# 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), ...

1answer
44 views

### How can one define the trace of a linear operator on any finite dimensional vector space, using the fact that $tr(A) = tr(P^{-1}AP)$?

Firstly, I had to prove that $tr(AB) = tr(BA)$ and deduce that the trace is an invariant of similarity i.e. that $tr(A) = tr(P^{-1}AP)$ for any $A$ and invertible $P$. I could prove the first part - ...
0answers
13 views

### Convert hermitian matrix to symmetric

Is there some simple transformation (or a simple way to find it) which would convert any given hermitian matrix $A$ to a symmetric matrix $B$ with the same spectrum as that of $A$ (so I guess that ...
0answers
24 views

### What is the error in Newton's Method for Matrix Inversion?

I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds. Suppose we have, as ...
0answers
48 views

### Invariant of matrix under elementary transformations

$\DeclareMathOperator{\rank}{rank}$ Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix  B = \begin{bmatrix} A & b \\ b^T & c ...
0answers
14 views

### when gives svd decomposition matrices,whose element are different from 0 and 1

i want to understand one basic question,what is a criteria that for given matrix,whose element are just only $1$ and $0$,it's svd decomposition gives me matrices, whose element are different from ...
2answers
36 views

### Linear Transformations: Scaling along the line $y=x$

What is the geometric meaning of scaling an object by a factor $k$ along the line $y=x$? What will be the shape of a square with vertices $(2,1)$, $(3,2)$, $(3,1)$, and $(2,2)$ if it is scaled by a ...
1answer
45 views

### Cancellation property in matrices.

I just found a question which is based on a doubt I have carrying for over 10 years. If $ACC^t=BCC^t$ : $C^t$ means transpose of $C$ Is $A=B$ $AC=BC$ Sorry if this is a trivial question. ...
0answers
36 views

### When is a matrix called well-ordered?

We have a quick question, looking for information and/or references links. Is there a more specific mathematical definition/criteria for when matrices can be called well ordered or totally ordered? ...
1answer
31 views

### Name of type of Matrix

I am struggling to remember the name of this type of Matrix. Could anyone assist?  \begin{bmatrix} 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 ...
5answers
159 views

### matrices $A$ of order $2 \times 2$ that satisfy the equation $A^2-5A+6I = O$

Find all matrices $A$ of order $2 \times 2$ that satisfy the equation $A^2-5A+6I = O$ My try:: Given $A^2-5A+6I = O\Leftrightarrow A^2-3A-2A+6I^2 =O$ So \$A=3I = \begin{pmatrix} 3 & 0\\ 0 & ...
0answers
15 views

### How will an orthogonal array look for 3 levels and 3 factors?

I understand that an orthogonal array with 3 factors (parameters) will have 3 columns and if there are 3 levels, then it means each parameter can have 3 values. However, when using a selector from ...
1answer
59 views

### Prove that $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is positive definite

How to prove if A is a positive definite matrix, then $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is also positive definite? Here $X'$ denotes the transpose of $X$. $A$ is square and $X$ is \$n\times ...
1answer
25 views

### On triangular decomposition of square matrix

Let $L\in Gl_n(\mathbb{C})$ and define $A=LL^*$. Let us consider another decomposition such as $A=L_1L_1^*$. What is the relation between $L$ and $L_1$. One obvious relation is $L_1=LU$ where $U$ is ...
2answers
84 views

### The relationship between eigenvalues of matrices $XY$ and $YX$

If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove that $\lambda (XY) = \lambda (YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$? Here, ...
1answer
47 views

### Linear algebra, Schur set

Can you guys give me some hints on how I can start this problem? Thanks in advance! Let $U(n) \subseteq M_n(\mathbb C)$ be the set (group) of all $n \times n$ unitary matrices. Let \$ T ...
1answer
19 views

### Expressing a transformation matrix

Let $B=\{v_1,...,v_n\}$ and $C=\{w_1,...,w_n\}$ be bases to $V$. Suppose: $w_i=m_{i1}v_1+...+m_{in}v_n$ for $m_{ij}\in F, 1\le i,j \le n$. $M$ is an invertible matrix whose ($i,j$) member is $m_{ij}$. ...
2answers
66 views

### A matrix is normal, if and only if?

Let $A \in M_n(\mathbb C)$. Let $\langle \; \cdot\; , \; \cdot\; \rangle$ be the standard inner product in $\mathbb C^n$, viewed either as row vectors or as column vectors. Let $r_j$ be the $j$-th ...
0answers
47 views

### Name of a special matrix

I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like A = \left[ \begin{array}{c c c c} a & b & c & d \\ e & f & g & c \\ h ...
2answers
28 views

### Is the linear dependence test also valid for matrices?

I have the set of matrices $\begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}$ \$ \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ ...
1answer
39 views

### How would I find this eigenvalue?

I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on ...
1answer
75 views

### Limit of matrix powers.

Consider an arbitrary matrix $A$ with eigenvalues within the unit circle. Is there a nice formula for $A^\infty = \lim_{n \rightarrow \infty} A^n$? In particular, maybe there is a formula which ...
1answer
53 views

### problem with 4x4 matrix with big elements

I have a homework for my linear algebra class at my university the thing is that we get a 4x4 matrix A then we have to find it's Transpose which is pretty easy and then find the matrix B=(A^T)*A also ...
0answers
16 views

### Geometric effect of homogeneous transformation

Describe the geometric effect of the following homogeneous transformation matrix: \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 ...
2answers
26 views

### Quick question about proofs of theorem concerning Jordan basis

I have a question about proofs of this theorem: Let $K$ be an algebraically closed field, $V$ be a finite-dimensional space over $K$ and $f : V → V$ be a linear operator. Then there exists a Jordan ...
3answers
66 views

### Matrix being not diagonalizable in F2

We were talking about how the symmetric matrix $A=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ is not diagonalizable in the field consisting of only 0 and 1, since the eigenvalues are 0 and ...
1answer
70 views

### Condition number of a $9\times9$ matrix

would like someone to look over this and assure me I'm not making a silly mistake.... Given a $3\times9$ matrix $V$:  \small\begin{bmatrix} 1.0814 & -0.1251 & -0.1726 & -1.4443 & ...
1answer
53 views

### Linear Algebra: Least-Squares Approximation & “Normal Equation”

I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
1answer
32 views

### I need to diagonalize this matrix but I'm not sure it can be

This is the matrix I need to diagonalize: $A=\left[\begin{matrix}3&2\\0&3\end{matrix}\right]$. So I found the eigenvalue by taking the determinant of $(A-\lambda I)$ and solving for ...
1answer
46 views

### Find all eigenvalues and corresponding eigenvectors for the matrix?

Find all eigenvalues and corresponding eigenvectors for the matrix $$\left(\begin{array}{cr} 0&-1 \\ 2&3 \end{array}\right)$$ Not looking for a answer but i dont know what a "eigenvalues" is ...
1answer
43 views

### How to combine covariance matrices?

I have a data set of points in three dimensions. I'm calculating the barycenter (mean) and $3\times3$ covariance matrix from this data set. I store the average, the $3\times3$ matrix (where really ...
0answers
19 views

### conditional matrix form

I am deriving a matrix form for part of an equation which demands a conditional form but I have trouble in making it so that to be acceptable in scientific communities. Let's assume that we have ...
2answers
34 views

### Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
1answer
38 views

### how to calculate the “variance OF the covariance” matrix : E[vech(x x') vech(x x')'] for normal distributed x?

Supposing a vector x follows normal distribution. I want to calculate the expectation of the "variance Of the covariance matrix" (not variance-covariance matrix) in a vector form, meaning E[vech(x ...
0answers
54 views

### Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
2answers
50 views

### Matrix Algebra (Elementary)

I have $\hat\xi =\lambda_1\textbf{1V}^{-1} + \lambda_2\textbf{rV}^{-1}$ and sub it in to my two constraints, namely, $\xi\textbf{1}^T = 1$ and $\xi\textbf{r}^T = \mu$. My lecture notes then say set ...
1answer
63 views

### Eigenvector with eigen value of 1

How is an eigenvector with eigen value of 1, say v, multiplied by its transpose the identity matrix? v' * v = I?
1answer
120 views

### How to prove that $D := ABC$ is also positive definite?

Let $A,$ $B$ and $C$ be symmetric, positive definite matrices and suppose that $D := ABC$ is symmetric. How might I prove that $D$ is also positive definite?
2answers
49 views

### Volume of a parallelepiped, given three vectors

I want the volume of a parallelpiped and I have the three vectors $$4e_1+2e_2-e_3$$$$e_1-3e_2-2e_3$$$$2e_1-e_2+3e_3$$ that coinciding with three of the parallelpipeds sides. HON-base I made it into a ...
1answer
43 views

### Is the sum of two singular matrix also a singular matrix

If $A$ and $B$ are singular and both $n\times n$, is $A+B$ also singular?
1answer
42 views

### Proof of adjoint(ab) = adjoint(b)adjoint(a)

So I'm trying to prove whether adjoint(AB) = adjoint(B)adjoint(A). My tactic is to look at the ijth entry of both matrices and compare, but the fact that AB is being transposed is confusing me. ...
0answers
26 views

### Subgradient of matrix $l1$-norm

Let $X$ be a square matrix, what is the sub-gradient of $f(X) = ||AX-XB ||_{l1}$? $A$ and $B$ are both constant matrix. I am very confuse about the chain rule on matrix derivatives.
2answers
48 views

### Identifying matrix vector multiplication

I have the following question in a book: According to the book, the answer is (D). But I don't understand how. Isn't this just scalar multiplication? The solution in the book says that I have to ...
1answer
32 views

### finding the P matrix (diagonalization of a matrix)

I'm trying to find the diagonalization of a matrix : this is my matrix :  A =\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 1 & 1 & 1 \\ ...
1answer
43 views

### diagonalization of a matrix

I want to diagonalize this matrix : $$A =\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 1 & 1 & 1 \\ \end{pmatrix}$$ the first step I ...
1answer
23 views

### Definition of minimal and characteristic polynomials

I have defined the characteristic and minimal polynomial as follows, but have been told this is not strictly correct since det$(-I)$ is not necessarily 1, so my formulae don't match for $A=0$, how can ...
1answer
63 views

### Get code words from generator matrix

I have some issue regarding the generator matrix. Please can some body can explain me "How to get Codebook from Generator matrix?" Following is my issue Generator matrix has 3 code words. Then ...
1answer
17 views

### determining the associated matrix has an endomorphism of R ^ n.

What is the method for determining the associated matrix has an endomorphism of R ^ n. for example: f is defined on i³ with : ∀(x,y,z) ∈ i³ , f(x,y,z) = (x+2z, 2y, 2x+z) The canonical basis of i³ ...
1answer
44 views

### Solution of a system of linear equations with n variables

I have a system of linear equations with n variables \begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = \frac{1}{2}x_1\\[4pt] a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = \frac{1}{2}x_2\\[4pt] ...
0answers
20 views

### Improve Upper Triangular Matrix Computation for Gauss Jordan method?

I am solving Simultaneous Equations using the Gauss Jordan method. I am having a problem in computing the Upper triangular Matrix with sufficient accuracy for no of variables >50. Some of the elements ...
2answers
61 views

### Rank of the difference of matrices [duplicate]

Let $A$ and $B$ be to $n \times n$ matrices. My question is: Is $\operatorname{rank}(A-B) \geq \operatorname{rank}(A) - \operatorname{rank}(B)$ true in general? Or maybe under certain assumptions?
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