For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0answers
23 views

What is permuted block-diagonal matrix?

I have been searching about permuted block-diagram matrixes, since far there is Block matrix. But it seems i cannot find any permuted block-diagonal matrix explanation/example... Can anybody explain ...
1
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0answers
67 views

Inverse of a $2\times2$ matrix with noncommuting entries

I want to invert a $2 \times 2$ matrix with quaternionic entries. Since non-commutativity, the determinant is not well defined and I've seen in http://en.wikipedia.org/wiki/Quaternionic_matrix that, ...
0
votes
1answer
64 views

At least one solution to equation

Given edit: an equation in matrix form $$\underline{\underline{A}} \ \underline{x} = \underline{B}$$ and A is known, what must be true for $\underline{B}$ for the equation to have at least one ...
1
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1answer
87 views

Basis of column/row space of $A$: using pivot columns of $A$ vs. $\text{rref}(A)$?

When we have column vectors and want to check which ones are linearly dependent to take them out and form a basis for the column space of $A$, we put them as column vectors in the matrix. Then, we ...
3
votes
3answers
218 views

For a 2x2 matrix A satisfying $A^k=I$, compute $e^A$

For a 2x2 matrix A satisfying $A^k=I$, compute $e^A$ Oh, the exponential of a matrix is: $e^A=\sum_{i=0}^\infty\frac{1}{i!}A^i$ I thought I'd solved the $e^A$ form but I actually did something ...
2
votes
1answer
57 views

What is “cyclic shift unitary” on $M_{n}(\mathbb{C})$?

Let $M_{n}(\mathbb{C})$ be the $n\times n$ complex matrices, and what is the "cyclic shift unitary of order $n$" on $M_{n}(\mathbb{C})$ ? (Maybe it is a very basic concept in functional analysis or ...
3
votes
2answers
100 views

Normal subgroup proof

I'm preparing for an algebra exam later this month and am trying out the exercises from my textbook. Sadly I got stuck with this one: Let $G$ be a group of all regular upper triangular matrices $2 \...
2
votes
2answers
68 views

Symmetric bilinear forms, quadratic forms and matrices

I have computed B=$ \left( \begin{array}{ccc} 0 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 1 \end{array} \right) $ Is this correct? If so, even though I may have achieved the correct ...
0
votes
2answers
60 views

Order of $A \in GL(n,\mathbb Z_p)$ cannot exceed $p^n-1$ ?

If $A \in GL(n,\mathbb Z_p)$ then is it true that order of $A$ cannot exceed $p^n-1$ ?
0
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1answer
52 views

Eigenvectors, bilinear forms and orthonormal bases

I have calculated (a) to be $(1,-2,2)^t, (-2,1,2)^t, (2,2,1)^t$. For (b) I have made all of these of unit length ie taken 1/3 of each vector. I have verified these are orthonormal by checking $<v1,...
0
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1answer
251 views

Solve a viscous Burgers' equation with a Newton-GMRes method

I implemented a preconditioner for a GMRes method. To test this preconditioner I want to solve this one dimensional viscous Burgers' equation $$\partial_t u(x,t) + u(x,t) \partial_x u(x,t)-\...
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2answers
74 views

Order of $GL(n, \mathbb Z_p)$ [duplicate]

Let $G$ be the group of all $n\times n$ matrices with entries of the matrices from the field $\mathbb Z_p$ , $p$ prime, such that determinant of every matrix is not $[0]$ , w.r.t. matrix ...
1
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0answers
82 views

Concerning the problem of finding the number of invertible nxn random {1,0} matrcies

In a few more words, if we look at the space of all nxn matrices (over a field of characteristic 0) with only 1 or 0 as an element in them ("binary matrices"), how many of them are invertible for each ...
0
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2answers
37 views

Prove, using Wronskian, that $e^\left(3x\right)$, $e^\left(2x\right)+x$, $e^\left(x\right)+1$ are linearly independent

I am currently working on a problem that asks to prove, using Wronskian, that $e^\left(3x\right)$, $e^\left(2x\right)+x$, $e^\left(x\right)+1$ are linearly independent. I went through the general ...
1
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2answers
358 views

Finding the determinant of a matrix given the adjoint

My attempt: Knowing that $$A(AdjA) = IdetA$$ I took the determinant on both sides: $$det(A)det(AdjA) = det(det(A))$$ So, $$det(A)det(AdjA) = (det(A))^3$$ $$det(AdjA) = (det(A))^2$$ $$det(A) = (AdjA)^{...
0
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1answer
172 views

Proving if $A$ is an $n\times n$ positive semi-definite matrix, A is Hermitian with non-negative eigenvalues.

I have a test on Monday and the professor hinted that this question might be relevant to the exam, unfortunately, I'm at a loss. As the title states, I would like to prove that if $A$ is an $n\times ...
1
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1answer
40 views

For a matrix ($2 \times 2$) $A$ satisfing (1.), (2.), (3.), show that $-2\leq a+d\leq 2$

I would appreciate if somebody could help me with the following problem: Question: For a matrix ($2 \times 2$) $A$ satisfing (1), (2), (3): (1.) $A=\begin{pmatrix} a& b\\ c&d\end{pmatrix}...
0
votes
1answer
422 views

Constructing a 2x2 matrix R which represents reflection in the x-y plane,

Construct a 2x2 matrix R which represents reflection in the x-y plane through the line $$(cos(\theta)x+(sin(\theta)y=0$$, where $\theta$ is any real number. (Let's call this line "L".) Write an ...
0
votes
1answer
305 views

Proving if A is an Hermitian matrix with nonnegative eigenvalues, A is positive semidefinite.

I'm trying to show that if A is an Hermitian matrix with non-negative eigenvalues, then A is positive semi-definite. The only thing I've thought of so far is using Spectral Theorem. I know I want to ...
2
votes
1answer
242 views

How to find eigenvalues of this 3x3 Jacobian Matrix

I am having to learn how to do jacobian matrices, determinants, and finding eigenvalues on my own and I cannot seem to find reasonable eigenvalues for this jacobian matrix. When I try to solve it I ...
1
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3answers
88 views

Prove that $\mathrm{span}\{ I,A,A^2… \} = \mathrm{span} \{ I,A,A^2,…, A^{k-1}\}$

Let $A\in M_n(F)$ and $k=\deg(m_A)$ where $m_A$ is the minimal polynomial of $A$. Prove that $\mathrm{span}\{ I,A,A^2... \} = \mathrm{span} \{ I,A,A^2,..., A^{k-1}\}$ So we have that $m_A = a_0 + ...
0
votes
1answer
43 views

What is the minimal polynomial of $A^2$?

Let $A\in M_n(\mathbb{C})$. The minimal polynomial of $A$ is $m_A = x^6 - 4x^4+3x^2 +1$. What is the minimal polynomial of $A^2$? I'd be glad for an hint/direction. Thanks!
0
votes
1answer
104 views

Setting up a matrix from a recurrence relation to find diagonal matrix?

Considering the recurrence $F_n= F_{n−1}+3F_{n−2}−2F_{n−3}$ where $F_0=0$, $F_1=1$ and $F_2=2$, use diagonalization to find a closed form of the expression. If the sequence is continued the numbers ...
1
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3answers
36 views

Tell if $A$ is diagonalized using it's characteristic and minimal polynomials

$$A= \left( {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right)$$ I already calculated that $f_A(x) = (x-3)(x-1)$. Also, the minimal polynomial must be also $(x-3)(x-1)$. How can I use ...
1
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2answers
41 views

Manifold of all 2x2 Hermitian Matrices

Is it true that the manifold of all $2\times 2$ Hermitian matrices is $\mathbb{R}^4$?
0
votes
1answer
34 views

Prove $A$ is scalar matrix

Let $A\in M_n(F)$ and let's assume $A$ has only one eigenvalue. Also, $A$ is diagonalized. Prove that $A$ is a scalar matrix. My Try: $${P^{ - 1}}AP = \left( {\matrix{ \lambda & {} & 0 ...
-3
votes
4answers
81 views

How $a_{13}=0$ in $\begin{bmatrix} {2}&{1}&{0}\\ {1}&{3}&{5} \end{bmatrix}$?

I'm reading Artin's Algebra. $$\begin{bmatrix} {2}&{1}&{0}\\ {1}&{3}&{5} \end{bmatrix}$$ It says that $a_{ij}$ is the matrix entry such that $i$ is the horizontal coordinate and $j$ ...
1
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2answers
49 views

Bounds of Sparse Matrix Multiplication

Does anyone know a good reference for bounds on sparse matrix multiplication? I'm interested in bounds of the number of scalar products required and bounds of the sparsity of the product. I know that ...
0
votes
1answer
86 views

Adjoint of a matrix and inverse of a matrix

As everyone know that we can use a matrix $A$ to represent an operator $T$. The adjoint of a matrix $A$ is denoted as $A^*$, which takes complex conjugate of $A$ and then transpose. My problem ...
0
votes
1answer
96 views

Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times n$...
1
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0answers
19 views

Matrix derivatives for the HJB and ARE relationship

How does one take the derivative of these matrix equations? (Backround:{My professor used them in the proof showing that the Hamilton-Jacobi-equation equivalently solves the free end-point ...
0
votes
1answer
45 views

Obscure Rotation Matrix

The points of the shape after the shear are $(-0.25, 1)$, $(1.75, 1)$, $(0.25, -1)$, $(-1.75, -1)$. Other than that the only other information given is that the vertical edges of the original shaded ...
2
votes
1answer
91 views

How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
0
votes
1answer
44 views

What does this determinant mean?

I have the following Jacobian matrix for an equilibrium of an SIR model $$J=\left( \begin{array}{cccc} -\text{$\alpha $N} & 0 & \zeta & 0 \\ \text{$\alpha $N} & -\beta -\rho & ...
2
votes
2answers
45 views

Find all 2 x 2 skew-symmetric matrices A [closed]

Find all $2 \times 2$ skew-symmetric matrices $A$, if any, such that $A^2 + I_2 = 0$ Please help me! Thanks!!
1
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1answer
97 views

Give a general formula in terms of $n$ for the determinant of the following matrix.

Let $M_n$ denote the $n$ x $n$ matrix over $\mathbb{R}$ of which the entry in the $i$-th row and the $j$-th column equals $1$ if $|i-j|\leq 1$ and $0$ otherwise. For example: $M_6=$ \begin{pmatrix} 1&...
1
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0answers
24 views

Compute new inverse when old inverse and new and old matrix known

Say I have a matrix $M$ and know its inverse $M^{-1}$. Then every element changes so that $M'=M+(M'-M)$. Is there a fast way to find $M'^{-1}$ from this information? That is without computing the new ...
0
votes
1answer
130 views

How to calculate this matrix in component-form? (Undergrad)

If ${A}_{ab} = \delta_{ab} + \varepsilon_{abc}n^c$ and $B^{ab} = \frac{1}{1+n^2}(\delta^{ab} + n^an^b - \varepsilon^{abc}n_c)$ what is the correct way to evaluate $$C^{ab} = (AB)^{ab} $$ Here, $\...
0
votes
2answers
52 views

be J be a matrix so: JJ^-1 = I and A a matrix so: A^t JA = J. prove that A is invertible so that AA^-1 = I

be J be a matrix so: $JJ^{-1}$ = I and A a matrix so: $A^tJA^ = J$. prove that A can invertible so that $AA^{-1} = I$ the big question here is: what are the properties of A transpose, that allows ...
1
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1answer
58 views

A is normal if and only if every matrix unitarily equivalent to A is normal

I need to prove that if $A$ and $B$ are unitarily equivalent, then $A$ is normal if and only if $B$ is normal. The proof is as follows: Suppose $A$ is normal and $B = U^*AU$, where $U$ is unitary. ...
1
vote
1answer
36 views

Matrices, Transition matrix

I have a matrix $B:= \begin{bmatrix}0 & 1\\-1 & -\lambda\end{bmatrix} $ I need to diagonalise it and work out the transition matrix. I have worked out that the eigenvalues are $ \mu_± = \...
2
votes
1answer
50 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
1
vote
1answer
99 views

Bound spectral radius of a certain matrix.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. $\xi^\...
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votes
1answer
42 views

$2\times2$ Matrix Problem and Recurrence relationship

For the $2\times2$ Matrix $A$, $a=1, b=1,c=1$ and $d=0$ Find a diagonal matrix $D$ and an invertible matrix $T$ such that $A = TDT^{−1}$. Hence solve the recurrence relation $f_{n+1} = f_n + f_{n−1}$...
0
votes
1answer
77 views

How to rewrite a derivative w.r.t. tensor as w.r.t. vector

I'm stuck on a (probably very simple) problem I've come across. Take a function $f(A)$ where $A$ is a 2-tensor. Now suppose $A=vv^T$ for an $\mathbb{R}^n$ vector, $v$. I want to rewrite the object $\...
3
votes
1answer
291 views

The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the Perron–...
0
votes
1answer
69 views

Diagonalisation and Kronecker Product

If $A$ is a $n\times n$ matrix with complex numbers for elements, and $C$ the $2\times2$ matrix defined by $$\begin{bmatrix} -2&4\\-3&5 \end{bmatrix}.$$ How do you prove that the Kronecker ...
2
votes
1answer
82 views

Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
0
votes
3answers
448 views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
1
vote
1answer
41 views

“a matrix is positive semi-definite” not necessarily equavalent to “all leading principle minors are nonegative”?

Have a look at this matrix: $$ H = \left( {\begin{array}{*{20}{c}} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{array}} \right).$$ All the leading ...