Tagged Questions
0
votes
1answer
23 views
Bilinear form matrix coordinates
I don't know how to solve this problem - I have to transform the coordinates of a bilinear form that has matrix
$$
\begin{pmatrix}
1 & 2 & -2 \\
2 & -2 & 3 \\
...
1
vote
1answer
45 views
Eigenvalues and eigenvectors of AB and BA, proof.
$A$ is an $n \times k$ matrix and $B$ is an $k \times n$ matrix.
If $v_1, ..., v_l$ are linearly independent eigenvectors of $BA$ corresponding to a single nonzero eigenvalue $c$, then $Av_1, ..., ...
0
votes
1answer
27 views
Is there a dot product with which the following linear operator becomes Hermitian
Given the linear operator
$A \in L(M_2(\mathbb{C}))$
$A \begin{bmatrix}a & b \\ c & d \end{bmatrix}=\begin{bmatrix}a-b & -a+b \\ d & -c \end{bmatrix}$
Is there a dot product where ...
0
votes
1answer
28 views
find matrix such that $ Ax=(1,1,1)^t$ has exactly three distinct solutions
Does there exist a matrix $3\times 3$ order such that $ Ax=(1,1,1)^t$ has exactly three distinct solutions? If so, find $A$.
I have no idea in this question please help.
0
votes
0answers
30 views
to find the eigenvalues and eigenvectors from linear transformation
Find the eigenvalues and eigenvectors of the linear transformation $T$:$R^3\to R^3$ defined by $T(x_1,x_2,x_3)=(x_1,x_2,x_3)$?
Please tell me how to find the matrix and then I can find the ...
0
votes
1answer
17 views
Schur decomposition of an $n-$by$-n$ matrix
$(\lambda, x)$ is a simple (with multiplicity 1) eigenpair of $A\in \mathbb C_n$ with $x^Hx=1$, $H$ denotes Hermitian.
Use Schur decomposition to show that there exists a nonsingular matrix $(x\ \ ...
2
votes
5answers
76 views
Finding the determinant of $2A+A^{-1}-I$ given the eigenvalues of $A$
Let $A$ be a $2\times 2$ matrix whose eigenvalues are $1$ and $-1$. Find the determinant of $S=2A+A^{-1}-I$.
Here I don't know how to find $A$ if eigenvectors are not given. If eigenvectors are ...
2
votes
1answer
35 views
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$, consider the linear map $T:M_2(\mathbb{R})\to M_2(\mathbb{R}):=B\to AB$ Then which of the following are true?
$T$ is ...
2
votes
1answer
58 views
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
Diagonalizable
Positive semidefinite
$0,3$ are only eigenvalues of $J$
Is positive definite
$J$ has minimal polynomial $x(x-3)=0$ so 1, ...
-3
votes
1answer
49 views
Fantastic Determinant (all $b$ plus multiple of $I$) [duplicate]
$$f(a,b)=\operatorname{det}~\begin{pmatrix} a & b & b & \cdots & b \\ b & a & b &\cdots & b\\ b & b & a &\cdots & b\\ \vdots & \vdots & \vdots ...
0
votes
1answer
49 views
Prove: symmetric positive matrix multiplied by skew symmetric matrix equals 0
My teacher gave me this task as preparation for the exam but I'm stuck and not sure if it's true anymore.
1
vote
1answer
27 views
Symmetric Matrices Using Pythagorean Triples
Find symmetric matrices A =$\begin{pmatrix} a &b \\ c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$.
Alright, so I've posed this problem earlier but my question is in regard to this ...
1
vote
2answers
17 views
Determinant of product of symplectic matrices
In optical ray tracing it's possible to use symplectic matrices. I have a problem with them.
If a matrix $M$ is symplectic, this means that for $M$ the following equation hols:
$$M^T\Omega M=\Omega$$
...
1
vote
2answers
72 views
Upper and Lower Triangular Matrices
Given the matrix A=$ \left( \begin{array}{ccc}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8\\
1 & -1 & 2 & 3 \\
2 & 1 & 1 &2\end{array} \right) $, write it in the ...
2
votes
1answer
24 views
Minimal polynomial matrix
I want to show that $ x^n-1$ is the minimal polynomial of the permutation matrix $P:=(e_2,e_3,....,e_n,e_1)$ where $e_i$ is the i-th unit vector written as a column vector.
And now I have to show ...
3
votes
0answers
44 views
Eigenvalues of a tridiagonal trigonometric matrix
Let $A$ be the diagonal matrix w/alternating in sign diagonal entries:
$$ A =
\begin{pmatrix}
\pm \tan(\frac{\pi}{2n+1}) & 0 & 0 & \ldots & 0 \\
0 & \mp ...
2
votes
1answer
35 views
inequality applied to Matrix possible?
My question is this : when is it possible to apply (if at all) a polinomial inequality like this little inequality conjecture ,for example, to a $n\times n$ Matrix $A$ (change the variable $x$ with ...
2
votes
1answer
31 views
Matrices manipulation
I am having difficulty with the following question
I have to determine if the following claim is true or not.
If it is true I have to proof it else I need to give an example
I believe it is not ...
0
votes
1answer
31 views
Can the second term of the Schur complement of a symmetric matrix be undefined?
Given the next symmetric matrix conformably partitioned
$$\begin{bmatrix}
A &B \\ B^T &C
\end{bmatrix}$$
I know that $A$ and $C$ are positive definite matrices.
The Schur complement is ...
2
votes
2answers
23 views
Calculating the centralizer of a matrix in a general linear group.
Let $G = GL(3,\mathbb{R})$ be the general linear group over the reals , of order $3$ , and let
$A\in G$ be :
$$
A=\begin{pmatrix}
-1 & 0 & 0 \\\
0 & 1 & 0 \\\
0 & 0 & 2
...
1
vote
1answer
20 views
How to show that every complex matrix with orthonormal columns can be supplemented into an unitary matrix?
Show that every matrix $A \in M_{n,k}(\mathbb{C})$ whose columns are orthonormal vectors in $M_{n1}(\mathbb{C})$ can be supplemented with additional n-k columns to an unitary matrix $U \in ...
2
votes
1answer
34 views
Why $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ is equivalent to $A^* Ax=0$ if and only if $Ax=0$?
Let $A \in M_{m\times n}(F)$ and $x \in F^n$.
$A^*$ is the adjoint of $A$.
Why is $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ equivalent to $A^* Ax=0$ if and only if $Ax=0$?
2
votes
1answer
70 views
How prove that $\;(1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4\;\;?$
Let $A=\begin{bmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}
\end{bmatrix}$ be an orthogonal matrix with $a_{i,j}\in \mathbb R$, where $\det(A)=1$
...
2
votes
0answers
41 views
Proof of Sum, Difference, Scalar Multiple of Diagonal Matrices
Assumming A and B are diagonal matrices of the same size, please prove that the following are diagonal matrices as well.
a) $A+B$
b) $A-B$
c) $kA$ , for a scalar $k$
It's not homework- just a ...
1
vote
1answer
36 views
$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices
Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
1
vote
0answers
31 views
Basis of kernel and image of a linear transformation - verification
The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$
Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated?
$$\begin{pmatrix} 1 & ...
1
vote
0answers
22 views
How to compress a linear operator and have the lossless composition property.
Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
0
votes
0answers
25 views
Quadratic form of block matrix
If one has a block matrix $\tilde A = \left[ {\begin{array}{*{20}{c}}
D&{{0_{n \times n}}}\\
{{0_{n \times n}}}&{{0_{n \times n}}}
\end{array}} \right]$ where $D\in {R^{n \times n}}$ is a ...
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 ...
0
votes
0answers
20 views
Inner product space an two orthonormal basis. [duplicate]
Let $V$ be an inner product space. And let $v_1,...,v_n$ and $w_1,...,w_n$ be two orthonormal basis of $V$. How one could show that $[Id]^{v_1,...,v_n}_{w_1,...,w_n}$ is unitary matrix.
3
votes
2answers
25 views
Matrix Equation, Solving for Variables.
I'm going through my exercises, and came across a problem that wasn't covered in our lectures. Here's the question:
$
\begin{align}
\begin{bmatrix}
a-b & b+c\\
3d+c & 2a-4d
\end{bmatrix}
...
0
votes
1answer
21 views
What functions are solution to a homogeneous system of differential equations?
Given a vector $\vec{u} \in \mathbb{R}^n$. For what functions $\psi(t)$ can $\vec{x}(t) = \psi(t)\vec{u}$ be a solution of $\dot{\vec{x}} = A \vec{x}$ for some $n \times n$ matrix $A$?
I'm trying to ...
2
votes
2answers
120 views
Diagonalizability in $\mathbb{R}$ and $\mathbb{C}$
Give an example of a matrix $A\in M_{n\times n}(\mathbb{R})$ that is not diagonalizable, but A is diagonalizable viewed as a matrix over the field of complex numbers $\mathbb{C}.$
-1
votes
0answers
31 views
orthonormal basis linear transformation
A linear transformation
which takes an orthonormal basis into another orthonormal basis
is orthogonal. (T)
I got True for the answer. But can't think of clear explanation of why that is true.
Why it ...
2
votes
1answer
49 views
QR computation only in square matrix A?
I thought the following was true. But the answer is False.
Why so? Could anybody give me some counterexample?
For any matrix A, one can find Q and R such that A = QR
, where Q is an orthogonal matrix
...
0
votes
1answer
39 views
Dimension of vector space and symmetric matrix [duplicate]
Why the following statement is true?
I am so frustrated that I could not have any clue on this problem.
The dimension of the vector space of all symmetric 4 by 4 matrices is 10.
Please help me.
1
vote
1answer
33 views
orthogonal matrix and elementary matrix
Answer is False. But I can't think of the counter example... Could anybody have it?
Let A be an orthogonal 4 x 4 matrix such that $$ Ae_1 = e_2, Ae_2 = e_3, Ae_3 = e_1$$ Then $$Ae_4 = e_4 $$
2
votes
4answers
96 views
Diagonalizable matrices in $M_{2\times 2}(\mathbb{F}_2)$
List all diagonalizable $2\times 2$ matrices over the a field $F$ consisting of two elements $0$ and $1$.
I want to try and do this using C++, but perhaps this isn't the place to ask. I have an idea ...
0
votes
3answers
59 views
Diagonalizable Operators: An Operational Extension
Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator
$$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$
on $V$ is also diagonalizable for any scalars $a_1, ...
0
votes
1answer
55 views
How to show this matrix is invertible?
Let $f:H \times H \to \mathbb{R}$ be a mapping with $H$ a Hilbert space.
Let $A$ be a matrix with entries $a_{ij}=f(b_i, b_j)$ with
$$a_{ii}=f(b_i, b_i) \geq C\lVert b_i\rVert_{H}^2.$$
Suppose $b_i ...
4
votes
3answers
97 views
How to find 3 x 3 matrix inverses
Is there a way of finding the inverse of a $3 \times 3$ matrix without forming an augmented matrix with the identity matrix? Also, is there a quick way of checking that a $3 \times 3$ matrix's ...
2
votes
1answer
37 views
How to frame this set of linear equations?
I have the following set of equations, as an example
$2x + 1y + 2z = A$
$0x + 2y + 2z = A$
$1x + 2y + 1z = A$
I assume this can be rewritten as a matrix? How can I check if a solution exists such ...
3
votes
1answer
72 views
Property of the trace of matrices
Let $A(x,t),B(x,t)$ be matrix-valued functions that are independent of $\xi=x-t$ and satisfy $$A_t-B_x+AB-BA=0$$ where $X_q\equiv \frac{\partial X}{\partial q}$.
Why does it then follow that ...
1
vote
1answer
39 views
Special linear transformations
Special linear transformations are matrices with determinant equal to 1.
What additional properties do such transformations have compared to "regular" linear transformations?
2
votes
1answer
50 views
If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?
We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm.
Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 ...
1
vote
1answer
41 views
Proof is needed for a lower bound of the maximal eigen-value of a non-negative, irreducible, integer matrix
$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eigen vector ($x$). It ...
1
vote
1answer
27 views
Composition of systems of equations
Suppose $$2x + 3y = u$$ $$x - 4y = v$$
and further that
$$3u - 5v = c$$ $$2u + 3v = d$$
Express c and d in terms of $x$ and $y$ by matrix multiplication.
It's quite easy by direct substitution but ...
1
vote
0answers
28 views
Using a matrix to organise values into groups
Let's say I have a matrix of size 6 x 6.
Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
5
votes
5answers
89 views
Symmetric Matrices of $I_{2}$
Find $10$ symmetric matrices $ A = \begin{pmatrix}
a &b \\
c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$
(I'm going to call matrix A the "square root" of $A^{2}$. If this is the incorrect ...
3
votes
2answers
30 views
Matrix multiplication related to complex numbers?
Evaluate and simplify the product
$\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
