Tagged Questions
1
vote
4answers
29 views
Prove that if $p(A)=0$ where A is a matrix of a linear operator ($A \in L(V)$), $p(\lambda)=0$ if $\lambda \in \sigma(A)$
I think it's all in the title. $p$ is some random polynomial.
I don't know how to approach this one. I've tried taking the roots of $p$, placing them on the diagonal of a new matrix and reasoning that ...
0
votes
0answers
34 views
Vandermonde matrix and polynomials
Question attached as image, deals with polynomials of order N and determinant of
Vanderbilt matrix.
0
votes
1answer
23 views
Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$
I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
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 ...
5
votes
0answers
58 views
My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible
Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial.
Here's my ...
1
vote
1answer
31 views
If $f(X) = a_0 + a_1 X + a_2 X^2 \in \mathbb{F}[X]$ then show $f$ is uniquely determined by $f(x)$, $f(y)$, $f(z)$?
This is the exact question:
It's part(ii) that I don't understand - what does it mean and what is it asking me to do? How would I go about constructing a proof? Any help would be much appreciated.
1
vote
1answer
162 views
Similar matrices and minimal polynomial
I guess that I'm missing here something...
How to prove that, if two matrices are similar, then their minimal polynomials are the same one.
Thanks!
1
vote
0answers
32 views
Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?
I come across a theorem that
$f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda
^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$
where $a_{i}$ is the sums of all ith order ...
4
votes
2answers
44 views
On similar matrices and polynomial matrices
Let $A,B,P\in M_n(F)$. Suppose that $A$ and $B$ are similar, thus $A=P^{-1}BP$.
If $p(x)=a_0+\ldots+a_nx^n$, and $T:V\to V$ be a linear transformation. Defining $$p(T)=a_0I+\ldots+a_nIT^n$$
How to ...
2
votes
2answers
37 views
why this is correct: $\det(C+Di)$ is not zero, then there exists some real number $a$ such that $\det(C + a D)$ is not zero
I wonder why the following statement is correct:
supposing $C$ and $D$ are two real matrix, if the determinant of the complex matrix $C + D i $ is not zero, then there exists some real number $a$ ...
2
votes
0answers
35 views
What does it mean that Singular Values of two in principle unrelated matrices are really similar?
I have two matrrices $Q_1$, $Q_2$ of size $5\times8$.
They should not be related to each other, they just describe the 8 coefficients of 5 bivariate polynomials, that should also not be related, apart ...
2
votes
1answer
49 views
How can I solve a polynominal of degree 2 with more than one variable?
(Sorry if the title is not informative)
How can I find the value of matrices $F$ and $d$, in the following equation:
$$y'Ay+b'y+c'c = (y-d)'F(y-d)$$
Given $A:n \times n$, which is positive definite ...
0
votes
1answer
32 views
polynomial matrices
How do we show that A=\begin{pmatrix}a &b &c \\ d& e &f \\g& h& i\end{pmatrix} is an polynomial martix over R[x], then the dervative of its determinant (considering the entries ...
2
votes
3answers
109 views
Correlation between polynomial equations and matrix determinants
Expanding
$p(x)=(ax-b)(cx+d)$
we get
$acx^2+(ad-bc)x-bd$.
Notice the determinant of the matrix
$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$
is $ad-bc$ exactly like the constant of $x$ ...
3
votes
0answers
68 views
Prove (*) by induction on k.
Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m ...
0
votes
2answers
105 views
Matrix polynomial - is there a trick?
Is there a trick for easily solving a matrix polynomial like
$$
p(A) = \left( 7\cdot A^4 - 4\cdot A^3 + 6\cdot A - 5\cdot E \right)
, A = \left(\begin{matrix}2 & -1 \\ 3 & 5\end{matrix}\right)
...
-3
votes
4answers
93 views
Complete instead of Complex, Irregular instead of Imaginary
Will the terms complex and imaginary ever be replaced? At least within beginning classes?
I imagine it is more of a kind of hazing into the "mathemitician's club" to allow the terms to confuse ...
2
votes
2answers
595 views
Minimal polynomials and characteristic polynomials
I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices.
When are the minimal polynomial and characteristic polynomial the ...
2
votes
1answer
71 views
The derivative of characterestic polynomial?
Let $A\in M_{n}(R)$ and $f(x)$ be the characterestic polynomial of $A$. Is it true that $f'(x)=\sum_{i=1}^{^{n}}\sum_{j=1}^{n}\det(xI-A(i\mid j))$ which $A(i\mid j)$ is a submatrix of $A$ obtained by ...
0
votes
1answer
159 views
Method of finding inverse of a Matrix using minimal polynomials
Using a piece from my last question I want to show how to find $A^{-1}$ as a polynomial expression in $A$ of degree < $\deg m_A$ where the leading coefficient of the polynomial is ...
1
vote
0answers
67 views
Spectrum of Transition Matrix for Random Walk
Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
1
vote
1answer
114 views
Rank of a Vandermonde Matrix with additional weighted columns
A Vandermonde matrix:
$\left(\begin{array}{ccc}
1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\
1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\
\vdots & \vdots & \ddots & ...
3
votes
0answers
158 views
Closed-form expression for sum of Vandermonde matrix elements
Given the Vandermonde matrix:
$$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\
2^0 & 2^1 & 2^2 & ... & 2^n \\
\vdots & \vdots & \vdots & \ddots & ...
2
votes
0answers
110 views
Smith normal form of a polynomial matrix.
I have the following matrix: $P(s) :=$
\begin{bmatrix}
s^2 & s-1 \\
s & s^2
\end{bmatrix}
How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
2
votes
1answer
127 views
When does a higher order polynomial have complex roots?
I try to say it all in the title.
I'm wondering under what conditions a matrix will have complex eigenvectors and eigenvalues. That question, I think, reduces to whether the characteristic ...
1
vote
2answers
223 views
Ordered quadruples in a grid
Moderator Note: This question is from a contest which ended on 22 Oct 2012.
Consider $(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ such that the ordered quadruple satisfies the following:
...
5
votes
2answers
95 views
Does the sign of the characteristic polynomial have any meaning?
The characteristic polynomial of a matrix $A \in \mathbb{C}^{n \times n}$, $p_A (\lambda) = \det(A-\lambda \cdot E)$ can be used to find the eigenvalues of the linear function $\varphi:\mathbb{C}^n ...
3
votes
0answers
121 views
Positive definite completion of a matrix
Suppose we have a real, symmetric matrix $A(x_1,x_2,x_3)$ given by
\begin{pmatrix}
a_{1,1} & a_{1,2} & x_1 & x_2 \\
a_{2,1} & a_{2,2} & a_{2,3} & x_3 \\
x_1 & a_{3,2} & ...
6
votes
1answer
166 views
What does the degree of a matrix minimal polynomial encode?
Let $\mathsf{F}$ be any field. Let $A$ be an $n \times n$ matrix over $\mathsf{F},$ whose rank is $r \le n.$ Let $\mu \in \mathsf{F}[x]$ be the minimal polynomial of $A.$
What does $\deg(\mu)$ ...
2
votes
1answer
43 views
Is there an explicit way to determine $\mathrm{Mat}_n(R[X_1,\dots,X_m])\simeq\mathrm{Mat}_n(R)[X_1,\dots,X_m]$?
For a commutative ring $R$, let $\mathrm{Mat}_n(R[X_1,\dots,X_m])$ denotes the matrix ring with entries from $R[X_1,\dots,X_m]$, and let $\mathrm{Mat}_n(R)[X_1,\dots,X_m]$ denotes the polynomial ring ...
4
votes
3answers
123 views
Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?
The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
1
vote
4answers
127 views
Elementary matrix/polynomial problem involving inequalities
I expect this probably has an easy answer but I'm feeling stupid and I'm stumped.
A book I'm reading, if I've understood it correctly, includes the following assertion. Let $M$ be a $3\times3$ ...
0
votes
0answers
179 views
Solving polynomial matrix equations over finite fields
The concrete problem is this: Find triplets of distinct matrices $(A,B,C)$ of dimension $6\times 6$ over the field $\mathbb{F}_{2^2}$ such that:
$A^2B=AB^2$
$C^2A=CA^2$
$B^3C=BC^3$
However, I'm ...
3
votes
2answers
150 views
Is $f:M_n(\mathbb{C})\longrightarrow M_n(\mathbb{C})$ continuous?
I want to know whether this is absurd question or reasonable to ask: Let
$f:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be given by $f(A)= B$, where $B$ is a diagonal matrix having the same eigenvalues as ...
0
votes
0answers
22 views
Matrix SNF confusion
This question has to do with this question. I am getting nowhere with finding the Smith Normal Form of the following matrix over $\mathbb Q[x]$ despite having read the Wikipedia article J.D. kindly ...
4
votes
1answer
220 views
Holomorphic function of a matrix
A statement is made below. The questions are:
(a) Is the statement true?
(b) If it is, does it appear in the literature?
Here is the statement.
For any matrix $A$ in $M_n(\mathbb C)$, write ...
5
votes
3answers
284 views
An application of Vandermonde determinant
Let $\lambda_1,\ldots,\lambda_n$ be complex numbers such that for each positive integer $k\geq 0$,
$$\sum_{i=1}^n \lambda_i^k=0.$$
Here I am supposed to show that $\lambda_i=0$ for each $i\in ...
1
vote
1answer
183 views
Characteristic and minimal polynomial - leading coefficient and norming
When calculating the characteristic polynomial as $$\det \; (A−t E_n)$$ I get the same polynomial as when I calculate the characteristic polynomial as $$\det\;(t E_n−A).$$ Only the signs are changed.
...
3
votes
0answers
389 views
quick ways to »verify« determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …
What are easy and quick ways to »verify« determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after caculating them?
So if I calculated determinant, minimal ...
5
votes
2answers
214 views
On rank of a matrix whose entries are polynomials
(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.)
How do you define the rank of a matrix whose entries are polynomials in $K[X]$?
If you assign some ...
1
vote
1answer
217 views
roots of minimal and characteristic polynomial
Why is it, that for the matrix
$A \in Mat(n\times n, \mathbb{C})$
the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have same roots?
Since $\chi_A(t) = \mu_A(t) \cdot ...
2
votes
2answers
150 views
Linear Algebra: Diagonalisability
Problem
Let A be the matrix
$\Bigg(\begin{matrix} 0&0&1\\ 1&0&0 \\0&1&0 \end{matrix} \Bigg)$
Giving brief justifications, determine whether A is ...
3
votes
5answers
245 views
How to compute the characteristic polynomial of $A$
The matrix associated with $f$ is:
$$ \left(\begin{array}{rrr}
3 & -1 & -1 \\
-1 & 3 & -1 \\
-1 & -1 & 3
\end{array}\right) .
$$
First, I am going to find ...
11
votes
2answers
273 views
Characteristic polynomials exhaust all monic polynomials?
Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse:
Let ...
2
votes
1answer
53 views
Expressing the number of non-zero rows of a binary matrix as a polynomial
Let $X$ be an $m\times n$ matrix, such that all of its elements are binary, i.e., for every $1\leq i\leq m$ and $1\leq j\leq n$ holds $x_{ij}=(X)_{ij}\in\{0,1\}$.
Is there any possible way to express ...
5
votes
1answer
462 views
Are the primes found as a subset in this sequence $a_n$?
Below is a introduction that contains some background to my question. The question is found at the bottom.
By calculating the eigenvalues of the matrix defined by the recurrence:
$\displaystyle ...
2
votes
2answers
129 views
Computation of characteristic polynomial fails for me
For a matrix $A\in\mathbb{K}^{n\times n}$ where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ the characteristic polynomial is defined as
$$\chi_A(\lambda) := \text{det}(A-\lambda I_n) = \sum_{k=0}^n c_k ...
2
votes
2answers
172 views
Given any polynomial p(x) over Z, can one construct a graph with characteristic polynomial p(x)?
Given any polynomial $p(x)$ over $\mathbb{Z}$, can one construct a graph with characteristic polynomial $p(x)$? [Edit: Title question added to post.}
Further questions include:
Are there classes of ...
4
votes
1answer
118 views
Understanding an algorithm for computing a matrix polynomial
I'm trying to understand this algorithm by Charles Van Loan for evaluating a matrix polynomial $p(\mathbf A)=\sum\limits_{k=0}^q b_k \mathbf A^k=b_0\mathbf I+b_1\mathbf A+\cdots$ (where $\mathbf A$ is ...
2
votes
2answers
268 views
Division matrix by polynomial
there is a polynomial:
$$p(x)=1\cdot x^3+bx^2+cx+d$$
And there is a matrix of form - Toeplitz matrix with coeffcients of $p(x)$ on main diagonal:
...
