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.
0
votes
2answers
12 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
12 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
56 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
24 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
19 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
30 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
22 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
19 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?
1
vote
2answers
25 views
Computing eigenvalues for $\mathrm{Sym}^2(\mathrm{Sym}^3 V))$ for $V = \Bbb C^2$
Given $V = \Bbb C^2$ the standard representation of $\mathfrak{sl}_2\Bbb C$, on page 157 of Fulton and Harris's Representation Theory they state
Since $U = \mathrm{Sym}^3 V$ has eigenvalues $-3, ...
4
votes
1answer
37 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
42 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 $$ x = Ee^{rt}$$
...
1
vote
2answers
50 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
27 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
40 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
44 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 ...
