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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1answer
25 views

Relation between eigenvalue and the kernel

Consider the $0$ matrix. It has one eigenvalue: $0$, and the dimension of its eigenspace is $n$, since it sends everything to $0$. If I have $n$ distinct eigenvalues, one of them zero, then the ...
0
votes
1answer
33 views

can two different matrices have same eigenvalues and eigenvectors

Let $A$ and $B$ $\in M^{n\times n}$ be matrices with $n$ distinct eigenvalues $\{\lambda_1,\ldots ,\lambda_n\}$ and corresponding eigenvectors $\{v_1,v_2,\ldots,v_n\}$. Is it necessary that $A$ and ...
1
vote
1answer
80 views

Lights out variation proofs?

I would like some help solving these questions regarding a specific variation of the lights out game where all lights are initially off. The game can be played here (by double clicking edit then ...
2
votes
1answer
24 views

A $3 \times 3$ matrix having integral entries

A is a $3 \times 3$ matrix having integral entries such that $det(A)=120$, number of such matrices is? Could someone help me as how approach this question?
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1answer
67 views

How to prove that $\sum_{i=1}^n\|x_i\|_2^4\leq d\|\sum_{i=1}^nx_ix_i^T\|_2\max_{i=1}^n\|x_i\|_2^2$ for any $x_i\in R^d$?

In the question, $\|\cdot\|_2$ means $l_2$ norm for vectors OR spectral norm for matrices. Thanks in advance. The question is reformulated from page 7 of paper http://arxiv.org/pdf/1506.00898v2.pdf.
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2answers
27 views

A diagonal matrix is commutative with very square matrix

If $A$ is a diagonal matrix of order $3\times3$ is commutative with every square matrix of order $3\times3$ under multiplication and $trace(A) =12$, then the value of $|A|$ is : Could someone give me ...
1
vote
1answer
35 views

Confusion about orthogonal matrices

given an orthogonal transformation $T: \mathbb R^{n}\longrightarrow\mathbb R^{n}$. We also know that the representation matrix $A$ of $T$ is orthogonal if the $A^TA =I$ or the columns of A are an ...
2
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0answers
24 views

Perpendicular Vectors in 3D space

I was wondering whether given two Vector's v0 and v1 whether I could find the two perpendicular vectors at a given distance, d, from v1, perpendicular to the v0/v1 line. I know that v0 and v1 will ...
0
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1answer
35 views

Formal expansion of matrices using Maple or Mathematica

I'd like to evaluate some powers of sum of matrices, say I would like to evaluate $\left(A+B\right)^{n}$ with $A$ and $B$ some matrices. Since I'd like to go to high order, I'd like to use Maple or ...
0
votes
1answer
33 views

General conditions for submatrices in regards to determinant

What are the most general conditions on sub-matrices A,B,C,D st det[A B;C D] = det(AD-BC) Obviously this is how determinant is defined for a regular square 2x2 matrix, but I don't understand how to ...
2
votes
0answers
48 views

When is this matrix positive semidefinite?

Let us fix dimension $n$. Consider the $n \times n$ matrix \begin{equation} S_n=\begin{bmatrix} 1 & z & z & \cdots & z \\ \bar{z} & 1 & z & \cdots & z\\ \bar{z} & ...
0
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0answers
15 views

If a matrix is of rank one, and let $v$ be the eigen vector corresponding the eigen value $d$. Then, $A=dvv^T$

If a symmetric matrix is of rank one, and let $v$ be the eigen vector corresponding the eigen value $d$. Then, $$A=dvv^T$$ How to prove this??
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votes
3answers
96 views

$trace (A^2)=trace(AA') \iff A= A'$

Let A be an $n\times n$ matrix then, $$trace (A^2)=trace(AA') \iff A= A'$$ where $A'$ is the transpose of $A$. only if part is easy by observing that $A=A'$ implies that $A^2=AA'$ and trace of ...
0
votes
1answer
31 views

Trace zero means matrix is nilpotent?

I have to prove or disprove: If $A$ is an $n \times n$ matrix in $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$ and the trace of any power of $A$ is $0$, then the matrix is nilpotent: $A^k = 0$ ...
1
vote
1answer
23 views

Describing product of diagonal terms in Smith normal form in term of minors

Suppose $A$ is an $n\times n$ integer matrix. Then $A$ is equivalent to a matrix with diagonal entries $d_1,d_1,\dots,d_n,$ where $0<d_i\mid d_{i+1}$ for $1\leq i\leq n-1.$ I'm asked to describe ...
2
votes
0answers
30 views

Up-to-date Matrix Cookbook

My copy of the Matrix cookbook is dated November 15, 2012, and is the newest copy I've been able to find. Identities may not change overtime, but the approach to an error-free presentation can be ...
0
votes
0answers
15 views

The existance of Schur Complement Inverse

A block matrix $\mathbf{M}=\left[ \begin{array}{ccc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^T & \mathbf{C} \end{array} \right]$ is invertible if $\mathbf{A}$ and ...
1
vote
1answer
9 views

How to find basis for orthoogonal complement basis for the following condition?

Let $W=$span$\{\begin{bmatrix}1&1\\0&0\end{bmatrix}\begin{bmatrix}0&0\\1&1\end{bmatrix}\}$ and suppose the span is orthogonal under certain Hermitian inner product space (just ...
1
vote
1answer
13 views

How to prove columns of matrix $A$ are linearly independent $\implies$ $C$ must be invertible for the following condition?

Suppose $A=BC$, where $B$ is an $m\times n$ matrix and $C$ is an $n\times n$ matrix .How to prove columns of $A$ are linearly independent $\implies$ $C$ must be invertible? In my opinion, I feel like ...
0
votes
0answers
28 views

When can $Tr(X^TAY)\geq 0$ when $A$ is positive semidefinite and $X,Y \neq 0$?

Under what conditions on $X,Y$ is $Tr(X^TAY)\geq 0$ when $A$ is square positive semidefinite but not the Identity matrix and $X,Y \neq \mathbb{0}$ (as in not equal to all matrix of all zeros)? $X,Y$ ...
0
votes
0answers
24 views

Looking for a Matrix permutation operation

Let's say I have five boxes. On one side, two boxes are filled with respectively {9, 11} balls, and on the other side, three boxes have to be filled respectively ...
0
votes
0answers
31 views

Solving system of differential equations using matrix exponentiation

$$x_1^{'} = 2x_1 + x_2$$ $$x_2^{'} = 2x_2$$ Since this is a non-diagonalizable matrix, I have no idea how to approach this problem. We have not learned the nilpotent, so I cannot use that to help me ...
0
votes
0answers
12 views

Block LU decomposition

Could you give me a hand with finding the LU decomposition of the following matrix : enter image description here where B belongs to R nxn and prove that det(A)>0. I tried in following way: tried ...
0
votes
0answers
34 views

Unconstrained minimisation problem Newton's method

min f(x) = $ x_1^4 + 2x_1^2x_2^2 + x_2^4 $ is an unconstrained min problem. The first question asks to show that $(0,0)$ is the unique minimiser. I have done the following.. Would I need to add ...
2
votes
1answer
34 views

Reading a Matrix

This is a softer question, but I'm having trouble keeping straight all of the information that a matrix provides you with straight in my head. All I know is that the rows correspond with the codomain ...
4
votes
1answer
220 views

Principal Minors of $B(AB)^{-1}A$ and Cauchy-Binet Terms

I am looking for a proof for the following conjecture. I think the result follows from applying a generalization of the Cauchy-Binet formula to the matrix $\mathbf{M}$ defined bellow. I've tested it ...
-2
votes
0answers
23 views

Matrix laws of operation

Question Please help me on this Matrix problem. Regards!
0
votes
1answer
23 views

Determine dependencies, might be cramer's rule

I have the following problem, Determine how $x_1(\alpha)$ depends on $\alpha$ when $x_1(\alpha)$ is the first component of the solution of the system $Ax = b$, where A = $\begin{bmatrix}2 & 1 ...
0
votes
1answer
48 views

finite classes of similarity relation on $n \times n$ matrices over $M_n(\mathbb{F})$

Let $\mathbb{F}$ be an infinite field. Assume the similarity relation on $M_n(\mathbb{F})$ and let $C$ be a finite class of this equivalence relation. Prove that $C$ has exactly one member. ...
0
votes
0answers
19 views

representation of a map with respect to 2 bases

From Heffron, p.231 Consider the two linear functions $h:$ ${R}^3$ $\longrightarrow$ $\mathcal{P}_2$ and ${g}: \mathcal{P}_2 → M_{2x2}$ $ \left( \begin{array}{ccc} a \\ b \\ c \end{array} ...
-1
votes
1answer
59 views

The diagonal of a symmetric matix $A\in M_n(\mathbb{Z}_2)$

Let $A \in M_n(\mathbb{Z}_2)$ be a symmetric matrix. Prove that the diagonal of $A$ is in row space of $A$. Any help to solve this problem would be appreciated.
2
votes
0answers
20 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $\pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] \cdot ...
1
vote
1answer
38 views

Rank of upper triangular matrix

Show that the rank of an upper triangular matrix is at least as large as the number of non-zero main diagonal entries. What I do not understand with this statement is how can one have a ...
-1
votes
1answer
27 views

Find coordinate vector in matrix vector space

How do I do this question? I don't understand the notation that describes B what is the superscript ij? what is E?
-1
votes
0answers
19 views

Proof of wronskian?

Can someone please provide a (simple) proof of this: I tried proving the the rank of the matrix was always 1 but I needed the determinant to always=0 for that to work
0
votes
1answer
19 views

Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix

I want to ask a question about invertible matrix. Suppose there is a $n\times n$ symmetric and invertible matrix $M$, and we know its Cholesky decomposion as $M=LL'$. Then do we have an efficient way ...
0
votes
2answers
23 views

Properties of the Product of a Square Matrix with its Conjugate Transpose

Let $A$ be a square complex matrix. What special properties are possessed by $AA^H$, where $^H$ denotes the conjugate transpose? One property I am aware of is that $AA^H$ is Hermitian, i.e. ...
-1
votes
1answer
41 views

How to know if the matrix is diagonalizable?

I put the matrix into a row echelon form but I got the matrix without $a$-s. What should I do ? M= $\pmatrix{1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0}$ It is not OK. I need to ...
1
vote
4answers
36 views

Find the limit a matrix raised to $n$ when $n$ goes to infinity

Let $ A $ be a $ 3\times3 $ matrix such that $$A \left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right),~~~A \left( \begin{array}{ccc} ...
0
votes
1answer
39 views

Let $A$ be a $5\times 5$ complex matrix with characteristic polynomial that is $p(x)=(x-\lambda)^5$.

Let $A$ be a $5\times 5$ complex matrix with characteristic polynomial that is $p(x)=(x-\lambda)^5$. I've found that $$\dim\ker(A-\lambda I)=2$$ $$\dim\ker(A-\lambda I)^2=3$$ $$\dim\ker(A-\lambda ...
0
votes
2answers
36 views

Why does matrix exponentiation work with transposes?

Let $A^T$ denote the transpose of $A$. Is it true that $(A^n)^T = (A^T)^n$?
1
vote
1answer
45 views

Finding a Canonical Basis for Nilpotent Linear Mappings

I've read in my Algebra book what seems to be like an easy way to find a canonical basis given a nilpotent mapping $N:V\to V$, and the corresponding cycle tableau. But I'm not sure I completely ...
0
votes
0answers
21 views

Why does Correlation Coefficient concern about the mean of the vector?

$r = \frac {\sum_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\sum_{i=1}^n(Xi-\bar X)^2} *\sqrt{\sum_{i=1}^n(Y_i-\bar Y)^2}}$ This is exactly the cosine of degree of the angle between vector $X-\bar ...
1
vote
0answers
30 views

minimize trace(AX) over X with a positive semidefinite X

I want to minimize trace(AX) over X, under the constraint that X is positive semidefinite. I guess the solution should be bounded only for a positive semidefinite A, and it's zero, or the solution ...
-8
votes
1answer
229 views

Showing that a matrix is symmetric positive definite

I am trying to show that, if the SPD matrix $$K = \begin{bmatrix} A & B \\ B^T & C \\ \end{bmatrix} \in \mathbb{R}^{n \times n},$$ where $A \in \mathbb{R}^{m \times m}$ is also SPD, then ...
0
votes
0answers
19 views

Eigenvalues of coupled system of nonlinear 2nd order ODEs

System of Equations I am trying to determine natural frequencies of system in which a compound pendulum is attached to a slider which in turn is attached to rigid support through a spring. I have ...
0
votes
0answers
23 views

System of equations given a function as an input

I’m having a set of equation $f_1(y,t), f_2(y,t)… , f_n(y,t)$, I also know that $y(t)=e^{At}v$ where $A_{nxn}$ is a matrix and $v$ is a $nx1$ Vector. Can I bound $f(y(t))$ somehow? Or say anything ...
1
vote
1answer
22 views

How to find the projw(x)

Sorry about formatting, new to this. Subspace w = \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} such that $x_1 = x_3$ of R3. x = \begin{pmatrix}-2\\1\\3\end{pmatrix} Question is to find $proj_w ...
0
votes
2answers
24 views

Show that Jordan block is idempotent if it is 0 or 1

An idempotent matrix $M$ is one such that $M^2 = M$. A Jordan block has its eigenvalue $\lambda$ on its diagonal and 1 on the superdiagonal. I figure that in order to ensure that $M^2=M$, it makes ...
1
vote
2answers
42 views

$n\cdot tr(AB)=tr(A) \cdot tr(B) $ $A$ is a scalar matrix

Let $A\in M^{n\times n}(\mathbb R)$. prove that if for every other $B\in M^{n\times n}(\mathbb R)$: $n\cdot tr(AB)=tr(A) \cdot tr(B) $, $A$ is a scalar matrix.