# Tagged Questions

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), ...

48 views

### To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz [duplicate]

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 ...
190 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 ...
140 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 semi-...
261 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 ...
119 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 $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space n_1+n_2=n$...
101 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 ...
1k 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 ...
232 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 ...
47 views

41 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 ...
56 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 ...
106 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?
260 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 ...
48 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 ...
72 views

289 views

112 views

25 views

### How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
140 views

### Practical examples/implementation details for Gauss-Seidel method

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with n>=1000, ...
38 views

### AAK theorem for finite dimensional Hankel matrix

Does the AAK theorem hold for finite dimensional Hankel matrix? Or maybe similar analysis exists? (From a quick look of the proof, it seems like the AAK solution has to be infinite dimensional ...
72 views

### Given a matrix M, is there a name for the matrix MM^T?

One can make a symmetric square matrix out of any m-by-n matrix $M$ by computing the matrix $MM^T$ (or $M^T M$). Is there a name for this operation? I want to call it "symmetrizing" the matrix, but I ...
63 views

### transofrmations (a,b,c) to (x,y,z)

I'm not 100% sure linear algebra will crunch this problem, but hopefully so. This may just be a case of matrices, which would be good cause I like those. Imagine we have a robot with a camera ...
53 views

52 views

127 views

### why similarity over $\bar{\mathbb{F}}$ of $A,B\in M_n(\mathbb{F})$ implies similarity over $\mathbb{F}$?

A classic problem in linear algebra is to determine if two matrices $A,B\in M_n(\mathbb{F})$ are similar one to another. When $\mathbb{F}=\bar{\mathbb{F}}$, we know that $A,B$ are similar if and only ...
24 views

### For which value of m are these 3 vectors linearly dependent?

In one of my revision worksheets there is a question which goes as follows: The vectors u=mi+j+k, v=i+mj+k and w=i+j+mk, where m is a real constant, are linearly dependent for either m=0, m=1, m=2, m=...
Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy$$ How can I find the matrix $A$ that maximizes expression \$K(A)...