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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0
votes
1answer
54 views

Matrix addition and eigen values/vectors

If I start with matrix A given by $A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and I express it as a sum $A = \begin{bmatrix} w & x \\ y & z ...
1
vote
1answer
28 views

Remainder modulo matrices.

Is it possible to have a consistent definition of $A\bmod_{left} B$ that respects matrix multiplication from left where $A$ and $B$ are two square matrices with non-negative entries? Is there a ...
0
votes
1answer
26 views

Regarding element-wise derivative of matrices

Let $X=(x_{ij})$ be a real n-by-n matrix where $x_{ij}$ are in a range such that $X$ is insvertible. Let $Y=X^{-1}=(y_{pq})$, and regard $y_{pq}=y_{pq} (x_{11},x_{12},\ldots,x_{nn})$ as a function of ...
2
votes
2answers
17 views

Dumb question: Orthogonal complement of kernel = Row space

I'm really confused when trying to prove the following: $\mathrm{kern}(A)^\perp = \{y \in \mathbb{R}^n \mid y = z^T A, \ z \in \mathbb{R}^n \}$ The $\supseteq$ direction is easy: Let $z^T A \in$ ...
3
votes
2answers
95 views

What kind of $n^{th}$ order polynomials are solvable by a square matrix with integer entries?

Consider a polynomial (monic for simplicity): $$x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ Here we assume the roots are complex numbers. $a_k$ are integers. Now consider the corresponding matrix ...
3
votes
1answer
72 views

Is there a name for matrices that are symmetric along the cross diagonal? [duplicate]

Something like $$ A= \begin{bmatrix} a & b & c\\ b & d & e\\ c & e & f \end{bmatrix} $$ would be a symmetric matrix because the values are reflected along the ...
1
vote
3answers
118 views

Prove that there are not two matrices 2x2 such that: $AB-BA=I_2$

I tried this question by multiplying explicitly the matrices but I think I'm not getting anything, so I think, well let's suppose false so $C(AB-BA)=C$ and find a contradiction but also I'm not ...
23
votes
7answers
2k views

Is there such a thing as a matrix of functions?

Do we ever put functions as entries of a matrix? If so, are these matrices used in linear algebra or do they have some other special use? There have been minor not neccessarily conflicts per se, but ...
6
votes
0answers
93 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
0
votes
1answer
47 views

Matrix with all entries N

Is there a specific name for a matrix where all entries are the name number? I am writing a program where I want to be able to describe a matrix like this in the same way I would the identity matrix, ...
10
votes
3answers
102 views

Can one factor matrices?

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there ...
1
vote
2answers
49 views

For matrices $C, D$, show that $(CD)^{100} \neq C^{100} D^{100}$

The question is to prove this is false: $(CD)^{100} = C^{100}\cdot D^{100}$, where $C$ and $D$ are matrices. I looked through my textbook and could not find a proof for this.
0
votes
2answers
47 views

Orthogonal Matrix 4 [duplicate]

Let $M_{32}$ be vector space with inner product of $AB$ given by $\text{tr}(B^TA)$. The question is to find a non-zero matrix B orthogonal to $$A=\left[\begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 &...
1
vote
2answers
70 views

How to determine which of the following matrices are similar?

If we have the following three matrices: $$ A=\begin{bmatrix} 7 &1 \\ -5 &3 \end{bmatrix},\;\; B=\begin{bmatrix} 5 &-1 \\ 1 &5 \end{bmatrix},\;\; C=\begin{bmatrix} 5 &1 \\ 1 &...
0
votes
1answer
37 views

Orthogonal matrices 5

The question is to find in the space $\mathrm{Mat}_{3\times 2}(\mathbb{\mathbb{R}})$ a non-zero matrix that is orthogonal to $$ A= \begin{pmatrix} 1 & 2\\ 3 & 4\\ 5 & 6\\ \end{...
0
votes
0answers
8 views

lipschitz continuity on matrix product

If $ f(t,x)=A(t)g(t,x)B(t) $ where $ A(t), g(t,x), B(t)$ represents square matrix functions. If $ A(t), B(t)$ are bounded and $ g(t,x)$ is Lipschitz continuous. Then is it correct to consider $ |f(t,x)...
-1
votes
0answers
57 views

$2 \times 2$ block matrix related

let $A$ be any matrix of order $n$, $J$ is matrix of order $n$ whose all entries are $1$, and $I$ is an identity matrix of order $n$, then how to find eigenvalues of following block matrix? $$M=\...
0
votes
0answers
33 views

Inverse of the sum of identiy matrix and a symmetric matrix

Is there a simple way to solve $(I + A) X = B$, where $I$ is the identity matrix, and $A$ is a symmetric matrix?
0
votes
1answer
31 views

For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold?

For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold? Here the absolute value signs are the determinant ...
1
vote
2answers
29 views

Finding the limit of a Matrices determinant

The problem is as follows: I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
0
votes
0answers
28 views

First steps in derivation of matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
0
votes
1answer
38 views

Does encountering a zero pivot during Gaussian elimination imply that the matrix is singular?

I was reading a problem about Gaussian elimination and pivots of a matrix, say $A$. The question is: During the Gaussian elimination process without pivoting a zero pivot has been encountered. Is ...
1
vote
2answers
37 views

Can we define component of a matrix which is orthogonal to another matrix?

Given two vectors $A$ and $B$ one can easily find component of $A$ along $B$ and component of $A$ perpendicular/orthogonal to $B$ and vice versa. This is possible as we can define dot product of two ...
3
votes
3answers
95 views

For which $a$ and $b$ is this matrix diagonalizable?

For which $a$ and $b$ is this matrix diagonalizable? $$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$ How to get those $a$ and $b$? I calculated ...
3
votes
0answers
14 views

Skew-Symmetric Parts of Stochastic Matrices

It's easy to see that the set $\{W - W^T : W \in \mathbb{R}^{n \times n}\}$ is precisely the set of real skew-symmetric matrices. This continues to be the case if we restrict to (entry-wise) non-...
2
votes
2answers
82 views

Prove that the number: $z = \det(A+B) \det(\overline A-\overline B)$ is purely imaginary.

Problem: Let $A_{n\times n}$ and $B_{n\times n}$ be complex unitary matrices, where n is an odd number. Prove that the number: $$z=\det(A+B) \det(\overline A-\overline B)$$ is purely imaginary. My ...
3
votes
3answers
164 views

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \...
2
votes
0answers
53 views

Rotation Matrix which maps a point to an specific point

How can I compute the rotation matrix which rotates an $n$-dimensional vector $\vec{A}$ around an $n-D$ vector $\vec{O}$, and maps it to a vector $\vec{B}$ (while $\vec{A}, \vec{B}, \vec{O}$ are known)...
1
vote
1answer
36 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
1
vote
3answers
68 views

Uniqueness of solution for a tridiagonal system

I have a claim I've been conjecturing. Not sure if it's true or not. Context: I'm doing some calculations with finite difference schemes. Say I have the following real $n$ x $n$ tridiagonal matrix $A$...
-2
votes
0answers
17 views

Orthogonal matrix B [closed]

The question is to find a non-zero 3x2 matrix B which is orthogonal to A= (12 34 56) (this is a 3 by 2 matrix space represents columns) Any help would be ...
0
votes
1answer
25 views

Basis for a matrix [closed]

Find a basis for the space M32 ? (3 by 2 matrix). I want to know basis for a matrix in general rather than being given a matrix to work with. Any help would be appreciated.
0
votes
0answers
36 views

Finding an orthogonal matrix for a 3x2

I know how to find an orthogonal matrix for a $2\times2$ or $3\times3$ matrix. However I have been stuck on how to do this for a $3\times2$ matrix. The question is how to find a non-zero $3\times2$ ...
0
votes
0answers
16 views

Visualization of residual sum of squares in matrix notation

I am trying to understand how to pass from \begin{equation} RSS(\beta) = \sum_{i=1}^n (y_i - x_i^T\beta)^2 \end{equation} to \begin{equation} RSS(\beta) = (y - X \beta)^T (y - X \beta) \end{...
0
votes
0answers
9 views

Terminology for operation on matrices to check psd

This is just a question of terminology. For defining positive definiteness or negative definiteness of a square $n\times n$ matrix $A$ (say if all entries of $A$ are real numbers) one asks whether $...
2
votes
2answers
42 views

Diagonalizable by orthonormal matrix

Given the matrix $$A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$ Explain why $A$ can be diagonalized by an orthonormal matrix and find an ...
0
votes
0answers
25 views

How to represent the summation in matrix equation

I have two vectors $a=[a_1, a_2,..a_n]$ and $b=[b_1,b_2,..b_n]$. A vector $c=[c_1...c_n]$ where $$c_i=\sum_{j=m}^l\alpha_j a_j+\sum_{j=k}^ h\beta_j b_j$$ In which, $i=1, \cdots, n; m,k \ge 1, m \le l ...
0
votes
1answer
41 views

Matrix power relation to prove coefficients exist

Let M be a 2x2 Matrix and $n$ an integer $2\geq\ n$. Prove that there exist integers $a_n$ and $b_n$ such that $$M^n=a_nM+b_nI$$ To begin, what I did was to use the Cayley-Hamilton Theorem then use a ...
3
votes
1answer
60 views

Binary matrices with rank $n$

I'm stuck doing this problem Let $A$ be a matrix of order $n \times n$ with entries in $\{0,1\}$, which has exactly two $1$'s on each row and on each column. Which conditions are necessary and ...
1
vote
2answers
51 views

How to find the standard matrix A for T [closed]

Let $T: \mathbb R^2 \rightarrow \mathbb R^2$ be the linear transformation that first rotates points clockwise through $30$ degrees and then reflects points through the line $y = x$ Find the ...
3
votes
4answers
103 views

Is there an alternative way to represent the $\operatorname{diag}$ function?

In optimization, it is common to see the so called $\operatorname{diag}$ function Given a vector $x \in \mathbb{R}^n$, $\operatorname{diag}(x)$ = $n \times n$ diagonal matrix with components of $x$ ...
0
votes
0answers
26 views

The gas cloud covering problem

I'm faced with problem described below. My goal in posting this here is having you guys lead me in the right direction. Maybe there is a scientific article that treats a similar problem? Maybe a ...
1
vote
1answer
59 views

Can someone come up with a better way to write $V = \operatorname{diag}(x_1,x_2)(Y-\mathbf{1}X^TY)$

$\newcommand{\diag}{\operatorname{diag}}$Let $X = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $Y= \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ I have a vector: $$V = \begin{bmatrix} x_1(y_1 - \...
3
votes
3answers
127 views

Find a matrix $B$ such that $B^3 = A$

$$A=\begin{pmatrix} 1 & -1 \\ -2 & 1 \end{pmatrix}$$ Find a matrix $B$ such that $B^3$ = A My attempt: I found $\lambda_1= 1+{\sqrt 2}$ and $\lambda_2= 1-{\sqrt 2}$ I also found ...
4
votes
1answer
45 views

About transpose matrix transformation problem.

I have this problem that I don't understand so I can't solve. I wish someone could explain me it or solve it. Let $M_2(\mathbb{R})$ the vector space generated by all the square matrices of $2\...
3
votes
0answers
47 views

How to approximate the cumulative distribution function of the normal by a product of functions?

Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{...
4
votes
3answers
63 views

Matrix-by-matrix derivative formula

I need to derive $\frac{\delta(X^{T}MX)}{\delta X}$, where $X$ and $M$ are $n \times n$ matrices. I know that $\frac{\delta(AXB)}{\delta X}=B^{T} \otimes A$ but am having a hard time deriving what I ...
-2
votes
2answers
109 views

Is a correlation matrix with positive determinant PSD?

Please note: I'm not interested in the difference between positive definiteness and semi-definiteness for this question. A correlation matrix is a symmetric positive semi-definite matrix with 1s down ...
2
votes
1answer
37 views

Determinant of a matrix and chech whether it is non negative definite or not

Let $V = \{ f : [0,1] \to \mathbb R | f$ is a polynomial of degree less than or equal to n $\}$. Let $f_j(x) = x^j$ for $0\leq j \leq n$ and let $A$ be the $(n+1) \times (n+1)$ given by $a_{ij} = \...
1
vote
1answer
47 views

derivation of fibonacci log(n) time sequence

I was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the ...