2
votes
2answers
51 views

If A+tB is nilpotent for n+1 distinct values of t, then A and B are nilpotent.

Suppose A and B are $n\times n$ matrices over $\mathbb{R}$ such that for n+1 distinct $t \in \mathbb{R}$, the matrix A+tB is nilpotent. Prove that A and B are nilpotent. What I've tried so far: ...
0
votes
1answer
16 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
0
votes
1answer
12 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
1
vote
1answer
35 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
-5
votes
0answers
44 views

How to prove the following rank problems [on hold]

I am quite confused with this question. How can I initiate and approach a solution. Thanks!
1
vote
2answers
44 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
2answers
102 views

How to find exponential of triangular matrix

I'm studying for an exam and I can't find this in my notes or in the book, but it's on a past exam... Given $A = \begin{bmatrix}-1 & 1\\0 & -1\end{bmatrix}$, $e^{tA} = \begin{bmatrix}e^{-t} ...
2
votes
0answers
40 views

What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
0
votes
0answers
31 views

How to further simplify this equation?

Given that V is an invertible $n$x$n$ matrix and $\Sigma$ is a diagonal rectangular $m$x$n$ matrix, U is an $m$x$m$ matrix, b is an $m$x1 matrix and $\lambda$ is a positive number, how do u further ...
3
votes
2answers
58 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
2
votes
1answer
71 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
0
votes
1answer
28 views

What is the number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My guess ...
1
vote
1answer
20 views

matrix function onto and 1-1

I have just started a linear algebra paper and we are doing 1-1 and onto functions. I understand in theory what they mean, I just don't know how to prove them. For example: Define $f: ...
2
votes
1answer
52 views

Invertibility of $I-AB$ [duplicate]

I got a question in linear algebra: 1) Let A and B be $n\times n$ matrices. If $I - AB$ is an invertible matrix, then prove that $I - BA$ is invertible. Can someone tell me how to solve this ...
0
votes
1answer
27 views

How to write this system in the form Ax=b

Given the following system of N equations with N unknowns, with $\lambda$ known and the $a_{ij}$'s also known entries of an m*n matrix A. How would you express the system in the form A x=b? x is of ...
2
votes
2answers
97 views

Prove that if the sum of each row of A equals s, then s is an eigenvalue of A. [duplicate]

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: ...
1
vote
0answers
25 views

Iterative methods for solving a linear equation system

There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating $x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But ...
0
votes
0answers
24 views

solving equation also involving unknown matrix in trace

Given two real $m$ x $k$ matrices $A_1$ and $B_1$ and two $k$ x $k$ real matrices $A_2$ and $B_2$ I want to solve the following equation for $Q$. $Q$ is an orthogonal matrix, i.e. $Q^TQ=I$. ...
2
votes
1answer
48 views

Basis for space of matrices in $\mathbb M_2(\mathbb R)$

Given that $G=\left\{ \left(\begin{array}{cc} a & -a\\ b & c \end{array}\right):a,b,c\in\mathbb{R}\right\} $ and $H=\left\{ \left(\begin{array}{cc} x & y\\ z & -z ...
2
votes
3answers
318 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
1
vote
1answer
31 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
0
votes
0answers
40 views

A problem on matrices over a commutative ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...
2
votes
3answers
65 views

Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
2
votes
1answer
25 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
1
vote
0answers
57 views

Integer matrices whose $m$-th power are identity matrix

How can one find all the matrices with integer entries of size $n \times n$ such that $A^{m}=I$ where $m$ is fixed integer and the matrix does not have fixed point in $\mathbb{Z}^n$ (except zero of ...
3
votes
1answer
48 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
0
votes
0answers
14 views

Matrix multiplier for ODE

I have matrix C with dimensions $3 \times 3 $ and it is skew symmetric too C is given by $C(0,0)=0,C(1,1)=0,C(2,2)=0 \tag 1$ $C(1,0)= sc_0+ px (c_1-c_0),C(0,1)=-C(1,0) \tag 2 $ $C(0,2)= ...
1
vote
0answers
23 views

Solving tridiagonal matrices where the top left element is zero

If I have a matrix like this: $$ \left[\begin{array}{rrrrrrrrr|r} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & ...
1
vote
1answer
27 views

Express summation in terms of matrix norm

Express the following $$\sum _{ i=1 }^{ n }{ ({ \beta }_{ 1 }x_{ i }+{ \beta }_{ 0 }-y_{ i })^{ 2 } }$$ To become something of the form: $∥Ax−b∥^{ 2 }$ where $A$ is an $m$−by−$n$ matrix and $b$ is ...
0
votes
1answer
26 views

Matrix transpose times itself

We define A to be a matrix in $R^{m*n}$ Does $A^TA$ have any particular structure? When is $A^TA$ invertible?
1
vote
1answer
31 views

Find the Jacobian of F

Given that $A \in \mathbb{R}^{m\times n}$, and $b \in \mathbb{R}^{m}$, we define: $$F:\mathbb{R}^{n} \rightarrow \mathbb{R} = \left\| Ax-b \right\|^2$$ Find the Jacobian of $F$, and show that it is of ...
1
vote
1answer
25 views

Reduced Row Echelon form without scalar multiplication?

Is it possible to transform any matrix to row reduced echelon form without using the row operation that multiplies a row by a scalar?
0
votes
1answer
13 views

Determine if matrix D belongs to Vect(A,B,C)

So there are 4 matrices, A, B,C,D. They belong to field F5. Determine if D belongs to Vect(A,B,C). I have pretty much done all the calculations its just i fail to conclude/find the right value for the ...
4
votes
1answer
58 views

What is the name of the matrix that is created by a vector times its transpose.

I am looking for the name of the matrix created by the following operation: $Z = z*z^T$ I know it should create a symmetric matrix with an element $Z_{ij} = z_{i}z_{j}$
4
votes
0answers
62 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
1
vote
2answers
39 views

Matrix Semi-Definite Inequality [duplicate]

Does the following inequality hold? If matrix $A$ is a $n \times n $ positive semi-definite, $A \succeq 0$, and $U$ is one $n \times k$ unit column-orthogonal matrix ($k \leq n$), $U^{T}U=I$, do we ...
0
votes
3answers
24 views

How do you know that rows are independent and what are the 120 terms?

I am having trouble with the question below, help me out;
0
votes
0answers
26 views

Eigenvalue bounds for a positive semidefinite matrix

I have a symmetric $(p\times p)$, positive semi definite matrix $\Omega$. If somebody says: find the eigenvalue bounds of the matrix such that $$w_1I \le \Omega \le w_2I$$ where $I$ is the identity ...
2
votes
0answers
36 views

Top bound on the value of an algebraic adjunct to elements of a nonnegative irreducible matrix

Let $A = ||a_{i j}||_1^n$ be nonnegative irreducible matrix with maximum eigenvalue $r$. Let $A_{i j}(\lambda)$ be an algebraic adjunct for the element $\lambda \delta_{i j} - a_{i j}$ in determinant ...
0
votes
2answers
54 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
2
votes
1answer
43 views

Let $A$ be an $n \times n$ matrix with real entries. [closed]

Which of the following is correct? (a) if $A^2=0$, then $A$ is diagonalisable over complex numbers (b) if $A^2=I$, then $A$ is diagonalisable over real numbers (c) if $A^2=A$, then $A$ is ...
1
vote
1answer
27 views

Equality case in the Frobenius rank inequality

In many linear algebra books, the following rank inequalities are found: Frobenius inequality Let $A$, $B$ and $C$ be three matrices such that the product $ABC$ is defined. Then ...
0
votes
2answers
46 views

Jordan chain when matrix has only one eigenvalue.

A $12\times 12$ matrix has sole eigenvalue $3$. It is given that the kernels of $A-3I$, $(A-3I)^{2}$, $(A-3I)^{3}$ and $(A-3I)^{4}$ have dimensions $4$, $7$, $9$ and $10$ respectively. What ...
0
votes
1answer
32 views

Equality in the Collatz-Wielandt-formula

Let A be a matrix with positive entries. The perron-frobenius-theorem states that A has a positive dominating simple eigenvalue, called the perron-frobenius-eigenvalue. I denote it with p(A). The ...
1
vote
1answer
27 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
3
votes
2answers
68 views

Matrix with all eigenvalues $0$ but not triangular?

Is the situation described in the title achievable? I am looking for a $3\times 3$ case specifically.
1
vote
1answer
29 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
0
votes
0answers
20 views

Power iteration sequemce for a special nonnegative irreducible imprimitive matrix

Let $A \in \mathbb{R}^{n \times n}$ be nonnegative irreducible matrix with maximum positive eigenvalue equal to 1. Let's assume $A$ has $h$, $h > 1$ eigenvalues $\lambda_1, \dots, \lambda_h$ with ...
1
vote
0answers
39 views

diagonal of pseudoinverse of laplacian matrix

I have to find the diagonal of the pseudoinverse of a laplacian matrix evaluated on a directed and weighted graph. My laplacian is defined as: L = D - A where: D is a diagonal matrix; Di,i the sum ...
0
votes
1answer
28 views

How the Wronskian works

To prove linear independence of a set of functions, we say that given their Wronskian matrix W, Wx = 0 implies trivial solution (0,0,0,...) if the value (determinant) of the Wronskian is identically ...