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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2
votes
1answer
15 views

Exponential of Matirx

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc) to calculate this exponential e^At with A (0 9) (-1 0) I'd ...
2
votes
0answers
10 views

Show that $EA$ is obtained from an elementary row operation on $A$

Suppose $E$ is an elementary $n \times n$-matrix. Prove that if $A$ is any $n\times n$-matrix and $E$ is any elementary matrix, then $EA$ is a matrix obtained by carrying out a single elementary row ...
1
vote
0answers
17 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
-1
votes
0answers
9 views

Number of possible graphs from a reachability matrix?

I need to know how to work out how many possible different digraphs can be drawn from a given reachability matrix. It needs to be with the minimum number of arcs between the nodes within the graph ...
0
votes
1answer
20 views

Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), ...
0
votes
0answers
17 views

Can I reform this to a tensor/matrix product?

so I have the following vector matrix product: $$v = A w$$ Now I have this $n$-times: $$v^{(n)} = A^{(n)} w^{(n)} \quad \forall n$$ Is there any way to write this without $\forall$. Maybe somthing ...
0
votes
0answers
20 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
2answers
16 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 ...
1
vote
1answer
16 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
9 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
25 views

Proving a theorem 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
34 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 = ...
-1
votes
0answers
16 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
32 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
27 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
31 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
22 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
31 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
51 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
9 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
23 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 ...
1
vote
5answers
109 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
43 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
43 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
27 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
49 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
32 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) ) = ...