Number associated to a linear operator from a vector space $V$ to itself: $\lambda$ is and eigenvalue of $T\colon V\to V$ if the map $x\mapsto $\lambda x-Tx$ is not injective.
5
votes
1answer
38 views
Convergence of eigenvalues for sequence of compressions of a compact operator
Suppose $H$ is a separable Hilbert space, $A$ is a Hilbert Schmidt operator on $H$, and $P_n$ is an increasing sequence of finite rank orthogonal projections of $H$ (so $P_nx\rightarrow x$ for all ...
0
votes
1answer
67 views
How to show all eigenvalues are positive?
Could you help me to show that the following matrix has all its eigenvalues positive?
$$H=
\begin{bmatrix}
\sum_{k=1}^ng_1(x_k)^2 & \sum_{k=1}^ng_1(x_k)g_2(x_k) & \cdots & ...
0
votes
0answers
34 views
eigenvalue and independence
Let $B$ be a $5\times 5$ real matrix and assume:
$B$ has eigenvalues 2 and 3 with corresponding eigenvectors $p_1$ and $p_3$, respectively.
$B$ has generalized eigenvectors $p_2,p_4$ and $p_5$ ...
1
vote
3answers
50 views
Eigenvalue calculation.
I am getting confused by this simple eigenvalue calculation.
Calculate the eigenvalues of $\begin{bmatrix} 5 & -2\\ 1 & 2\end{bmatrix}$.
Firstly, I row reduce it, to go from ...
1
vote
1answer
31 views
Show that the following matrix is diagonalizable
My question is related to this question discussed in MSE.
$J$ be a $3\times 3$ matrix with all entries $1\,\,$. Then prove that $J$ is
diagonalizable.
Can someone explain it in terms ...
4
votes
3answers
58 views
Geometric multiplicity of repeated Eigenvalues
I am still finding it difficult to determine the geometric multiplicity for repeated eigenvalues and the resultant eigenspace. For example, I am not quite sure what to do with the following matrix, ...
0
votes
2answers
26 views
Let T : $\Bbb{R}^n\to \Bbb{R}^n$ be the linear operator with $T(e_i) = (1,…,1)$ for all $i = 1,\ldots,n$. Find an eigenbasis for $T$.
I know that if the matrix is an $n\times n$ matrix, the eigenvalues will be $n$ with alg. multiplicity $1$ and $0$ with alg. multiplicity $n-1$. I am having a hard time generalizing the eigenbasis ...
3
votes
0answers
81 views
+50
nonegative inverse eigenvalue problem
I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form:
$$
\mathbf{M} = \begin{vmatrix}
\mathbf{A} & \mathbf{b} \\
...
1
vote
0answers
19 views
Why is Lax's statement of the minimax principle stated only for real vector spaces?
I am reading Peter Lax's Linear Algebra. On page 116 he states the minimax principle as follows: for a real symmetric operator $H$ on an $n$-dimensional real vector space, suppose we label the ...
1
vote
2answers
28 views
Eigenvectors of inverse complex matrix
For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ ...
1
vote
1answer
36 views
Eigenvectors from eigenvalues doesn't add up
I having some trouble understanding how to find eigenvectors of a matrix.
I have the following matrix:
$A=a\left[ \begin{matrix}
-1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & ...
0
votes
1answer
38 views
diagonalizability and the Null space of eigenvector
If $A\in{\mathcal{C}^{n\times n}}$, and for each eigenvalue $\lambda$ of $A$, $N((A-\lambda I)^2)=N(A-\lambda I)$, prove that $A$ is diagonalizable.
2
votes
1answer
31 views
Jordan block in a matrix with a complex eigenvalue
Could you tell me, in general, how many Jordan blocks there are in the Jordan form of a matrix whose eigenvalue is complex, for example the matrix is $6 \times 6 $ and the eigenvalue is $3i+2$?
What ...
0
votes
2answers
28 views
Verify my attempt to diagonalize matrix
Decide whether matrix $A$ is diagonalizable. If so, find $P$ such that $P^{-1}AP$ is diagonal.
We are given: $A = \begin{bmatrix}1 & 0 & 0 \\ 1 & 2 & 1 \\ 0 & 0 & ...
0
votes
2answers
44 views
Eigenvalues of a block matrix
For $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
, how are eigenvalues of X
related to eigenvalues of A
?
3
votes
2answers
45 views
Prove $T$ has at most two distinct eigenvalues
The question is from Axler's Linear Algebra text. The $\mathcal{L}(V)$ stands for the space of linear operators on the vector space $V$.
Suppose that V is a complex vector space with dim ...
3
votes
1answer
41 views
Intersection of planes by forming 3d lines
If I have $n$ known planes (known normal vector and a point on a plane) that intersect each other in such a way so as to form closely located 3D lines, then
(1). To get a common single 3D line to ...
0
votes
0answers
26 views
Conditions for this subtractions.
Suppose A is a positive definite matrix and B a non-negative definite matrix.
Both A and B are symmetric. Is A+B positive definite? Find some conditions
for B when A − B is positive definite.
0
votes
1answer
37 views
Order of Eigenvalues and Eigenvectors
I am having a hard time trying to figure out the order eigenvalues and eigenvectors result in when trying to diagonalize a $2\times 2$ matrix:
$$\left[\begin{matrix}
3 & 1\\
1 & 3
...
2
votes
1answer
22 views
Linear Algebra dependent Eigenvectors Proof
Problem statement:
Let $n \ge 2 $ be an integer. Suppose that A is an $n \times n$ matrix and that $\lambda_1$, $\lambda_2$ are eigenvalues of A with corresponding eigenvectors $v_1$, $v_2$ ...
6
votes
2answers
41 views
Finding the number of symmetric,positive definite $10 \times 10$ matrices having…
I was looking at old exam papers and I was stuck with the following problem:
What is the number of symmetric,positive definite $10 \times 10$ matrices having trace equal to $10$ and determinant ...
2
votes
2answers
87 views
Matrices with at most one negative eigenvalue
Suppose a vector $y$ and a symmetric matrix $M$ are given.
\begin{equation}
\forall x; \quad x^Ty=0 \implies x^TMx \ge 0
\end{equation}
Prove that $M$ has at most one negative eigenvalue.
2
votes
1answer
48 views
Eigenvectors of matrices which commute with a projection
Just a quick question. Cant seem to prove it or find any relevant references! Maybe it's really simple :\ Is the following statement true (for square matrices of the same finite dimension)?
If there ...
2
votes
2answers
40 views
Compute eigenvalues and eigenvectors problem
I really don't know how solve this problem:
Let $V$ be the space of real functions spanned by $\cos(x)$, $\cos(2x)$ and $\cos(3x)$. Let $T\in\mathcal{L}(V,V)$ con $T(\cos(x)) = 3\cos(x) + 2\cos(2x) - ...
1
vote
1answer
20 views
Problem from Roman: a lower bound for trace of |T|^2
I'm working on a problem from Stephen Roman's Linear Algebra text, #20 on p. 235: Suppose $\tau \in \mathcal{L}(\mathbb{C}^n)$ and let the characteristic polynomial $\chi_{\tau}(x)$ have roots ...
2
votes
1answer
48 views
Determine invariant subspaces
imagine that a matrix of an endomorphism has the characteristic polynomial
$(\lambda-2)^2(\lambda-3)$
now i was wondering whether all invariant subspaces can be determined by $0,V$ and $\ker(A-2)^2, ...
1
vote
0answers
74 views
Eigenvalues of Block Anti-Diagonal Matrix
In line with this answer, I am trying to find the eigenvalues of:
$\mathbf P\mathbf K\mathbf P^\top=\begin{pmatrix}& d_1 & & & & & & \\d_1 & & e_1 & ...
1
vote
1answer
66 views
Minimal spectral radius of a primitive matrix
Given the set of all primitive matrices of dimensions $m$ by $m$ that are non-negative and integer - which one is the matrix with the minimal spectral radius?
Edit (according to the first comment):
...
6
votes
4answers
193 views
What is the fastest way to find the characteristic polynomial of a matrix?
Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$.
I know that:
the coefficient of $\lambda^n$ is $(-1)^n$,
the coefficient of ...
2
votes
2answers
75 views
Regarding a Paper by Paul A. Clement on Tridiagonal Matrices
I've asked this question at MathOverflow and was told it'd be better suited for here.
In Paul A. Clement's (1959) paper,
A Class of Triple-Diagonal Matrices for Test Purposes, SIAM Review, Vol. ...
0
votes
1answer
27 views
Problem related with boundary value problem and eigenvalue, eigenfunctions
I was looking at previous year exam papers and was stuck on the following problem:
For the boundary value problem, $\,\,y''+\lambda y=0; y(0)=0,y(1)=0, \,\,\exists$ an eigenvalue $\lambda$ ...
2
votes
2answers
77 views
Onto and one-to-one
Let $T$ be a linear operator on a finite dimensional inner product space $V$. If $T$ has an eigenvector, then so does $T^*$.
Proof. Suppose that $v$ is an eigenvector of $T$ with corresponding ...
3
votes
2answers
45 views
Injectivity of $A-\lambda I$
I'm reading a paper on determinants and on one point the author states that:
A complex number $\lambda$ is called an eigenvalue of matrix $A$ if $A-\lambda I$ is not injective.
Why is this? Could ...
1
vote
1answer
49 views
Matrices AB, BA eigenspaces [duplicate]
Take two matrices $A$ $n\times m$ and $B$ $m\times n$. They both have a nonzero eigenvalue $\lambda$. How do you prove that the dimension of the eigenspaces of $AB$ and $BA$ corresponding to $\lambda$ ...
1
vote
1answer
57 views
Eigenvalues and eigenvectors of AB and BA, proof.
$A$ is an $n \times k$ matrix and $B$ is an $k \times n$ matrix.
If $v_1, ..., v_l$ are linearly independent eigenvectors of $BA$ corresponding to a single nonzero eigenvalue $c$, then $Av_1, ..., ...
2
votes
2answers
71 views
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$, consider the linear map $T:M_2(\mathbb{R})\to M_2(\mathbb{R}):=B\to AB$ Then which of the following are true?
$T$ is ...
2
votes
1answer
101 views
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
Diagonalizable
Positive semidefinite
$0,3$ are only eigenvalues of $J$
Is positive definite
$J$ has minimal polynomial $x(x-3)=0$ so 1, ...
1
vote
1answer
39 views
Show that eigenvalues are negative
I have to consider the eigenvalue problem:
$$ L[u] := \frac{d^2 u}{dx^2}= λu,x \in (0,1)\quad u(0)-\frac{du}{dx}(0)=0, u(1)=0.$$
I need to show that the eigenvalues are negative.
2
votes
1answer
36 views
Matrices manipulation
I am having difficulty with the following question
I have to determine if the following claim is true or not.
If it is true I have to proof it else I need to give an example
I believe it is not ...
0
votes
1answer
33 views
Two Lagrange multipliers with one equation
I have an equation as below,
$$Rw = \lambda_1R_aw + \lambda_2R_bw $$
where, $R$, $R_a$, and $R_b$ are positive definite at least semi-positive definite and Hermitian matrix. $\lambda_1$ and ...
1
vote
2answers
39 views
Upper bound on the difference between two elements of an eigenvector
Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = ...
1
vote
1answer
36 views
Multiplicity of an eigenvalue is equal to $\dim V_{\lambda}$
I am trying to prove that multiplicity of an eigenvaliue $\lambda$ = $\dim V_{\lambda}$ and I have problems with this inequality:
$\dim V_{\lambda} \le $ multiplicity $\lambda$.
I know that ...
3
votes
2answers
96 views
REVISTED$^1$: Circumstantial Proof: $P\implies Q \overset{?}{\implies} Q\implies P$
To prove that if a matrix $A\in M_{n\times n} ( F )$ has $n$ distinct eigenvalues, then $A$ is diagonalizable is enough to show that the opposite holds? That is, if $A$ is diagonalizable, then $A$ has ...
0
votes
1answer
165 views
I want help with $4\times 4$ symmetric matrix
I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
5
votes
1answer
65 views
Generalized eigenspaces of a compact operator are finite dimensional
Let $T : H\rightarrow H$ be a compact operator on a Hilbert space $H$. Say that $\lambda \in \mathbb C$ is a generalized eigenvalue of $T$ if there is some $n \geq 1$ such that $(\lambda - T)^n$ is ...
1
vote
3answers
35 views
Eigenvector Proof $(I+A)^{-1}$.
Show that the eigenvectors of the $n \times n$ matrix A are also eigenvectors of the matrix $$M = (I+A)^{-1} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues.
My Work:
...
0
votes
0answers
23 views
Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector
Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries.
The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...
1
vote
1answer
39 views
Solve a System with Variable
Given these matrices, how does one find two real solutions?
$dx/dt$ =
$\begin{bmatrix}
3 & -5\\
5 & 3
\end{bmatrix}x$
with $x(0) = \begin{bmatrix}
2\\
-3
\end{bmatrix}$
1
vote
1answer
44 views
Same eigenvalues, different eigenvectors but orthogonal
I am using a two different computational libraries to calculate the eigenvectors and eigenvalues of a symmetric matrix. The results show that the eigenvalues calculated with both libraries are exactly ...
1
vote
2answers
37 views
Is $\varphi:x \mapsto A\cdot x$ an orthogonal projection for M
I got the transformation
$\varphi:x \mapsto A\cdot x$
and the matrix
$M = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}$.
I have to check whether $\varphi$ is the orthogonal projection for ...
