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
1answer
226 views

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

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
2
votes
3answers
154 views

Square root of a $3\times3$ matrix

Here is $3\times3$ matrix$$\begin{pmatrix} 0& 0& 1\\ 0 & -1 & 0\\ 1& 0 & 0\end{pmatrix}$$ How can I solve this by using Cayley-Hamilton? I know how to ...
0
votes
0answers
53 views

Eigenvalues of a matrix formed by derivatives of a polynomial

Let $f=f(x,y)$ be a polynomial with $x\geq 0,\ -2x\leq y\leq -x/2$ (so when $x=1, -2\leq y\leq-1/2$). Denote \begin{equation*} \begin{split} f_1=&\frac{\partial \ln f}{\partial x}(1,y),\\ f_2=&...
0
votes
1answer
26 views

Hesse-matrix notation

I'm currently trying to solve an exercise and I'm having a notational issue here. Maybe someone knows that kind of notation and could help me out. If we want to compute $$\frac{\partial}{\partial t} ...
0
votes
2answers
145 views

Linear combinations of three-dimensional vectors

There is a three-dimensional vector $v$. Show that $v$ can be expressed as a linear combination of $v_1$, $v_2$, and $v_3$, where $v_1 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$, $\quad v_2 = \begin{...
9
votes
1answer
5k views

Evaluating eigenvalues of a product of two positive definite matrices

Let $A,B\in M_n(\mathbb{R})$ be two symmetric positive definite matrices, i.e.: $$\forall x\in\mathbb{R}^n, x\neq 0, (Ax,x)>0, (Bx,x)>0,$$ where $(\cdot,\cdot)$ is the usual scalar product in $\...
0
votes
1answer
37 views

Has anyone seen $N(M)$ in the context of linear algebra/matrix notation?

To be honest, I can't shed too much context other than that it's related to matrices, as this is some feint memory for a test I did a few months ago. I wrote down after the test certain notation ...
1
vote
0answers
84 views

linear independence of normal matrix's eigenvectors

We know that a normal matrix will have, or can be made to have (by orthogonalization, if not all its eigenvalues are distinct), orthogonal eigenvectors. This means that they are also linearly ...
2
votes
3answers
703 views

What is Homogeneous Coordinates? Why is it necessary in 2D transformation?

What is Homogeneous Coordinates? Why is it necessary in 2D transformation of objects in computer graphics? The concept of homogeneous coordinates in effect converts the 2D system a 3D one. So, why ...
1
vote
1answer
70 views

How can I compute higher powers of a large matrix quickly?

Is there a block matrix technique, assuming that the matrix has a lot of zeroes in it? I want to compute its nilpotence degree. Thanks, (Even with a lot of zeroes in the matrix, there's still a lot ...
4
votes
2answers
75 views

Equality of two matrices

If we have a diagonal matrix D which verify $D = A^*MA = B^*MB$ where $^*$ denotes the conjugate transpose, with A, B and M being unitary matrices Plus, B is symetric, M is real, and none of these ...
4
votes
1answer
517 views

What's the solution to $X = AXA$?

I have given a general $n \times d$ matrix $B$, and need to compute two similarity matrices $R \in \mathbb{R}^{n \times n}$ and $S \in \mathbb{R}^{d \times d}$ as follows: $R = BSB^\top$ and $S = B^\...
1
vote
0answers
28 views

Number of square matrix

Let $A$ and $B$ be $n\times n$ complex matrices. What is upper bound for the number solutions to $B^2=A$ if you know that is finite for a given matrix $A$. The solutions are given here Linear ...
2
votes
0answers
46 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
1
vote
1answer
74 views

If $A$ is nipotent, how to prove that $A+A^*$ is not nilpotent?

If $A\neq0$ is nipotent, how to prove that $A+A^*$ is not nilpotent? $A,A^*$ are nilpotent, but I have no idea how to continue
2
votes
1answer
44 views

Analytical Solution to a Eigenvalue-like Problem

In my research, the stationary condition of my optimization problem is the following: $\frac{1}{2} C_1 B C_2 = \lambda B$. $C_1 \in R^{m \times m}$, $C_2 \in R^{n \times n}$, $B \in R^{m \times n}$. ...
0
votes
1answer
30 views

Image and Kernel of matrices that have no rows of zeroes

Suppose we have two matrices $A_T, A_V \in \mathbb{R}^{n\times s}$ with $Image(A_T) = Image(A_V)$ and both have no rows of only zero elements. The image of the matrices are generated by the non-...
3
votes
2answers
77 views

Sum of identity and idempotent (projection) matrix

Let $P$ be an idempotent $n \times n$ matrix ($P^2 = P$). What is $(I + P)^{-1}$? I've been thinking about this problem for a while, but can't find an answer. I tried a few examples, but I'm not sure ...
1
vote
1answer
55 views

Prove or disprove: $|\det(Q)|=1 \Longrightarrow Q$ is unitary.

I wonder whether the statement of above can be written as an equivalence. So far I could prove the other direction $(\Longleftarrow)$: If $Q$ is unitary, then $1=\det(I)=\det(Q^HQ)=\det(Q^H)\det(Q)=...
2
votes
0answers
35 views

positiveness of product of matrix

If $A$ is a positive definite matrix, B is not sure but $tr(B)>0$ where $tr$ is trace, will $tr(AB)>0$ ? that is the trace of the product of those two matrices. B is not a diagonal matrix or ...
0
votes
2answers
48 views

Is the row echelon form of a system of equations unique?

I'm talking about the row echelon form, not the reduced row echelon form. If it isn't can you give me some examples?
0
votes
1answer
87 views

eigenvectors that span to the same eigenvalue

I recently solved a problem in which i found a matrix to have four eigenvalues, three eigenvalues of 2 and one eigenvalue of 3. What is the significance of having repeating answers of eigenvalues. Don'...
1
vote
0answers
34 views

Find the term p with matrix .

how I can solve this. Find $p$. Let $x,y,z,p\in\mathbb{R}^{n}$ and $a,b,c$ are constants, such that $$2\langle p,x-y\rangle=(\vert\vert x\vert\vert^2-\vert\vert y\vert\vert^2)-(a^2-b^2)$$ $$2\langle ...
2
votes
5answers
263 views

Rotation matrix

HI I am wondering if there is a unique matrix that maps $(x_1,y_1,z_1)$ into $(x_2,y_2,z_2)$. These two vectors have equal magnitude and are defined in orthogonal 3-D basis. If there is a unique ...
1
vote
6answers
111 views

If the product of two matrices, $A$ and $B$ is zero matrix, prove that matrices $A$ and $B$ don't have to be zero matrices

I can give an example where product of two non-zero matrices is zero matrix, $$ A= \begin{bmatrix} 3 & 6 \\ 2 & 4 \\ \end{bmatrix} $$ $$ B= \begin{...
2
votes
0answers
59 views

Bounds on the product of a matrix exponential and a vector

I have a control system with a state matrix $S = -B^{-1} A \in \mathbb{R}^{n\times n}$, where: $B$ is a strictly positive diagonal matrix $A$ is positive definite $M$-matrix I know that all the ...
0
votes
2answers
31 views

Matrix Algebra to trace workflow precedents

what I'm trying to achieve is the following. I have a production chain, say for example: ...
0
votes
0answers
679 views
1
vote
1answer
694 views

A proof that an orthogonal matrix with a determinant 1 is a rotation matrix

Reading proof(starting on page 5) for item 1 of "Rotation Matrix Theorem" in this doc i'm stuck at understanding its last step. Matrix A being an orthogonal Matrix, at this step the conclusion that A ...
2
votes
2answers
122 views

If $A$ is an $m\times n$ matrix, $B$ is an $n\times m$ matrix and $n<m$, then $AB$ is not invertible.

The question was given in the early chapters of Linear Algebra by Hoffman & Kunze, so I am trying to give a proof with only the tools given to me so far - which are mainly row reduction and ...
1
vote
1answer
33 views

Singular values and singular vector pair for the matrix

What are the singular values and singular vector pairs for a matrix $1_{1\times q}\otimes w_{p\times 1}$? Here $1_{1\times q}$ is the row vector of all ones, $w_{p\times 1}$ is an arbitrary column ...
0
votes
2answers
48 views

Question about a “sort of” skew symmetric matrix

it has main diag elements = 0, non-zero off diag elements like A12 = 1/A21 etc. That's why I called it "sort of" skew symmetric in the title. What is this type of matrix called and where can I learn ...
3
votes
2answers
40 views

Bases s.t. matrix for $T: V \rightarrow W$ is diagonal

Let $T: V \rightarrow W$ be a linear map between finite-dimensional vector spaces. Show that there exist bases ${e_i}$ of $V$ and ${f_i}$ of $W$ such that the matrix of $T$ has entries $\alpha_{i,j} = ...
1
vote
0answers
19 views

Condition of the matrix

I have a problem is find the condition $$K_\infty (B_n)$$ where $$ B_n = \begin{bmatrix} 1 & -1 & -1 & \cdots & -1 \\ 0 & 1 & -...
1
vote
1answer
76 views

How can we classify the eigenvalues of this 3x3 matrix?

Suppose we have the matrix $$ \left( \begin{array}{c c c} -(A+B) & A & 0 \\ C & -(C+D) & 0 \\ 0 & E & -F \\ \end{array} \right) $$ where $0<A,B,C,D,E\in \mathbb{R}$. I am ...
8
votes
1answer
972 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
1
vote
1answer
124 views

Verifying orthogonality between two binary sequences

I have studied that for orthogonality to exist between two binary sequences: [Number of bit agreements - Number of bit disagreements]/sequence length=0 Eg, for an orthogonal matrix X given by: \...
1
vote
1answer
113 views

Matrix integration by parts

It seems to me that the integration by parts rule carries over simply to the matrix case. This can be seen by applying: $(AB)' = A'B + AB'$ and then integrating for square (time dependent) complex ...
0
votes
1answer
139 views

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
8
votes
4answers
3k views

Determinant of rank-one perturbations of (invertible) matrices

I read something that suggests that if $I$ is the $n$-by-$n$ identity matrix, $v$ is an $n$-dimensional real column vector with $\|v\| = 1$ (standard Euclidean norm), and $t > 0$, then $\det(I+tvv^...
2
votes
1answer
68 views

show that a matrix is invertible

Let $A$ be an $n \times n$ matrix such that $|a_{ii}|>\sum_{j=1,j\neq i}^n|a_{ij}|$ for each $i$. Show that $A$ is invertible. $(complex matrix) The straight forward way is to show that the ...
1
vote
1answer
16 views

Proving that $L_{22}L_{22}^T=S$ is the Schur complement of a cholesky factorization

Let $A$ be an $n+m \times n+m$ symmetric positive definite matrix. $A=\begin{bmatrix}A_{11} & A_{12}\\ A_{12}^T & A_{22}\end{bmatrix}$ where $A_{11}$ is an $n \times n$ matrix, $A_{12}$ is an ...
0
votes
1answer
19 views

Given a parametric solution $\vec{x}(t)$ to $Ax = b$, how can I choose the parameter $t$ so that all entries in $\vec{x}(t)$ is between 0 and 1?

Given a solution to the matrix equation $A\vec{x} = \vec{b}$ on the form $\vec{x}(t)$, how can I choose the parameter t such that all entries in $\vec{x}$ are squeezed between 0 and 1? That is, for ...
1
vote
0answers
38 views

Decomposing a matrix as the product of rotations

I'm reading an article about joint diagonalization algorithms. The article states without proof that any nonsingular $n \times n$ matrix $Q$ can be decomposed as \begin{align*} Q = \prod_{1 \leq p <...
0
votes
0answers
258 views

How to solve series of 8 equations with 8 unknowns?

In this article http://www.fmwconcepts.com/imagemagick/bilinearwarp/FourCornerImageWarp2.pdf they speak of solving for a0,a1,a2,a3,b0,b1,b2,b3 but I want to know ...
0
votes
1answer
45 views

How to calculate projection matrix for quadrilateral transform?

I have a square and its 4 corner coordinates $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. And I have a quadrilateral with corner coordinates $(x_1',y_1'),(x_2',y_2'),(x_3',y_3'),(x_4',y_4')$ where $(x_i,...
2
votes
2answers
64 views

$rk(A^2)=rk(B^2) \implies rk(A)=rk(B)$ is it true?

The original statement is this: given A,B matrices $n \times n$, if $A^2$ is "Left-Right equivalent" to $B^2$ then A is LR equivalent to B (is it true or false?) I know that A is LR equivalent to B ...
0
votes
1answer
65 views

Find the minimum of a function for only positive values of the vector variable

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, \...
0
votes
1answer
64 views

looks like Vandermonde determinant [duplicate]

Calculate the determinant of $M = \left( {\begin{array}{*{20}c} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{array}} \right)\;$. How can one calculate this? Is there a ...
0
votes
1answer
47 views

What is this matrix doing

I am trying to find out what this matrix is doing. I am trying to follow the guide: https://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/2D_affine_transformation_matrix.svg/512px-...