Tagged Questions
1
vote
1answer
34 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
86 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 ...
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
38 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
34 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 ...
0
votes
1answer
24 views
Finding the x value after a matrix multiplication?
I have the following solution of a problem, and I was wondering about a hopefully quite simple thing in it:
I was wondering how do they get from [5,10,5] to 5x? I am pretty sure there is a simple ...
1
vote
1answer
22 views
Eigenvalues of $\sum_{i=1}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}$
Consider the cuadratic form
$$
\mathbf{x}^{\intercal}Q\mathbf{x} = \frac{x_1^2}{\lambda_1} + \sum_{i=2}^n \frac{(x_i - x_{i-1})^2}{\lambda_i}\ .
$$
Is it true that the eigenvalues of $Q$ are ...
3
votes
2answers
74 views
Diagonalising a $2 \times 2$ and $3 \times 3$ matrix
For each of the following matrices $A$, find an invertible matrix $P$ over $C$ such that
$P^{-1}AP$ is upper triangular:
$$A = \begin{bmatrix}4 & 1\\-1 & 2\end{bmatrix} \quad \text{ ...
0
votes
1answer
33 views
Normal matrices with orthogonal basis
we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} ...
0
votes
2answers
47 views
Can you have a matrix with no zero rows that has an eigenvalue equal to zero
Can you have a matrix with no zero rows that has an eigenvalue equal to zero?
So for $A \in \mathbb{R}^{n \times n}$ with $0$ as an eigenvalue, is the $\text{rank}(A)$ always smaller than $n$?
2
votes
1answer
52 views
Linear Algebra - Can I use RREF to solve augmented matrix for eigenvector?
I am solving David Lay's 4th edition 7.1 number 16. So here's the problem. The original matrix: $$\begin{bmatrix}-7 & 24 \\ 24 & 7\end{bmatrix} \ ,$$ with eigenvalues $\pm 25$.
I am having ...
8
votes
5answers
167 views
Matrices with eigenvalues 0 and 1
How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1?
My attempt:
I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
1
vote
1answer
50 views
Image of linear transformation equal to eigenspace
Question is as follows:
Suppose $A^2 = I$ (the identity matrix) and F = Q, R or C. Eigenvalues of A are then $\lambda=1$ or $\lambda=-1$. Show that $\ker(L (I+A))=E(-1)(A)$ and that $im(L ...
2
votes
2answers
93 views
How find this matrix has eigenvalues $\lambda_{j}=4\sin^2{\dfrac{j\pi}{2(n+1)}}$
Show that the $n\times n$ tridiagonal matrix
$$A=\begin{bmatrix}
2&-1&0&0&0\\
-1&2&-1&0&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
...
1
vote
0answers
46 views
Vandermonde question
I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem):
I have ...
0
votes
2answers
34 views
Does a constant eigenvalue of a linearly parameter-dependent matrix have a constant eigenvector?
Let $A(\alpha)=A_0+\alpha A_1$, with $A_0,A_1\in\mathbb R^{n\times n}$ and $\alpha\in\mathbb R$, such that there exists a $\lambda\in\mathbb C$ with the property that
for all $\alpha$: $\det(\lambda ...
3
votes
1answer
148 views
Sum of eigenvalues and singular values
How one can prove that for a matrix $A\in \mathbb{C}^{n\times n}$ with eigenvalues $\lambda_i$ and singular values $\sigma_i$, $i=1,\ldots,n$, the following inequality holds:
$$ \sum_{i=1}^n ...
0
votes
1answer
37 views
$\lambda_{min}\left (\frac{A+A^*}{2} \right )\leq \sigma_{min}(A)$
For $A \in \mathbb{C}^{n \times n}$, how to show that
$\displaystyle \lambda_{min}\left (\frac{A+A^*}{2} \right )\leq \sigma_{min}(A)$?
2
votes
2answers
63 views
Minimum eigenvalue and singular value of a square matrix
How to show that the relationship $\left | \lambda_{min} \right | \geq \sigma_{min}$ holds between the minimum eigenvalue and singular value of a square matrix $A \in \mathbb{C}^{n \times n}$?
0
votes
1answer
66 views
Largest and smallest eigenvalues of a hermitian matrix
How to show that the largest and smallest eigenvalues of a hermitian matrix $A \in \mathbb{C}^{n \times n} $ can be found as:
$\displaystyle \lambda_{max} = ...
1
vote
2answers
89 views
The relationship between eigenvalues of matrices $XY$ and $YX$
If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove that $\lambda (XY) = \lambda (YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$? Here, ...
1
vote
1answer
39 views
How would I find this eigenvalue?
I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on ...
2
votes
1answer
61 views
Show AB and BA have the same eigenvalues [duplicate]
If $A$ and $B$ are $n$ by $n$ matrices show that $AB$ and $BA$ have the same eigenvalues. I see why this is true if both are nonsingular. But does it still hold if they are not invertible?
Thanks!
0
votes
2answers
26 views
Quick question about proofs of theorem concerning Jordan basis
I have a question about proofs of this theorem:
Let $K$ be an algebraically closed field, $V$ be
a finite-dimensional space over $K$ and $f : V → V$ be a linear operator.
Then there exists a Jordan ...
0
votes
0answers
22 views
Inverse of Eigen value
What is the physical meaning of inverse square root of the eigen value? Is it possible to use it as stretch factor to decorrelate the data.
0
votes
1answer
63 views
Eigenvector with eigen value of 1
How is an eigenvector with eigen value of 1, say v, multiplied by its transpose the identity matrix? v' * v = I?
-1
votes
1answer
25 views
diagonalizing a matrix $A$: can $P$ be bigger than $A$?
can you have a P bigger than the original A matrix? in other words after I found the eigenvalues I then found all the eigenvectors so when I constructed the P vector turns out to be bigger than my ...
0
votes
1answer
20 views
Perturbation parameters in Eigenvalue question [easy]
I'm solving an eigenvalue/eigenvector question of the matrix:
\begin{bmatrix}
2 & 1 \\
0 & 2 + \varepsilon
\end{bmatrix} where $\varepsilon$ is the perturbation parameter. Would I just solve ...
0
votes
0answers
33 views
Diagonalizing the sum of a matrix and a multiple of the identity matrix
Suppose we have a matrix $A = B+\lambda I$, where $B\in \mathbb{R}^{n\times n}$, $I$ is the identity matrix and $\lambda\in \mathbb{R}$. If I know the eigenvalues and eigenvectors of $B$, what can I ...
0
votes
2answers
25 views
Matrix multiplication and eigen vectors
If $a$ is a right eigenvector of $S$ and $R^T$ with eigenvalue $1$. How would determine $a^TRSa$? Is $Sa$ simply $a$? Any hints that apply here would be greatly appreciated.
0
votes
0answers
21 views
Properties of eigenvectors
If a is a right eigenvector of Z and b is a right eigen vector of Y, is a * b' a right eigenvector of Z * Y?
4
votes
1answer
39 views
To prove that the dimension of $V$ is $d_1^2 + \ldots + d_k^2$
Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial
$$(x - c_1)^{d_1} \cdots (x - c_k)^{d_k} , $$
where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n \times n$ ...
1
vote
1answer
48 views
Differential Equation: Complex Eigenvalue
For the following system
$$ x'=\left( \begin{array}{ccc}
\frac{-1}{2} & 1 \\
-1 & \frac{-1}{2} \end{array} \right)x $$
To find a fundamental set of solutions, we assume that $$ x = Ee^{rt}$$
...
2
votes
2answers
60 views
Eigenvalues and Eigenvectors of $X'X$ and $XX'$
I am trying to derive (or prove) the relationship between the eigenvalues and eigenvectors of the matrices $X'X$ and $XX'$. It is fairly intuitive that they are related but I cannot derive the ...
0
votes
3answers
31 views
How to find solutions for linear recurrences using eigenvalues
Use eigenvalues to solve the system of linear recurrences
$$y_{n+1} = 2y_n + 10z_n\\
z_{n+1} = 2y_n + 3z_n$$
where $y_0 = 0$ and $z_0 = 1$.
I have absolutely no idea where to begin. I understand ...
3
votes
2answers
55 views
Proof of the linear independence of the generalized eigenvectors of a square matrix
I'm currently stuck on this problem:
Let $V$ be a finite dimensional vector space. If $S: V\rightarrow V$ and $T: V\rightarrow V$ are linear maps and $ST=TS$, prove every eigenvalue of $ST$ is a ...
3
votes
2answers
47 views
Linear algebra, eigenvectors problem
Suppose you know that A is $2x2$ and symmetric.
Assume the eigenvalues are $4$ and $7$.
An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$?
So first we let ...
0
votes
1answer
23 views
Linear algebra eigenvalues and limits problem
Suppose a matrix $A$ has eigenvalues of $-0.9$, $0.8$, $\pm 0.5$ and $0.9 \pm 0.2i$. What can you say about $\displaystyle\lim_{k \to \infty} A^k$ when $k$ approaches infinity ?
Thanks.
1
vote
0answers
40 views
Help find a proof : $ \lambda $ is $f$'s eigenvalue then $f|_{V_{\lambda}} $ has Jordan's basis
Could you help me find a fairly simple proof of the following theorem?
$f: V \rightarrow V, \ \ \dim V < \infty, \ \ \lambda$ is $f$'s eigenvalue $\Rightarrow \ \ f|_{V_{\lambda}}: \ V_{\lambda} ...
0
votes
1answer
19 views
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in\mathbb{C}^n$ the matrix $E(v)=uv^*$. Give a polynomial $p$ such that $p(E)=0$.
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in
\mathbb{C}^n$ the matrix $E(v)=uv^*$.
Give a quadratic polynomial $p$ such that $p(E)=0$.
I know that the eigenvalue of ...
0
votes
1answer
24 views
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in \mathbb{C}^n$ the matrix $E(v)=uv^*$. Find all eigenvalues of $E(v)$.
Let $0 \ne u \in \mathbb{C}^n$ fixed and consider for every $v \in
\mathbb{C}^n$ the matrix $E(v)=uv^*$.
Find all eigenvalues of $E(v)$.
$E(v)=uv^*=
\begin{pmatrix}
u_1\\
\vdots \\
u_n
...
2
votes
1answer
104 views
What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?
Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
2
votes
3answers
63 views
Finding eigenvectors with square root eigenvalues
I have a matrix
$$\begin{bmatrix}1 &-1 &2\\2 &-2 &4\\0 &1 &1\end{bmatrix}$$
Its eigenvalues are $0$, $\sqrt{5}$ and $-\sqrt{5}$
(These are checked in MATLAB to be correct). I ...
1
vote
0answers
44 views
Linear Algebra: Linear transformation and eigenvalues [duplicate]
Hi could some one please help. I am having problems proving this.
Let $A$ be an $n \times n$ matrix with complex entries and let $f (t) =\det(A - tI)$ be its characteristic polynomial.
Prove ...
0
votes
1answer
42 views
Hermitian matrices [duplicate]
Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such
that
$\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H.
How to prove ...
1
vote
1answer
106 views
Hermitian Matrices are Diagonalizable
I am trying to prove that Hermitian Matrices are diagonalizable.
I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...
2
votes
2answers
57 views
Dimension of the corresponding eigenspace?
I'm studying for my linear exam and would appreciate any help for this practise question:
You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace?
A = ...
3
votes
1answer
69 views
Does this matrix have strictly positive eigenvalues?
Consider a square matrix $A$ with real entries and a diagonal matrix $B$ with strictly positive elements. Assume that the symmetric part of $A$, i.e. $\displaystyle \frac{A^T+A}{2}$, has strictly ...
0
votes
0answers
35 views
A Has characteristic polynomial that can be reduced to linear products $\Rightarrow$ A similar to upper triangular Matrix
Prove that if $A\in M_{n}\left(\mathbb{F}\right)$ matrix with a characteristic polynomial that can be written as a product of linear elements (?) ...
2
votes
2answers
37 views
Generalised eigenvalue is eigenvalue if it is in the field
I would like to prove the following assertion:
Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
