0
votes
0answers
13 views

Finding the least square fit for 3 parameters in Linear Algebra

I know how to find least square for $y = mx+b$ when we have two parameters. But this question has $3$ parameters, am trying to think of how to approach it but so far no success, I can't find any ...
1
vote
1answer
29 views

Finding a matrix projecting vectors onto column space

I can't find $P$, for vectors you can do $P = A(A^{T}A)^{-1}A^T$. But here its not working because matrices have dimensions that can't multiply or divide. help
2
votes
1answer
34 views

Checking connectivity of adjacency matrix

What do you think is the most efficient algorithm for checking whether a graph represented by an adjacency matrix is connected? In my case I'm also given the weights of each edge. There is another ...
2
votes
2answers
19 views

Decomposing a square matrix into two non-square matrices

I have a matrix $A$ with dimensions $(mxm)$ and it is positive definite. I want to find the matrix $B$ with dimensions $(nxm), (n << m)$, which follows the following expression: $$A = B'B$$ Here ...
0
votes
1answer
24 views

Getting linear combinations in linear algebra?

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;
2
votes
0answers
26 views

Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
0
votes
0answers
26 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
3
votes
4answers
82 views

Can you use row and column operations interchangeably?

Is it possible to use row and column operations "at the same time" on a matrix $A$? So, for example, first subtracting $row_1$ from $row_2$, and then choosing to multiply $column_3$ by a constant $c$? ...
0
votes
1answer
39 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
0
votes
0answers
23 views

When is the LU decomposition unique?

I want to find out when a matrix decomposition $A = LU $ (L lower and U upper matrix) is unique? Clearly, if $A$ is not invertible, there is no chance that this decomposition is unique. Hence, ...
-1
votes
1answer
32 views

Symmetric matrix problem

$A$ is a symmetric matrix and has a eigenvalue $\lambda$ of order m why $\lambda$ has m independent eigenvector
3
votes
1answer
42 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
1
vote
4answers
28 views

Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$

I came across the fact that the rank of a nxn-matrix A with $A^2=0$ is at most $\frac{n}{2}$. The easiest way to proof this is using the inequality $rank(A) + rank(B) -n \le rank(AB)$. With $A=B$ and ...
0
votes
1answer
34 views

Let 1r be the identity matrix… [on hold]

I'm just beginning this subject and finding it hard to get my head around some of the terms. Any help is appreciated.
0
votes
1answer
25 views

Cholesky factorization and non-positive definite matrices

When Cholesky factorization fails, is there an alternative method to obtain the $\mathbf{L}$ matrix in: $\mathbf{A}=\mathbf{L}\mathbf{L}^{*}$ I'm dealing with a matrix not guaranteed to be ...
1
vote
2answers
69 views

On the nilpotence of the matrix $AB-BA$ [on hold]

Given $n\times n$ matrices $A,B$ satisfy: $rank(AB-BA)=1$ Prove that $(AB-BA)^{2}=0$ Generalize the problem if possible. Any solution not mention Jordan canonical form would be appreciated!
4
votes
3answers
137 views

$A+A^2B+B=0$ implies $A^2+I$ invertible?

Let $A$ and $B$ be two square matrices over a field such that $A+A^2B+B=0$. Is it true that $A^2+I$ is always invertible ?
1
vote
1answer
35 views

Some “Product” of Positive Definite Matrices

I could remember that if $A,B$ are two positive definite matrices, then $(a_{ij}b_{ij})$ is positive definite also. But I could not see how to prove it then.
0
votes
3answers
62 views

About semipositive definite matrix

Suppose $A$ and $B$ are positive semidefinite matrices $A \ge B\ge 0$ Is the statement $A^2\geq B^2$ true or false? Why? $\geq$ means nonnegative pointwise
1
vote
1answer
32 views

Study endomorphism diagonalization

Given an endomorphism whose matrix is: $\begin{pmatrix} 1+a & -a & a \\ 2+a & -a & a-1 \\ 2 & -1 & 0 \end{pmatrix}$ How can I study if it's diagonalizable or not depending ...
0
votes
0answers
19 views

Explain why a matrix is orthogonally diagonalisable.

If people could tell me if I'm on the right track on and give me a push in the right direction for the ones I'm unsure of that would be much appreciated. Let A$\epsilon$M$_{3}$($\mathbb{R})$ and ...
0
votes
1answer
22 views

A and B are nxn matrices. A = $B^{T}B$ Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.

A and B are nxn matrices. A = $B^{T}B$ Prove that if rank(B)=n, A is positive definite, and if rank(B) My current understanding is that if rank(B)=n, then rank($B^{T}B$)=n then rank(A)=n, making A ...
1
vote
1answer
33 views

Find a 2x2 matrix with positive eigenvalues, but a negative quadratic form for some x in $R^{2}$

Find a 2x2 matrix with real and positive eigenvalues, but a negative definite quadratic form. Also, find a 2x2 matrix with real and positive eigenvalues, but an indefinite quadratic form. Isn't this ...
0
votes
1answer
58 views

Connection between Eigenvectors and linear equations

I'm trying to understand the connection between Eigenvectors/Eigenvalues and linear equations: $Ax=b$ If you are given the eigenvectors and eigenvalues of $A$, can you construct the solution for the ...
0
votes
0answers
33 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
8
votes
0answers
198 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrix $A,B,C\in M_{n}(C)$ is Hermitian matrix and is Positive definite matrices ,such $$A+B+C=I_{n}$$show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge ...
1
vote
1answer
34 views

Solving an augmented coefficient matrix so there are infinitely solutions

I am trying to figure out this math problem. For what values $a,b$ does the linear system have infinitely many solutions? This is the matrix $$ \left[ \begin{array}{ccc|c} ...
0
votes
0answers
17 views

Correlating random numbers seems to skew the data

I am trying to generate a series of correlated random numbers that represent currency exchange rates for a Monte-Carlo simulation. I am attempting to do this via a Cholesky decomposition of the ...
1
vote
2answers
51 views

For which values of $a, b$ does the system of equations NOT have any solutions?

I am trying to solve this math problem: For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution? $$ \left[ \begin{array}{ccc|c} ...
2
votes
1answer
25 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
0
votes
1answer
18 views

Difference between matrices with altered eigenvalues

Given two p.s.d. matrices $X_1$ and $X_2$ with eigen decomposition $X_1 = U_1V_1U_1^T$ and $X_2 = U_2V_2U_2^T$ and a constant $\lambda > 0$ Now consider an altered version of the eigenvalue ...
3
votes
3answers
54 views

Cross product: matrix transformation identity

How can one prove the following identity of the cross product? $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$ $a$ and $b$ are 3-vectors, and $M$ is an invertible real 3x3 matrix.
0
votes
1answer
48 views

Matrices and algebra

Given the matrix $A$ $=$ $$ \begin{pmatrix} -1 &3 & 5 \\ 1 & -3 & 5 \\ -1 & 3 & 5 \end{pmatrix} $$ and $X$ be the solution set of the equation ...
4
votes
2answers
97 views

Prove that $\det(M-I)=0$

$M$ is a $3 \times 3$ matrix such that $\det(M)=1$ and $MM^T $= I, where $I$ is the identity matrix. Prove that $\det(M-I)=0$ I tried to take $M$ $=$ $$ \begin{pmatrix} a &b & c \\ ...
1
vote
3answers
51 views

How find the matrix $K$ such $AKB=C$

Question: Find a matrix $K$ such that $$AKB=C$$ given that $$A=\begin{bmatrix} 1&4\\ -2&3\\ 1&-2 \end{bmatrix},B=\begin{bmatrix} 2&0&0\\ 0&1&-1 \end{bmatrix} ...
0
votes
1answer
22 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
1
vote
3answers
43 views

Tridiagonal Symmetric Matrix

Could anyone help me to find the determinant of a $N\times N$ tri-diagonal symmetric matrix, named "$A[i,j]$" with $i,j \le N$, that has all the elements in the super-diagonal and sub-diagonal equal, ...
0
votes
1answer
41 views

Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
1
vote
0answers
29 views

Matrix with trace zero [duplicate]

Question is that : Suppose a matrix $A\in M_n(\mathbb{C})$ is a commutator by which i mean $A=BC-CB$ for some $B,C\in M_n(\mathbb{C})$ then we see that Trace of $A$ is $0$ But Suppose a matrix is of ...
1
vote
0answers
21 views

Matrix that shows how close indices are to each other

I have a vector of n words, not all distinct. [the, quick, brown, fox, jumps, over, the, lazy, dog] From this, I want to make an n×n matrix that shows how ...
1
vote
4answers
55 views

TT* + I is invertible

I've the following exercise which I can't solve: Prove that: $$ AA^* + I $$ is invertible for all Matrix $ A $ in finite-dimensional field $V$ with inner product. $ A^* $ is the adjoint operator. ...
9
votes
2answers
96 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
1
vote
1answer
22 views

$A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$

How can i prove that $A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$ i stuck in this problem i know that if $A$ ...
0
votes
2answers
27 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
10
votes
4answers
331 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
7
votes
5answers
114 views

Positive semi-definite of a matrix composed of semi-definite blocks

Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or $$\begin{pmatrix} A ...
1
vote
1answer
25 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
7
votes
2answers
98 views

Proof of the inequality $\sqrt{\det X} \leq \frac{\operatorname{tr}X}{2}$

Let $A, B \in M_2(\mathbb{R})$ be symmetric and positive definite. Put $X:=AB$. then, we have the following inequality: $$\sqrt{\det X}\leq \dfrac{1}{2}\operatorname{trace}X.$$ and the equality ...
1
vote
0answers
33 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
1
vote
0answers
34 views

Schur decomposition of real-eigenvalue matrix

Is Schur decomposition of real-eigenvalue matrix a real orthogonal decomposition? If yes, why is it? Is it because all the eigenvectors are real? If I have $$ A^T+A^2=I $$ then I deduced ...