Tagged Questions

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
0answers
6 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
19 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
14 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 $h=\bar\delta$ ...
-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
28 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
0answers
25 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
27 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
17 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
21 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
36 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
25 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
63 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
49 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
6 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
22 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
16 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
40 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
102 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
25 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 ...
1
vote
1answer
40 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
41 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
45 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
45 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
23 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
22 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}\\ ...
1
vote
0answers
48 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Could you point out some references (undergraduate level) that give a geometric understanding of vector spaces, matrices, and linear applications? As far as I know, many textbooks start with ...
0
votes
1answer
14 views

Creating a matrix out of a column vector by matrix multiplication

Is there a way to transform a $(n*n) \times 1$ column vector into a $n \times n$ matrix that contains the entries of the vector in its columns using matrix products? Example: I want to transform ...
0
votes
1answer
26 views

Showing that the eigenvectors (when eigenvalue is 1) can be chosen to be integer valued

Suppose $A$ is an $d \times d$ matrix with integer entries. If there exists $\underline{n} \neq 0 \in \mathbb{R}^d$ such that $(A^T)^k \underline{n}= \underline{n}$. How can you show/justify that ...
-2
votes
1answer
19 views

Show that if one of the diagonal elements of a symmetric matrix is positive, then the matrix has at least one positive eigenvalue [on hold]

Show that if one of the diagonal elements of a symmetric matrix is positive, then the matrix has at least one positive eigenvalue.
1
vote
1answer
19 views

$A=(a_{ij})_{m\times n}$ real matrix, $n>m$ then I need to say which of the following are correct statements.

$A=(a_{ij})_{m\times n}$ real matrix, $n>m$ then I need to say which of the following are correct statements. $Ax=0$ has a solution $Ax=0$ has no nonzero solution $Ax=0$ has a nonzero solu ...
0
votes
0answers
12 views

Finding an overgroup or a subgroup in PGL

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$. Let $G=PGL_4(k)$. Let $H=\{ \small\left[\begin{array}{cccc} x & ...
0
votes
0answers
43 views

an interesting matrix [on hold]

I have a tri-diagonal infinite block matrix, as follows: $$S=\begin{pmatrix} 1 & & & &\\ & A_0 & & &\\ & & A_1 & &\\ & & & A_2 &\\ ...
0
votes
1answer
5 views

Show that the entries of the square of diagonal matrix are equal to the square of the entries of the diagonal matrix.

The question seems trivial which is why I have some trouble coming up with a proof that is mathematically correct. BTW I cannot yet use eigenvalues as we have not yet covered them in class. If ...
1
vote
1answer
26 views

Linear combination of vectors in orthogonal set

If an orthogonal set is linearly independent how can we get the linear combination of these vectors to form another vector that is in the orthogonal set? I thought linear independence meant you cant ...