0
votes
2answers
41 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
1answer
34 views

Relation on the determinant of a matrix and the product of its diagonal entries?

Let $A$ be a $3\times 3$ symmetric matrix, with three real eigenvalues $\lambda_1,\lambda_2,\lambda_3$, and diagonal entries $a_1,a_2,a_3$, is it true that \begin{equation*} \det ...
3
votes
1answer
49 views

$3 \times 3$ real matrix: relation with determinants

$A$ is a $3 \times 3$ matrix with real entries such that $\operatorname{det}(A+I_3)=\operatorname{det}(A+2I_3)$. Then is $2\operatorname{det}(A+I_3)+\operatorname{det}(A-I_3)+ 6 =3 ...
5
votes
0answers
101 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
0
votes
2answers
85 views

$\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let A be real square matrix. If $\det (A^2 - I) < 0$, then A has eigenvalue $\lambda \in (-1,1)$. How to prove this?
0
votes
2answers
75 views

Prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$

I want to prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$ If $A\in \mathbb{F}^{m\times n}$ and $B\in \mathbb{F}^{n\times m}$ It is easy to show that $0$ has algebraic multiplicity of at least $m-n$ ...
2
votes
0answers
41 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
1
vote
0answers
43 views

Prove that every proper principal submatrix of $\lambda I-A$ is nonsingular under certain assumptions

Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda ...
1
vote
1answer
49 views

Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
16
votes
10answers
2k views

Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
2
votes
1answer
60 views

Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
4
votes
2answers
88 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
4
votes
2answers
119 views

Determinant of rank-one perturbation of a diagonal matrix

Let $A$ be a rank-one perturbation of a diagonal matrix, i. e. $A = D + s^T s$, where $D = \DeclareMathOperator{diag}{diag} \diag\{\lambda_1,\ldots,\lambda_n\}$, $s = [s_1,\ldots,s_n] \neq 0$. Is ...
0
votes
3answers
49 views

Determinants and eigenvectors [duplicate]

Hello, I'm trying to work through this question. I define linearly independent as: $a_1*v_1+a_2*v_2+...+a_n*v_n = 0$ iff every $a_i=0$. I also know that an eigenvector is a vector $v$ such that: ...
2
votes
2answers
82 views

Does this matrix have negative eigenvalues?

Suppose I have the following square block-matrix $A= \begin{pmatrix} M M^\dagger & F \\ F^\dagger & M^\dagger M \end{pmatrix}$ where $\det(M M^\dagger)=0$. 1) Does the matrix A have a ...
1
vote
2answers
42 views

If we add $I$ to a matrix $M$, does that mean we always add 1 to each of $M$'s eigenvalues?

Title says it all, Suppose we have a matrix $\mathbf{M} \in \mathbb{R}^{N \ \text{x} \ N}$, with eigenvalues $\lambda_i$, for $\ i = 1, 2 ... N$. If we now add the identity matrix $\mathbf{I}$ to ...
0
votes
1answer
73 views

I have a 2x2 positive-semidefinite matrix. I am trying to find the equation of its elements.

So long story short. I have a matrix $A \in S^2_+$, that is, a symmetric, positive semi-definite 2x2 matrix. Here it is: $A = \begin{bmatrix} x & y \\y & z \end{bmatrix}$. Here is what it ...
4
votes
4answers
194 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
6
votes
5answers
233 views

If $A^T=-A$, then A is not invertible

Let $n \in \mathbb{N}$ be odd and $A \in$Mat$(n,\mathbb{R})$ with $A^T=-A$. Show that $A$ is not invertible. I have no idea how to start this...
2
votes
1answer
53 views

Finding Jordanizing matrix

Let $$A=\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ And I found B, A's Jordan form to be: $$B=\left(\begin{matrix}0&1&0 \\ 0&0&0\\ ...
2
votes
0answers
34 views

The trace of a matrix is the sum of its eigenvalues [duplicate]

If $A$ is a complex square matrix, I need to prove that the trace of $A$ is the sum of its eigenvalues. I've already proved that, if $p(x)$ is the characteristic polynomial, then ...
1
vote
2answers
80 views

About eigenvalues and complex matrix

If $A$ is a square complex matrix with $n$ rows, prove that the constant term of the characteristic polynomial is equal to $(-1)^ndet(A)$ and that the coefficient of degree $n-1$ is equal to $-Tr(A)$ ...
1
vote
1answer
92 views

Determinant of PSD matrices

I'm trying to show that the determinant of X is the product of the eigenvalues. How would I do this? I know I have to do eigenvalue decomposition but I'm not sure how to proceed.
1
vote
3answers
142 views

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue? I don't really know where to start with this one. I know that ...
3
votes
3answers
257 views

Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A

So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$ Let $u$ be the eigenvector for the eigenvalue -1. Let $S = A^3 + 2A^2 - A - 5I$ then ...
1
vote
1answer
185 views

Finding Eigenvalues of Block Matrix

I have a block matrix of size $3N \times 3N$ of the form: $B = \left[\begin{array}{cccc} A & C & \dots & C\\ \vdots & A & \dots & C\\ C & \vdots & \ddots & ...
1
vote
2answers
107 views

Show that $\det{(A + sxy^*)}$ is linear in $s$

Suppose that $A \in \mathbb{R}^{n\times n }$, and $x,y\in \mathbb{R}^n$. Show that there exist numbers $a,b$ so that $\det{(A + sxy^*)}=a+bs$ Show that if $\det{(A)}\neq 0$ then $a = \det{(A)}$ ...
0
votes
1answer
41 views

How to compute $\text{det}((K+D)^{-1}K)$

Given the eigendecomposition of the positive semi-definite matrix $K = Q\Lambda Q^T$, and a diagonal matrix $D$ with positive diagonal elements, is there an efficient way to compute the determiant ...
1
vote
0answers
194 views

How to determine that a certain eigenvalue is doubly degenerate?

Given a symmetric matrix $X$. I ask myself, how to determine that a certain eigenvalue $\lambda$ is (exactly) doubly degenerate? I thought about several approaches: Calculate the derivative of $ ...
3
votes
1answer
418 views

On the difference of two positive semi-definite matrices

I am relatively new to linear algebra, and have been struggling with a problem for a few days now. Say we have two positive semi-definite matrices $A$ and $B$, and further assume that $A$ and $B$ are ...
9
votes
4answers
1k 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 ...
4
votes
2answers
148 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 ...
2
votes
1answer
1k 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 ...
4
votes
2answers
188 views

Math hack for solving system of equations

Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
3
votes
1answer
108 views

eigenvalues of block matrix with the eigenvalues of one block already known

Give a matrix which can be decomposed into 4 parts $B = \left[\begin{matrix}A &I \\ -I &0\end{matrix}\right]$ where $I$ denotes the identity matrix and $0$ is a zero matrix. It's easy to ...
2
votes
4answers
1k views

Determinant of matrix exponential?

Suppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$, where $\sigma(A)$ is the multiset of eigenvalues of $A$? The ...
1
vote
0answers
109 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
5
votes
3answers
198 views

Formal proof of $\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i)$

I'm looking for a formal proof for: $$\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i).$$ I'm very new to matrix theory therefore please forgive me if you find this elementary. Your help in this ...
0
votes
2answers
256 views

Eigenvalues, minimal polynomials and characteristic equation

What is the difference between solving $\det(xI- A) = 0$ and $\det(A-\lambda I) = 0$ to find eigenvalues of a Matrix $A$? Is the only difference that the first equation will give you the ...
0
votes
2answers
153 views

Find the determinant

I am trying to find the eigenvalues of a matrix and I cannot remember how to find the determinant of $A-\lambda I$: \begin{equation} \pmatrix{1-\lambda& 2& 1 \\ 2& -\lambda & ...
1
vote
1answer
493 views

Eigenvalues of A and associated determinant

Let A be a 4 x 4 matrix. a) If the eigenvalues of A are 1,-2,3,-3, is it possible to determine det(A)? Why or why not? b) What if the eigenvalues are -1,1,2? c) What if the eigenvalues are -1,0,1? ...
5
votes
0answers
58 views

What about other symmetric functions of the eigenvalues? [duplicate]

Possible Duplicate: Identities for other coefficients of the characteristic polynomial Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
7
votes
4answers
587 views

Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$

Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
5
votes
1answer
195 views

What is $\frac{\det(A+tI)}{\det(B+tI)}$ as $t\to0$?

If $A$ and $B$ are two real $2\times 2$ matrices with $\det A = 0 $ and $\det B = 0 $ and $\mathrm{tr}(B)$ is non zero. then what will be limit of $$\lim_{t\to0}\frac{\det(A+tI)}{\det(B+tI)}$$ I used ...
4
votes
1answer
158 views

Can we say that there exist an integer n such $A+nB$ invertible?

If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exist an integer $n$ such that $A+nB$ invertible? I was trying by choosing n such that eigne values of $A+nB$ ...
4
votes
1answer
189 views

$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant

Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to $1$. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the ...
1
vote
2answers
296 views

Linear Algebra - Finding Eigenvalues of a Matrix

$A=\begin{bmatrix}3 & -2 & 5\\ 1 & 0 & 7\\ 0 & 0 & 2\end{bmatrix}$, Find the eigenvalues of A. I realized that if I swap columns I and II then I can make it an upper ...
3
votes
1answer
188 views

Proving that an $n\times n$ matrix has at most $n$ distinct eigenvalues

$A$ is a $n\times n$ matrix over the field $F$. How can I prove that there are at most $n$ distinct scalars $c$ in $F$ such that $\det(cI - A) = 0$? Thank you!
4
votes
1answer
928 views

Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal ...
2
votes
1answer
131 views

complexity cost for which one is greater : determinant or eigen values?

what is complexity cost for determining all of eigen values? what is complexity cost for calculating determinant ?