1,...,n\}$, and $\mathbf{B}_{m\times n}$ is an arbitrary matrix with singular values ...
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1answer
3k views

Find the Matrix of a Linear Transformation Relative to a Basis

Our book gives this problem: Find the $\mathcal{B}$-matrix for the transformation $\vec{x} \rightarrow A\vec{x}$ when the basis $\mathcal{B} = \{ \vec{b}_1, \vec{b}_2 \}$, where $A = ...
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3answers
165 views

Is $\det$ of $M_{i,j} = \min(x_i, x_j)^2 \left( 3 \max(x_i, x_j) - \min(x_i, x_j) \right)$ non zero?

Let $0<x_1<x_2<...<x_n<1$. Let us consider the matrix M defined such that: $$M_{i,j} = \min(x_i, x_j)^2 \left( 3 \max(x_i, x_j) - \min(x_i, x_j) \right)$$ I believe the determinant of ...
0
votes
1answer
210 views

Is there a generalization of Strassen algorithm?

Let $A$ and $B$ be $n \times n$ matrices. Strassen's algorithm for multiplication works on a partitioning of $A$ and $B$ into $2^2$ submatrices each. Is there any generalization of Strassen's ...
1
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1answer
275 views

Find a price vector p for various prices of industries.

( Leontief input-output model ) Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the $3 \times 3$ consumption matrix A = ...
4
votes
2answers
787 views

Determinants: multiply a row by a scalar, all is good; multiply a row by a square matrix?

When calculating determinants it can be nice to multiply a row by a number or to add one row to another (your basic row operations). Each has an easy to understand effect on the determinant. Today I ...
1
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2answers
189 views

Parameterization of modular group

Is there a parameterization by integers of the modular group SL(2,Z)? What I mean is, some expression for a matrix in terms of several variables (a,b,c,...) such that for each n-tuple of integers ...
0
votes
1answer
829 views

SVD for a complex matrix and approximation to a real matrix

Suppose M is a 20 by 3 complex matrix, and I'd like to SVD. For example, in Matlab, I can do easily with: [U, S, V] = svd(M); where U, S, and V are complex ...
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2answers
7k views

Absolute value of all values in a matrix

How do I express the matlab function abs(M), on a matrix M, in mathematical terms? I thought about norms or just |M|, but these return scalars, not another matrix of the same size as M. Sorry for ...
2
votes
2answers
969 views

Determine whether the set $H$ of all matrices form a subspace of $M_{2 \times 2}$

Determine if the set $Z$ of all matricies form $ \left[ \begin{array}{cc} a & b \\ 0 & d \end{array} \right] $ is a subspace of $M_{2 \times 2}$ (the set of all $2 \times 2$ matrices). % ...
14
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6answers
8k views

Matrix Inverses and Eigenvalues

I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
4
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1answer
464 views

How do you do a cross product of two $3 \times 3$ boolean matrices?

I have two boolean matrices: A = |1 1 0| |0 1 0| |0 0 1| and B = |1 0 0| |1 1 1| |0 0 1| What is the result of A x B and what are the steps ...
5
votes
1answer
925 views

Counting paths in a square matrix

Question: Consider a square matrix of order $m$. At each step you can move one step to the right or one step to the top. How many possibilities are to reach $(m,m)$ from $(0,0)$? I think ...
1
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2answers
509 views

Exporting a matrix in Mathematica in curly brackets format

I am trying to export a matrix in a file using Mathematica. I get the matrix and then use Export["myfile.dat", MyMatrix, "???Format???"] There are many formats. ...
2
votes
1answer
628 views

Is it possible to link the eigenvalues of a matrix to the Fourier transform of the matrix?

I'm trying to get insight into the eigenvalue spectrum of a square matrix (large N, symmetric,positive semi definite matrix) using Fourier transforms (I've tried transforming a bunch of thigngs: the ...
1
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3answers
150 views

Matrix with exactly one 1 in each row

Is there a name associated to rectangular matrices $M \times N$ that have exactly one entry equal to $1$ in each row and $0$ everywhere else?
2
votes
1answer
157 views

expansion of an expression

The Fokker-Planck equation for several variables is : $$\frac{\partial W}{\partial t} = L_{FP}W\qquad(1)$$ where $$L_{FP} = -\frac{\partial}{\partial x_i}D_i(\{x\})+\frac{\partial^2}{\partial x_i ...
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0answers
151 views

A form of the Baker-Hausdorff equation

I wonder how many different ways are there of writing the Baker-Hausdorff equation! This is a form which I recently encountered and haven't been able to figure out how it comes, $e^ae^Xe^b = ...
2
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1answer
238 views

Do these matrices have a name?

I'm wondering if these matrices have a name? (I'm somehow tempted to call them subunitary but it seems to be reserved for something else.) The matrix $M \in \mathbb{C}^{n\times n}$ is called ..., if ...
19
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2answers
6k views

Motivation behind Definition of Matrix Multiplication

I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It ...
3
votes
3answers
316 views

Find all invertible $n\times n$ matrices $A$ such that $A^2 + A = 0$

This was a question on one of our practice midterms: Find all invertible $n \times n$ matrices $A$ such that $$A^2 + A = 0.$$ I was told to expand $A^2$ and then solve, but that seems like a really ...
4
votes
1answer
1k views

Is this matrix diagonalisable?

Let $A$ be the following matrix: $$A = \left(\begin{array}{rrr} -1 & \hphantom{-}3 & \hphantom{-}0\\ 0 & 2 & 0\\ -3 & 3 & 2 \end{array}\right).$$ I've found that the ...
3
votes
0answers
121 views

Deducing linear combinations of column matrices

I've a $m \times n$ column matrix $A$ that can contain values {0,0.5,1}. i.e $\Omega = \{0,0.5,1\}$ $A$ = $\begin{bmatrix} 0 & 0.5 & 0.5 & 1 & 1 & \cdots & 0 \\ 0.5 & ...
5
votes
1answer
334 views

How to detect antitransitivity from an adjacency matrix?

I'm trying to derive a taxonomy from sparse "is-a" relationships and am looking for a linear algebraic solution. More specifically, the data is noisy and I want to detect relations that violate ...
3
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5answers
464 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 ...
6
votes
6answers
8k views

Finding a Rotation Transformation from two Coordinate Frames in 3-Space

The question I'm trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, ...
0
votes
1answer
2k views

Eigenvalue decomposition Singular value decomposition

following @Sivaram Ambikasaran's answer for SVD, I get the computing (using MATLAB)of: A = 2 1 3 1 2 5 3 5 4 [U,S,V] ...
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2answers
379 views

How to prove matrix identities involving the determinant, logarithm and derivatives

I'm talking about these specific relations, where g is the determinant of the metric tensor (so it's symmetric spscific), which is a function of $x^k$: $\frac{1}{2g}\frac{\partial g}{\partial ...
0
votes
2answers
3k views

How to compute SVD (Singular Value Decomposition) of a symmetric matrix

If I have only the upper triangular part of a symmetric matrix A How could I compute SVD, having this upper triangle makes easier the computing? $$\begin{pmatrix} 1 & 22 & 13 & 14 \\ ...
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votes
5answers
1k views

What's the point of orthogonal diagonalisation?

I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is. The definition is basically this: "A matrix A is ...
2
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1answer
323 views

Exact powers of an integer matrix

Assume that I have a large, non-symmetric matrix $\bf{A}$ of zeros and ones. If I want the exact diagonal elements of $\bf{A}^{k}$ (not the trace, but the elements themselves) where $k$ is some large ...
3
votes
1answer
221 views

How to find a representative equation for a linear system

This is for a graduate level course on computational linear systems. I want to do it myself, without help, but it's been ten years since I took linear algebra and I'm not sure I understand what the ...
6
votes
3answers
3k views

Solving non square matrix equations

Lets say we have: $\mathbf{A=BX}$ Where A and B are known matrices, X is unknown. In case B was square, a solution can be found by $\mathbf{B^{-1}A=X}$. But how do you attempt to solve for X when ...
3
votes
1answer
320 views

Minimizing the sum of the $k$ smallest elements of the diagonal of a matrix

I have a $N\times N$ symmetric positive semidefinite matrix $Q$, and am considering a class of symmetric positive definite matrices having all eigenvalues in a given bounded interval $[a, b]$. Is it ...
1
vote
1answer
225 views

Difficulty with last part of finding eigenvector

I need to be able to find the eigenvalues and eigenvectors of a matrix (sticking to $2 \times 2$ matrices at the moment). I'm fine with finding the eigenvalues, but my answers for the eigenvectors ...
1
vote
1answer
231 views

Rotation matrices for arbitrary dimensions

I initially asked this question here, and someone suggested this may be a better place to get an answer. I have a question about a rotation matrix, which can be represented in 2 dimensions as: ...
2
votes
1answer
345 views

Singular matrix

Suppose I have a singular matrix given by $$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{12} & a_{11} & a_{14} & a_{13}\\ a_{31} & a_{32} & a_{33} ...
2
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1answer
539 views

Conformability and Matrix Derivatives

What is the formula for the derivative of the product of two matrices when they are different sizes? Further, what is the formula for the derivative of a Hadamard product when the derivatives of the ...
3
votes
2answers
2k views

How to read an NxN matrix diagonally? After this how to write it diagonally?

I am having an NxN matrix . I want to read the elements of that NxN matrix diagonally and need to store it in an array.How? For example, I am having one 3x3 matrix $$\begin{bmatrix} A B C \\ D E ...
0
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1answer
408 views

How do I label the solution of a word problem using matrix multiplication?

The table on the left gives the birth and death rates (per million) by region. The table on the right gives the populations (in millions) in each region for a number of years. Use matrix ...
9
votes
2answers
644 views

Has there been a rigorous analysis of Strassen's algorithm?

According to Wikipedia, Strassen's Algorithm runs in $O(N^{2.807})$ time. Has anyone seen a more rigorous analysis displaying constants, possibly in a specific language such as C or Java? I ...
0
votes
1answer
188 views

Conjugacy class of a permutation from its matrix representation

Apologies if this is too basic, but given a permutation matrix $M$, is there any parameter or formula based on $M$ that gives the disjoint cycle decomposition, or at least the conjugacy class, of the ...
12
votes
2answers
482 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
1
vote
2answers
334 views

a question regarding the use of Moore–Penrose pseudoinverse

In the link http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse, it talks about solving $Ax=b$ by $x = A^+b + [I − A^+A]w$ for any vector $w$. Let's say $A$ is $m\times n$, and $b$ and ...