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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3
votes
2answers
34 views

If det $A = 0$ and $\det B \neq 0$ then show that $abc = -1$

This has been hurting my head for a while now.... If $$ \det\begin{bmatrix}a&a^2&1+a^3\\b&b^2&1+b^3\\c&c^2&1+c^3\end{bmatrix}=0 $$ And $$ ...
0
votes
0answers
5 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_{1}|\nu_{1}>+\lambda_{2}|\nu_{2}>$$ and $$\sigma=\gamma_{1}|\omega_{1}>+\gamma_{2}|\omega_{2}>$$ ...
0
votes
0answers
12 views

A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...
6
votes
2answers
129 views
+50

Determinant of a Certain Block Structured Positive Definite Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
1
vote
1answer
19 views

Find the maximum value of this form

Let $A,B$ be $n\times n$ real symmetric matrices such that $B$ is positive definite. Show that $G$ defined below attains a maximum value at an eigenvector related to $A$ and $B$. Also find the ...
0
votes
1answer
45 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
2
votes
1answer
41 views

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row?

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row(uniform distribution)? What sort of algorithm should I use to do this task? Brute Force algorithm- ...
2
votes
1answer
31 views

Scaling a svg image while keeping the offset position.

I have an svg image of a map that i have to scale up to make it zoom in. Javascript has a function to scale up SVG images. However the svg scale function uses the upper left corner as center when ...
0
votes
2answers
43 views

For what values of $a$, $b$, and $c$ the above system has: One solution. Infinitely many solutions. No solutions.

I am stuck with this now, I tried reducing the matrix to row echelon form, but it gets a bit hard. Is there not a simpler way? The system is: \begin{align*} a x + b y − 3 z &= −3\\ −2 x − b y + ...
0
votes
0answers
15 views

Multivariable polynomial matrix representations

This is a follow-up to matrix representation of parabola and matrix representation to generate monomials. I found a method to build such matrices to implement this type of functionality for one ...
0
votes
0answers
14 views

What is pseudospectra of matrix polynomials? .

What is pseudo spectra of matrix polynomials? Please guide me with some example or some reference regarding it. Thank You!
0
votes
1answer
142 views

parabola in homogeneous coordinates

So if I have the parabola $Y = X^2$, how do I go about representing this homogeneously? I know I can parameterize it as $F(t) = (t, t^2)$, but then what? The reason I ask is because I have a ...
1
vote
3answers
39 views

Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
0
votes
3answers
34 views

Matrix Multiplication: Both ways okay?

Say I have two matrices $A$ and $B$ where $A$ has dimensions of $1 \times 2$ ($1$ row, $2$ columns) and $B$ has dimensions of $2 \times 3$ ($2$ rows, $3$ columns) When you multiply these like so $(A ...
0
votes
0answers
7 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
0
votes
0answers
13 views

PCA of the large symmetric almost-diagonal matrix

I was doing factor reduction of the correlation matrix of the special form $\rho_{ij}=\rho+(1-\rho)e^{-\beta |i-j| }$, with $i,j \le n=100$, $\rho \ll 1$ and $\beta \le 1 $. $$ \begin{bmatrix} ...
1
vote
1answer
404 views

Determinants of Matrices det(4A) equals? [duplicate]

Suppose A is a 4 x 4 matrix such that $\det(A) = 1/64$. What will $\det(4A^{-1})^T$ be equal to? Here's my thinking, $\det(A^T) = \det(A)$ I has no effect on the determinant. And $\det(A^{-1}) = ...
-4
votes
2answers
54 views

Crout matrix decomposition [on hold]

In naive terms and step by step, how to to find the determinant of any NxN matrix by using LU Decomposition of Crout's method. Also, discuss its efficiency as compared to other LU decomposition ...
1
vote
1answer
27 views

Determine the isomorphism class of $\mathbb Z^3 / M$ for the subgroup $M$ of $\mathbb Z^3$generated by $(13,9,2),(29,21,5),(2,2,2)$

The problem seems not so hard. My confusion rise from the statement in the solution above that "This question is equivalent to reducing the matrix via row and column operations". Please see the ...
57
votes
4answers
76k views

Is a matrix multiplied with its transpose something special?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^T$ something special for any matrix $A$?
10
votes
3answers
171 views

Find this Determinant

I have to find this determinant, call it $D$ \begin{vmatrix} \frac12 & \frac1{3}& \frac1{4} & \dots & \frac1{n+1} \\ \frac1{3} & \frac14 & \frac15 & \dots & ...
0
votes
1answer
290 views

Computational complexity of Gaussian elimination

If it took me approximately 4 minutes to solve an equatian $Ax=b$ for $x$ (where $A$ is a $3\times3$ matrix and $b$ is a $3\times1$ matrix) using Gaussian elimination, how much longer would it take me ...
1
vote
1answer
43 views

Consequences of the positivity condition $v^t A v > 0$ for the eigenvalues of $A$

Let $A$ be an $n \times n$ symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive ...
1
vote
1answer
23 views

Transition matrix question,

In diagonalizing a matrix A, we use a matrix S, which consists of eigenvectors of A. To compute S, we simply take each eigenvector and write it as a linear combination of the standard basis. So if ...
0
votes
1answer
14 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, ...
0
votes
0answers
27 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
4
votes
1answer
64 views

Calculating the rank of a Boolean matrix and Boolean matrix factorization

I am interesting in some sort of algorithm for calculating the Boolean rank of small $M \times N$ Boolean matrices. Just to be clear, by Boolean matrices I mean matrices with entries $0$ or $1$ where ...
0
votes
2answers
341 views

Rank of a Matrix and Echelon Form to determine ranks.

What is the meaning rank of a matrix in terms of vectors, and how does Echelon form work in determining the rank of a matrix?
0
votes
1answer
93 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 ...
1
vote
6answers
92 views

Gradient of $x^{T}Ax$

I just came across the following: $\nabla x^TAx = 2Ax$ Which seems like as good of a guess as any, but it certainly wasn't discussed in either my linear algebra class or my multivar calc class. Is ...
-4
votes
1answer
18 views

double summations

Assume that w1=0.4; w2=0.5; w3 =0.1. Basing on the following matrix that provides values for xij : 7 4 9 6 4 12 3 2 17: Calculate the following value: 3 3 ∑∑wiwjXij i=1 ...
0
votes
2answers
23 views

Deriving a Formula for the determinant of a block matrix.

This is a follow up question to this. I want to solve the following problem: Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space ...
0
votes
3answers
41 views

Book for Linear Algebra and Matrix

my major is Electrical Engineering and I am new in linear algebra and I need to be familiar with matrix theory deeply because of my research topic which is Image Processing. But, I do not know from ...
3
votes
3answers
180 views

matrix representations and polynomials

I just investigated the following matrix and some of its lower powers: $$M = \left[\begin{array}{cccc} 1&0&0&0\\ 1&1&0&0\\ 1&1&1&0\\ 1&1&1&1 ...
1
vote
2answers
763 views

Limit of a sequence of matrices

I'm preparing or my exam in linear algebra and I'm stuck with a question. I've tried to find some help in my textbook (Linear Algebra and its applications, 4th Edition, By David C. Lay). I can't find ...
6
votes
1answer
51 views

Upper bound on the distance of orthogonal matrices

Dear math stackexchange users, I have a question on orthogonal matrices: suppose I have a matrix $X\in\mathbb{R}^{n\times n}$ and I consider the orbit of the orthogonal group $O(n)$ acting from the ...
8
votes
1answer
206 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
1
vote
1answer
43 views

When is the matrix $\mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T}$ a symmetric matrix?

let $\mathbf{A}\in\mathbb{R}^{m\times n}$ and $\mathbf{x}\in\mathbb{R}^{n\times 1}$. \begin{equation} \mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T} \end{equation} Can we say that ...
1
vote
1answer
22 views

Proving Properties of Discrete Time Markov Chain mathematically

I want to prove that the queue length at a store is not a Discrete Parameter Markov Chain (DPMC). Now I have the equation: $$Q_k = (Q_{k-1} - 1) + V_k$$ $Q_k$ is the queue length at time instant ...
2
votes
1answer
85 views

A conjecture about the eigenvalues of symmetric pentadiagonal Toeplitz matrix

Is there a way to find out the exact eigenvalues and eigenvectors of a real symmetric pentadiagonal Toeplitz $n\times n$ matrix with the form given below? $$ \begin{pmatrix} a & b & c & 0 ...
0
votes
1answer
36 views

Matrices inside matrix. Showing $det(M)=det(C)$

Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space n_1+n_2=n$ $$M=\begin{pmatrix}E_{n_1}&B\\O&C\end{pmatrix}$$ where $E_{n_1} ...
0
votes
0answers
6 views

What is pseudospectra of matrix polynomials?

What is pseudospectra of matrix polynomials? Please guide me with some example or some refrence regarding it. Thanks.
64
votes
10answers
16k views

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it ...
0
votes
1answer
31 views

Question on normal matrices

Hello all I was given this question in my linear algebra class which I have tried to solve but to no avail, and I would really appreciate any help. I am given a matrix $ A \in M_{nxn}(C) $ and am ...
1
vote
4answers
43 views

How to find a real matrix with complex eigenvalues,

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$. I would like hints only. So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which ...
1
vote
3answers
78 views

Eigenvalues of matrix $A^TA+I$ are real and greater than 1?

In this paper, the author states that the eigenvalues of the matrix $A^TA + I$ are real and greater than 1, since $A^TA$ is symmetric positive definite. But why?
1
vote
0answers
28 views

Is it okay to perform the same row operation twice on opposite rows?

I am trying to find the inverse of the following matrix: 1 2 3 2 1 4 1 0 2 I draw the identity matrix next to it and start performing row operations. ...
1
vote
2answers
27 views

Get the camera transformation matrix (Camera pose, not view matrix)

Let's say that I have an object and a camera (its representation) in a 3D world coordinate system. I have the camera pose to see the object (rotation matrix and translation (eye position)). If I apply ...
0
votes
1answer
19 views

Bounding the smallest eigenvalue of symmetric matrix product

Let $X = ABA^T$ where $B \in \mathbb{R}^{p \times p}$ and $B$ is positive definite matrix and $A \in \mathbb{R}^{q \times p}$ so that $X \in \mathbb{R}^{q \times q}$. My question is concerning an ...
2
votes
2answers
310 views

Derivative of Hadamard product

What is the derivative of Hadamard product of two matrices with respect to one of them? I.e. what is $D(AB)$ with respect to $A$?