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), ...

learn more… | top users | synonyms (2)

0
votes
0answers
11 views

Eigenvalues of a matrix with special form

Let $p,a_1,...,a_n\in(0,1)$ and $\sum_{i=1}^na_i=1$. Now consider the following matrix: $$ \left(\begin{array}{ccccc} (1-p) & \sqrt{p(1-p)}a_1 & \sqrt{p(1-p)}a_2 & ... & ...
0
votes
0answers
8 views

Computing orthogonal projection

The question asks: A vector u and a line L in R^2 are given, compute the orthogonal projection w of u on L. u=[3,4] and y=-x In one example i was given two ...
0
votes
0answers
7 views

Isometries of $\mathbb{R}^n$ as a sum of rotations and translations

I am reading the Wikipedia article on $O(n)$ and came across the claim that every $Q\in O(n)$ can be written as: $$\begin{bmatrix} \begin{matrix}R_1 & & \\ & \ddots & \\ & & ...
1
vote
1answer
13 views

upper bound on this matrix norm

What would be the upper bound on the 2-norm (or any norm) of the following matrix product ? Please consider the smallest upper bound. $\|\left(I+BA^T\right)\left(I+AA^T\right)^{-1}\|< ?$ where A ...
0
votes
0answers
6 views

Find a matrix and a vector using partial derivative and system of matrices.

Let $f(x)$:=[$f_1(x),...,f_d(x)]^T$ and suppose that |$\frac{\partial^2 f_i(x)}{\partial x_j \partial x_k}|$$\le$K for all $i,j,k$=1,...,d and $x\in\Re^2$. Show how to define an $dxd$ matrix $J(y)$ ...
0
votes
0answers
15 views

matrix two norm derivative with respect to X

What would be the result of the following derivative in terms of X? $\frac{d \|X\|_2}{d~ X}=?$
1
vote
1answer
13 views

Linear maps, inverses and associated matrices?

This is likely a very simple question but if we have a linear map $f$ with an associated matrix $A$ is it a necessary and sufficient condition that for $f$ to have an inverse then $A$ must also have ...
0
votes
1answer
16 views

How to merge similar terms to get a perfect square form?

There is a objective function that has the following form: $$ \alpha \|X^T AX\|_F^2-trace(B^T X) +\beta\|X-C\|_F^2 $$ where $\alpha,\beta$ are scalars, and $X,A,B,C$ are compatible matrices. ...
-2
votes
0answers
8 views

Finding inverse, determinant and adjoint of 3 by 3 matrix for mcq..

I am gonna attempt mcq paper in which these questions are asked? Therefore need a easy and short way to solve it due to less time.
1
vote
1answer
22 views

Continuity of $f(x)=(xI-A)^{-1}$?

Let $A\in \mathbb{C}^{n\times n}$ and $I_n$ be an identity matrix. If $z\in \mathbb{C}$ is not a eigenvalue of $A$, then $f(x)=(xI-A)^{-1}$ is a continuous function at $z$. Is that correct?
0
votes
0answers
21 views

Proving a theorem a about matrix derivations

Ok, so Im doing some research and I have to understand the following theorem. The theorem states: Let $h$ be a derivation on $Tn(R)$ with $h(e_{ij})=0,\,\, 1\le i \le j \le n$. Then $h=\bar\delta$ ...
1
vote
1answer
33 views

Is a vector of coprime ring elements column of an invertible matrix?

Given a commutative ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = ...
-2
votes
0answers
15 views

Transition Matrices for Jordan Form [duplicate]

Thought I would throw out my line one more time. I have this matrix $M$ $M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 ...
2
votes
1answer
31 views

Every matrix in $SU(2)$ can be written as: $P= I\cos \theta+ A\sin \theta$, $A$ on the equator.

How can I show that every matrix in $SU(2)$ can be written as: $P=I\cos \theta + A\sin \theta$, with $A$ on the equator?
0
votes
1answer
31 views

Jordan Canonical Form transition matrix

I have this matrix $M$ $M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 \end{bmatrix}$ And I was asked to put it into Jordan Canonical ...
0
votes
1answer
26 views

Help to in finding the Eigenvectors for the following $2\times2$ Matrix

Please help in finding the eigenvectors for the following $2\times2$ matrix. This is very urgent, required for my examination. Your help will be greatly appreciated. Thank you. Matrix $$ A = ...
-1
votes
0answers
29 views

The set of matrices with nonnegative determinant is not a subspace. [on hold]

Disprove using a counterexample: The set of all $3\times 3$ matrices with determinant $\ge 0$ is a subspace of $M_3(\Bbb C)$.
0
votes
0answers
18 views

Find a basis for the span of each set?

I found the span of the set. Then I used GJ to get the RREF, and used the row reduced rows to form the basis. I got the basis as <( 1 0 -2 ; 0 1 1 )> However, my lecturer went a different way, ...
0
votes
0answers
7 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
1
vote
1answer
23 views

Equation for minimum/maximum eigenvalue

It is well known that for a hermitian matrix $A$ we have $\lambda_{min}(A)=min{x\ne 0} <x,Ax>/<x,x>$, which we can see be diagonalizing $A$. Now here is my question about the following I ...
0
votes
1answer
39 views

Prove that $tr(A^-)=\sum_{i=1}^n\lambda_i^{-1}$ [on hold]

If $A$ is an n$x$n symmetric matrix with $r$ nonzero characteristic roots $\lambda_1,\lambda_2,...,\lambda_n$ and $A^-$generalized inverse of $A$ (not $A^{-1}$), then ...
0
votes
1answer
30 views

The spectral radius of a non-negative matrix [on hold]

Prove the following result: given a $n\times n$ non-negative matrix, let the spectral radius of this matrix be $\lambda$, if the matrix is irreducible, then $\lambda+1>n$.
0
votes
2answers
64 views

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____?

$\det (AB)=\det(A)\det(B)$ is possible when $A$ and $B$ are _____? This is a fill-in-the-blank problem that I found in my paper, but I don't have this answer.
0
votes
1answer
26 views

What is the purpose of Jordan Canonical Form?

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is ...
1
vote
1answer
50 views

How to put a matrix in Jordan canonical form, when it has a multiple eigenvalue?

I have a question that reads: Put the matrix \begin{bmatrix} 3 & -4\\ 1 & -1 \end{bmatrix} in Jordan Canonical Form. Moreover, in each case, find the appropriate ...
0
votes
0answers
16 views

Understanding math from a paper about attitude filters

I've spent the better part of an afternoon trying to understand the steps between equation 3 and equation 4 of this paper. He begins with the following matrix equation: $\bar{C}=C_{ref}^TC$ takes a ...
0
votes
0answers
8 views

Using the Perron-Frobenius theorem for non-negative irreducible matrices to estimate the dominant eigenvalue of a matrix??? [on hold]

Use the Perron-Frobenius theorem for non-negative irreducible matrices to fi nd an estimate of the dominant eigenvalue of A (without calculating the eigenvalues).
0
votes
0answers
12 views

Using the inverse of the matrix find all the solutions of the following systems of equations?

I found the inverse using row operations and the identity matrix but I dont know where to go from here. Can someone direct me please ?
1
vote
2answers
30 views

$rank(T^n) = rank(T^m)$ for any positive integer $m \geq n$

Let $T$ be a linear operator on a finite dimensional space $V$. Prove that if $rank(T^n) = rank(T^{n+1})$ for some positive integer $n$, then $rank(T^n) = rank(T^m)$ for all positive integer $m \geq ...
1
vote
1answer
23 views

Inverse matrix as a sum of matrix powers [duplicate]

I have matrix $ A\in \mathbb{C}^{n x n}$ and $A$ is invertible. How can I show that coefficients $c_0,...,c_{n-1}$ exist : $A^{-1} = c_0I+c_1A+...+c_{n-1}A^{n-1}$ I tried to solve it first by ...
0
votes
0answers
8 views

What is the Moore-Penrose pseudoinverse for a hermitian block-matrix with one zero block?

Given a block matrix of the form \begin{pmatrix} A & B^* \\ B & 0 \end{pmatrix} where $A$ is singular (otherwise one could simply use the well-known block matrix inverse), is there a ...
0
votes
1answer
22 views

Need help regarding Subspace of matrix and its basis

I need some kind of hint to get me going for this question as I'm so lost at it. Any sort of help would be appreciated. Let E be the set of all 2x2 matrices that have $v={(1,-1)}$ as an eigenvector. ...
0
votes
0answers
19 views

Find solution of signle element $y_i$ in vector $y$ subject to $Ay=c$

I have a interesting question about linear algebra problem. Assume that I have a matrix $A^{m \times n}$ and vector $c^{n \times 1}$ are known and I want to find the solution of vector y subject to ...
0
votes
1answer
42 views

Induced invariant linear map in the dual space

This is the problem that I am stuck on. Problem: Let $V$ be a finite dimensional vector space and $T: V\rightarrow V$ be a linear transformation. Suppose ...
1
vote
5answers
104 views

If $A^n=0$, then $I_n-A$ is invertible. [on hold]

How do I solve this problem? $A$ is $n\times n$ and $A^n=0$. Prove that $I_n-A$ is invertible.
0
votes
1answer
17 views
0
votes
0answers
29 views

An analytic characterization of eigenvalues of a Hermitan matrix.

If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. (where we have $\lambda_i > \lambda_{i+1}$) for a vector $v$ let its component along the corresponding ...
2
votes
1answer
42 views

If det(A) is zero, what is det(adj(A))?

I wanted to prove that det(adj(A))=det(A)^n-1 for an nxn matrix A. I separate the proof in two cases: singular and non-singular matrix A. For the non-invertible A, det(A)=0. In my head, I know that ...
0
votes
3answers
42 views

Evaluating a determinant for eigenvalues

I need to evaluate $$\left| {\matrix{ {3 - \lambda } & 1 & 1 \cr 2 & {4 - \lambda } & 2 \cr 1 & 1 & {3 - \lambda } \cr } } \right|$$ A direct computation ...
1
vote
1answer
26 views

A question about the properties of the pseudospectrum

Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by $$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$ ...
3
votes
2answers
47 views

Inverse 4x4 matrix

If I have a 4x4 matrix (consists of a lot of irrational numbers) and want to calculate the inverse, what is the easiest/fastest way? The calculator I am allowed to use (casio fx991 es plus) can handle ...
1
vote
0answers
46 views

Rank of a matrix $A$ such that $A + A^T = 0$ [duplicate]

I need to prove (using only elementary operations and induction) that rank of a matrix $A\in \Bbb C^{n\times n}$ such that $A + A^T = 0$ is an even number. I know that elementary operations doesn't ...
0
votes
1answer
29 views

To find a complex symmetric matrix

I need to find a complex symmetric matrix $A$ such that there is no unitary matrix $P \in U_n(\mathbb{C}) $ with $PAP^{*}$ diagonal. I couldn't find one easily. I know that the unitary matrix ...
1
vote
2answers
24 views

Cardinality of a set of matrices

Consider the set $S$ of $3\times3$ matrices with binary coefficients, that is the coefficients are integers modulo 2. Compute $|S|$ I am not sure what is this question trying to ask. Am I right to ...
0
votes
3answers
64 views

Construct an example matrix such that $\mathbf A \mathbf A^T$ is not invertible

One theorem says: A matrix $\mathbf A \in \mathbb R^{m \times n}$ is: full column rank iif $\mathbf A^T \mathbf A$ is invertible full row rank iif $\mathbf A \mathbf A^T$ is invertible (proof ...
0
votes
2answers
31 views

Use row operation to find the determinant?

Use row operations to find the determinant: Can someone give me a full answer please? Also can anyone tell me if the sign of the determinant matters ? Row operations : Det ( e(A) ) = ...
0
votes
0answers
26 views

Using generalized eigenvectors as a basis for eigendecomposition?

As I understand, the eigenvectors of a Diagonalizable matrix form a basis. Hence, for a diagonalizable matrix A, any vector x can be written as: $x=\alpha_{1}v_{1}+\alpha_{2}v_{2}..+\alpha_{n}v_{n}$ ...
1
vote
1answer
37 views

Linear Transformation and Matrices

I have been studying linear algebra for a while now, and I still can't understand the basic concept of linear transformation and the easy ''translation'' of them the matrices. I understand that every ...
0
votes
1answer
23 views

existence of LU factorization

I am given a matrix $M=E-\alpha X$, where $E$ is an identity matrix, $0<\alpha<1$, and $0 \le X_{i,j} \le 1$ and the sum of every column of $X$ is 1. Does the matrix $M$ always exist a LU ...
0
votes
1answer
14 views

Interpretation of market completeness: full row rank payoff matrix

Suppose that there are $K$ assets and $S$ states of nature. The assets' payoff is represented by the matrix $$ \underbrace{R}_{S\times K}=\begin{pmatrix} r_{11}&\cdots& r_{K1}\\ ...