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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

0
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1answer
580 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 ...
10
votes
2answers
1k 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 ...
1
vote
1answer
220 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 ...
14
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2answers
3k 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 ...
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vote
2answers
493 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 $x\...
1
vote
1answer
235 views

Finding roots for a Lie algebra g, wrt toral subalgebra h

I'm trying to find the root space decomposition of a lie algebra wrt a toral subalgebra h. Both a matrix lie algebras. I'm confused about how do I find the linear forms $\lambda \in \mathfrak{h}^*$ ...
4
votes
1answer
257 views

Name the matrix (not the game show)

I have a matrix of the following form: $ \begin{matrix} a_1 & 0 & \ldots & 0 \\ a_2 & a_1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ a_n & a_{n-...
3
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1answer
4k views

Proof - Square Matrix has maximal rank if and only if it is invertible

Could someone help me with the proof that a square matrix has maximal rank if and only if it is invertible? Thanks to everybody
3
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2answers
504 views

Find matrices $A$ and $B$ given $AB$ and $BA$

Given that: $$AB= \left[ {\matrix{ 3 & 1 \cr 2 & 1 \cr } } \right]$$ and $$BA= \left[ {\matrix{ 5 & 3 \cr -2 & -1 \cr } } \right]$$ find $A$ and $B$.
0
votes
1answer
327 views

Calculating axis angle from matrix without wrapping at PI

I need to get the axis angle from a matrix of the form: $$ \begin{matrix} \cos \theta & -\sin \theta & tx\\ \sin \theta & \cos \theta & ty\\ 0 & 0 & 1\\ \end{matrix} $$ ...
18
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3answers
118k views

shortcut for finding a inverse of matrix

I need tricks or shortcuts to find the inverse of $2 \times 2$ and $3 \times 3$ matrices. I have to take a time-based exam, in which I have to find the inverse of square matrices.
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3answers
22k views

Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks

Let $A$ be a block upper triangular matrix: $A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$ Show ...
5
votes
2answers
788 views

Knowledge about Graph Spectral Theory and correlation between a graph Weighted Adjacency matrix and its eigenvalues

I know that this question is some sort of bridge between Informatics and Mathematics, not knowing the best place where to post this question, I opted for this place because of the type of answer I ...
77
votes
7answers
90k views

Physical meaning of the null space of a matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it. E.g.: I think ...
5
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1answer
3k views

Orthogonal Decomposition of A Matrix

I'm trying to follow/understand a research paper that I have, and well, it's been a while since I've done this kind of math. At this point I have an nxn matrix H and from that construct an (n-1)xn ...
70
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4answers
44k views

Importance of matrix rank

What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller). Why is it a problem if a matrix is rank ...
4
votes
1answer
254 views

rotation matrix for particle track

Struggling with some equations here. I've got a particle I need to track in an absolute reference frame, but each step I move/rotate it relative to its own reference frame. I need to track it's ...
4
votes
2answers
9k views

What's the fastest way to take powers of a square matrix?

So I know that you can use the Strassen Algorithm to multiply two matrices in seven operations, but what about multiplying two matrices that are exactly the same. Is there a faster way to go about ...
3
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3answers
1k views

Positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?
4
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1answer
1k views

A direct proof of the properties of the matrix of minors

Let $A$ be an invertible $n\times n$ matrix. Define the matrix of minors $\Delta(A)$ of $A$ to be the matrix whose $(i,j)$ entry is the determinant of the minor of $A$ with the $i$th row and $j$ ...
0
votes
1answer
358 views

How to use parameterized solution to find solutions?

An electronics company produces three models of stereo speakers, models A, B and C, and can deliver them by station wagon, truck, or van. A station wagon holds $3$ boxes of model A, $3$ boxes of model ...
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1answer
236 views

Trouble deriving the Harris Corner Detection

I just started studying a small paper about the Harris Corner Detection. The problem is I don't understand how step 7 is derived from step 6. In step 7 the expression is expanded in a way that we get ...
1
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1answer
3k views

How to solve this problem using matrix

A company produces three combinations of mixed vegetables that sell in 1Kg package. Italian style combines 0.3kg of zucchini, 0.3kg of broccoli and 0.4kg of carrots. French style combines 0.6kg of ...
2
votes
1answer
13k views

How to parameterize a solution of a system of equations?

Give the parameterized solution for the dependent system represented by the matrix: $$\left(\begin{array}{ccc|r} 1 & 0 & 3 & 6\\ 0 & 1 & 2 & -4\\ 0 & 0 & 0 & 0 ...
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vote
0answers
127 views

Is that series-transformation known in the context of divergent summation

Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
26
votes
4answers
55k views

Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$

Given a matrix $A$ and column vector $x$, what is the derivative of $Ax$ with respect to $x^T$ i.e. $\frac{d(Ax)}{d(x^T)}$, where $x^T$ is the transpose of $x$? Side note - my goal is to get the ...
6
votes
2answers
2k views

Olympiad linear algebra problem

This is a problem from an olympiad I took today. I tried but couldn't solve it. Let $A$ and $B$ rectangular matrices with real entries, of dimensions $k\times n$ and $m\times n$ respectively. Prove ...
9
votes
1answer
2k views

Updating eigen-decomposition of symmetric matrix $A$ to eigendecomposition of $A+D$ where $D$ is low-rank diagonal

Given a symmetric positive definite matrix $A$ and a mostly-zeros non-negative diagonal matrix $D$, is there a way to cheaply update the eigenvalues and/or eigenvectors of $A$ to that of $A+D$? ...
1
vote
1answer
556 views

Matlab - Matrix Function with an Independent Variable?

I'm trying create a function that returns a matrix containing a variable "l" which is an independent variable to be swept for a plot later on. I would calculate "phi" based on user inputs which ...
3
votes
1answer
151 views

Constructing a projection

In our matrices class we were given a problem that I'm having trouble with. Let $E$ and $H$ be subspaces of $\mathbb{C}^n$ such that $\mathbb{C}^n = E \oplus H$. Construct a projection $\mathbf P$ ...
9
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2answers
5k views

Eigenvalues of product of a matrix and a diagonal matrix

My situation is as follows: I have a symmetric positive semi-definite matrix L (the Laplacian matrix of a graph) and a diagonal matrix S with positive entries $s_i$. There's plenty of literature on ...
4
votes
1answer
308 views

Matrix Analysis in $\mathbb{C}$

1.Let $A \in GL_n(\mathbb{C})$. Show that $\det(I+A)=1+\operatorname{tr}(A)+ \epsilon(A)$ where Modulas of epsilon(A) by norm of A=0 as A tends to 0,for any matrix norm. If I define J(A)= det(A) for A ...
3
votes
2answers
279 views

Matrix Diagonal Multiplication

I have a matrix-vector inner product multiplication $G = X D x$ where $D$ is a diagonal matrix. Now let's say I already know $E = Xx$. Is there a method that I can use to change $E$ into $G$ using $D$ ...
5
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2answers
1k views

$(A-λI)X=0$ works, but$ (λI-A)X=0$ does not when finding eigenvectors. Why?

Either works when trying to find the eigenvalues, but only the former works when trying to find corresponding eigenvectors. I can understand how it makes a difference, but what I don't understand how ...
6
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7answers
3k views

Best way to exactly solve a linear system (300x300) with integer coefficients

I want to solve system of linear equations of size 300x300 (300 variables & 300 equations) exactly (not with floating point, aka dgesl, but with fractions in ...
4
votes
3answers
12k views

Pseudoinverse matrix and SVD

I'm trying to solve an homework question but I got stuck. Let A be a m x n matrix with the SVD $A = U \Sigma V^*$ and $A^+ = (A^* A)^{-1} A^*$ its pseudoinverse. I'm trying to get $A^+ = V \Sigma^{-...
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3answers
260 views

Counting matrices based on determinant and trace

Let $p$ be an odd prime number and $T_p$ is the following set of $2 \times 2$ matrices: $$ T_p = \left\{ A = \begin{bmatrix} a & b \\ c & a \end{bmatrix}\ \Biggm|a,b,c \in \{0,1,2,...(p-...
4
votes
1answer
245 views

Invertible $N \times N$ matrix over ${\rm GF}(2)$ having on each row and column $N/2$ ones

As per the title, I'm looking for the name and for a way to construct a ${\rm GF}(2)$ square matrix of size $N$ with the following properties: All rows/columns should be linearly independent On each ...
20
votes
3answers
9k views

Integer matrices with integer inverses

If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it's not ...
3
votes
1answer
241 views

Inverting $(A + D_p)$ efficiently for many diagonal $D_p$ (and $A$ SPD)

At one step of an algorithm (i=1: something big) I have to draw a vector from the normal distribution $X_p^{(i)}\sim N((A^{(i)} + D^{(i)}_p)^{-1}m, (A^{(i)} + D^{(i)}_p)^{-1})$ for $p=1:P$, with a ...
2
votes
0answers
387 views

From a triangular matrix to a symmetric square matrix

I am a computational physics postgrad student, working with libraries like ATLAS and MAGMA. I have a matrix which is upper-triangular, and is the result of a Cholesky decomposition. I need to ...
2
votes
1answer
569 views

Matrix and Binomial Coefficients

Considering the construction of a matrix as follows. The $n$th row in the matrix is filled with the coeffcients of $x^r$ in the expansion of $(1+x)^n$ from the columns $2n$ to $3n$ inclusive and ...
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2answers
3k views

Normalizing the Eigenvalues so their sum is equal to min(n, p)?

If I compute the eigenvalues and eigenvectors using numpy.linalg.eig (from Python), the eigenvalues returned seem to be all over the place. Using, for example, the ...
8
votes
1answer
2k views

connection between graphs and the eigenvectors of their matrix representation

I am trying to learn graph theory and the linear algebra used to analyse graphs. The texts I have read through have lots of lemmas and theorems proved. The proofs are convincing but I fail to see the ...
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1answer
3k views

Question about the standard basis's relationship to the identity matrix

I'm in the process of reading my first Linear Algebra textbook, and was just wondering...Is the standard basis of a vector space in n dimensions equivalent to the row space of the n x n identity ...
7
votes
3answers
776 views

Showing that the path-connected component of the identity matrix in a subgroup of $GL_n(\Bbb R)$ is a normal subgroup

Let $M(n;\mathbb{R})$ denote the set of all $n \times n$ matrices with real entries (identified with $\mathbb{R}^{n^{2}}$ and endowed with its usual topology) and let $GL(n;\mathbb{R})$ denote the ...
6
votes
1answer
123 views

Verifying the distribution of an ensemble of random matrices

Suppose one is trying to devise a method to generate random matrices with a certain distribution. How does one verify that the generated matrices follow the desired distribution? In particular, I am ...
2
votes
1answer
85 views

Help me spot the error?

I have a determinant to expand which is $$\triangle = \begin{bmatrix} p& 1 & \frac{-q}{2}{}\\ 1& 2 &-q \\ 2& 2 & 3 \end{bmatrix} = 0 $$ But when I am expanding the ...
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vote
2answers
502 views

How to multiply vector and matrix expressions involving transposes

How do i expand and simplify the vector expression $(\vec{a}-\vec{b})^T.(\vec{a}-\vec{b})$ = ? And if there are matrices A and B instead of vectors a and b, how do i multiply and simplify this ...
7
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1answer
2k views

Compactness of the set of $n \times n$ orthogonal matrices

Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.