0
votes
1answer
21 views

Finding eigenvalues and “eigenmatrices”.

On the space of $2\times 2$ matrices, let $T$ be the transformation that transposes every matrix. Find the eigenvalues and "eigenmatrices" for $A^T =\lambda A$. By taking determinants on the left and ...
0
votes
0answers
32 views

Build up a not diagonalizable linear map

I need an hint for this problem. Let be $M = \begin{bmatrix}2 & 1 \\ -2 & 0\end{bmatrix} \in M_2(\mathbb{K})$ and $H=\{A \in M_2(\mathbb{K}) : AM=MA \} $ Build up a linear map $f: ...
1
vote
2answers
43 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
1
vote
3answers
40 views

Find the Eigenvector of a matrix

Find the eigenvectors of the matrix $$\displaystyle\begin{bmatrix} 0 &2 &3 \\ -2 &0 &5 \\ -3 &-5 &0 \end{bmatrix}.$$ So I start with $|A-\lambda I|=0$ ...
1
vote
0answers
16 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
0
votes
1answer
43 views

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
2
votes
1answer
33 views

Perron–Frobenius theorem

What is exactly the Perron–Frobenius theorem? In different books papers I read different statments, and I don't know what is the truth. In wikipedia there are also a lot of statements under this ...
8
votes
1answer
78 views
+50

Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
3
votes
1answer
43 views

Is it possible that a matrix depicts like this?

Is it possible for $A\in \mathbb{C}^{n\times n}$, that $$\frac{|Ax|}{|x|}>|\lambda_{max}|$$ where $\lambda_{max}$ is the biggest eigenvalue of A? I know this can not happen, if there is a basis of ...
2
votes
1answer
24 views

Finding eigenvectors for the largest eigenvalue vs one with the largest absolute value

If I want to solve a generalized eigenvalue problem such as: $$A x = \lambda x$$ The problem is to find eigenvectors corresponding to the largest eigenvalues (sometimes in an optimization problem ...
2
votes
1answer
26 views

Eigenvalues of ad (Adjoint action) in semisimple lie algebra?

Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
0
votes
1answer
33 views

Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
1
vote
1answer
64 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
0
votes
1answer
25 views

Quadratic form in canonical form relation [on hold]

The homogeneous quadratic form can be written as a matrix. It is also written as a canonical form by using orthogonal transformation. Why we are going for canonical form and what is the relation ...
-1
votes
0answers
21 views

Positive definite [closed]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
0
votes
1answer
35 views

Positive definite matrix. [on hold]

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
0
votes
2answers
72 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
3
votes
3answers
108 views

Eigen values of AB and BA

let A be a linear transformation from $R^n$ to $R^m$, and B be a linear transformation from $R^m$ to $R^n$, it's easy to show that AB and BA has same eigen-value(except $0$). But my question is how ...
3
votes
2answers
43 views

“Sandwich theorem” for eigenvalues of symmetric matrices

I am looking for a reference for the following result for symmetric matrices Let $A\in\mathbb R^{n\times n}$ be symmetric with eigenvalues $\lambda_n \leq\ldots\leq\lambda_1,\, M\subset \lbrace ...
3
votes
1answer
66 views
+100

Spectrum of matrix with single scaled row

Let $M$ be a real symmetric positive-definite matrix and $D_a$ the diagonal matrix $$D_a = \left[\begin{array}{ccccc}a & & & &\\& 1 & & &\\& & 1 & ...
1
vote
0answers
32 views

Eigenvalues of a certain $3\times 3$ matrix [closed]

could you help me to find the eigenvalues of the matrix P? Thanks in advance. Assume that three column vectors ${\bf x}_1$, ${\bf x}_2$, and ${\bf x}_3$ of dimension $N+1$ and with unitary Euclidean ...
11
votes
3answers
286 views

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
1
vote
2answers
62 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
0
votes
2answers
62 views

Eigenvalues of a matrix that is a product of a vector and transpose vector

Find eigenvalues, eigenvectors and rank of matrix $A$. $$\textbf{a}=\begin{bmatrix}a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix}, \quad \textbf{b} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ ...
0
votes
2answers
35 views

Diagonalizing a matrix. Which formulae is correct?

In my coursebook on linear algebra on some page I see that a diagonal matrix $D$ for a matrix $A$ that can be diagonalized ca be found as follows: $$\tag{1}D=T^TAT$$ But reading further I see that my ...
0
votes
0answers
20 views

How to get transformation matrix for Linear Discriminant Analysis from eigen values?

I am trying to implement Linear Discriminant Analysis. I have 2 questions. A)Can I directly use the matrix with eigen vectors of the product of between scatter matrix inverse and within scatter ...
0
votes
1answer
40 views

Trouble understanding the diagonal matrix theorem.

The Diagonal Matrix Representation Theorem states: Suppose $A=PDP^{-1}$, where $D$ is a diagonal $nxn$ matrix. If $B$ is the basis for $R^n$ formed from the columns of $P$, then $D$ is the $B$-matrix ...
3
votes
3answers
65 views

Prove that $T^n$ is diagonalizable.

Prove or give a counterexample: If $V$ is a complex vector space and $\text{dim V} = n$ and $T \in L(V)$, then $T^n$ is diagonalizable. In order to show that $T$ is diagonalizable I need to show ...
1
vote
0answers
14 views

Is there any restriction to the sum of eigenvalues for non-negative, irreduceble and square matrices?

I'm trying to find if there is a restriction in tr(A) or eigenvalues sum for a non-negative, irreducible square matrix A. As an additional information, the row sums and the order of the matrix is ...
1
vote
1answer
29 views

Lower bounds on eigenvalues of a symmetric matrix based on the diagonals

A symmetric matrix $A$ always has real eigenvalues. If I know the elements on the diagonals, is it possible to have a lower bound on the smallest eigenvalue? How sharp would this bound be? For now I ...
0
votes
1answer
26 views

How do get eigenvalues of a matrix B if add a row/column pair of a matrix A?

I have a matrix of size N×N of the form: where and A is N-1 x N-1 matrix, a=0. I known the eigenvalues of A. Any possible for getting eigenvalues of B from eigenvalues of A?
0
votes
1answer
51 views

Eigenvalues and eigenvectors of a non-symmetric matrix which is a product of 2 symmetric matrices?

I have a non symmetric matrix $AB$ where $A$ and $B$ are symmetric matrices. How can I find the eigenvectors and eigenvalues of $AB$? In a paper( Fisher Linear Discriminant Analysis by M Welling), ...
0
votes
0answers
21 views

How to get the transformation matrix for Linear Discriminant Analysis?

I am trying to implement Linear Discriminant Analysis. Is the eigen vectors of the product of within scatter matrix and between scatter matrix inverse (Sw*Sbinverse), the transformation matrix? Could ...
1
vote
3answers
32 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
0
votes
2answers
37 views

How does a matrix change the magnitude of a vector?

I have the following problem: $z=Ax$, in which $z$ and $x$ are $N\times 1$ vectors and $A$ is a $N\times N$ matrix. I am interested in how the magnitude of $x$ changes after applies $A$ on it. Is ...
3
votes
3answers
42 views

How to get the two eigen vectors for eigen =1

I have to find the eigen vectors for this matrix. \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{pmatrix} I end up with this matrix to plug in the eigen values. ...
0
votes
1answer
50 views

Relation between eigenvectors after transforming a nonsymmetric matrix to symmetric?

I need to find eigenvectors and eigenvalues of a matrix which is product of 2 symmetric positive definite matrix(SwInverseSbProd=SwInverse*Sb). Since SwInverseSbProd is non-symmetric and calculation ...
1
vote
0answers
16 views

How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...
1
vote
1answer
78 views

Finding eigenvalues of a block matrix

I have a block matrix of size $2N \times 2N$ of the form $$B = \begin{bmatrix} A_N & C_N \\ C_N & A_N \end{bmatrix}$$ where $A_N$ and $C_N$ are both $N \times N$ matrices. Specifically, $$A_N ...
0
votes
1answer
18 views

Eigenvectors and the relationship between variables in a system of equations.

I am learning about complex eigenvalues in Linear Algebra and I am confused with one problem. I have a matrix in $A-\lambda I $ form. For the eigenvalue $\lambda=3+2i$, $A-\lambda I=\begin{bmatrix} ...
1
vote
1answer
38 views

Need help understanding the proof: if v is a left singular vector of A then v is a unit eigenvector of $AA^{T}$

This is the proof in my textbook: What I don't understand it why " $AA^{T}u = 0u $ means that u is an eigenvector. Is this a theorem that I don't know? That if you multiply a matrix by a vector ...
2
votes
0answers
35 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
1
vote
1answer
34 views

Eigenvalues of a Matrix Using Diagonal Entries

I just started learning about complex eigenvalues and eigenvalues and one example in the book I am using says that the matrix $A = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$. The book then ...
1
vote
0answers
31 views

What is (are) the condition(s) for sum of a non-singular matrix and its transpose to be non-singular

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ be a real symmetric positive definite matrix (i.e., $\mathbf ...
0
votes
1answer
35 views

Eigen vectors of the matrix whose columns are eigen vectors of the original matrix

Consider a matrix $A$ of dimension $n$X$n$ whose eigen vectors are $y_1,y_2,y_3,...,y_n$ and are linearly independent. What are the properties of the eigen vectors of the matrix $P$ whose columns are ...
0
votes
1answer
31 views

How to prove that for any real n*n matrix, the eigenvalues are real or are a complex conjugate pair?

I'm trying to show that for any square matrix (whose entries are all real) the eigenvalues are real or are complex conjugate pairs. I've tried so far by stating that for 2*2 matrices, finding the ...
1
vote
2answers
104 views

How do you quickly find the eigenvalues of this matrix?

I have a final exam tomorrow, am sure a 3x3 eigen value problem like the one below is there. But I find it very hard to find eigen values without zeros in the matrix Show me how you do it quickly so ...
0
votes
1answer
31 views

Normal matrices with constraints on eigenvalues

I need to find two different transformations with an inner product that follows these rules: (I don't study math in english so I'll try my best to explain) 1) a. $TT^*=T^*T$ (normal) b. All ...
0
votes
2answers
40 views

Determinant Formula for Tri-Diagonal Matrix

for an assignment in numerical analysis, I need to find the eigenvalues of a matrix with values only in the diagonal, upper diagonal and lower diagonal. I guess there is an easy formula for this sort ...
1
vote
3answers
101 views

Show that $AB^{-1}$ and $B^{-1}A$ have the same eigenvalues

Let $A$ and $ B$ be two nonsingular matrices. Show that $AB^{-1}$ and $B^{-1}A$ have the same eigenvalues My attempt: $$ \begin{align} f(\lambda) &= | I\lambda -AB^{-1}| \\ &= ...