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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0
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1answer
27 views

How to determine the signs of the eigenvalues of a symmetric $3\times 3$ matrix?

This is a homework problem: Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. If $A=\begin{pmatrix} 1&b&c\\b&a&0\\c&0&1\end{pmatrix}$, then which of the ...
0
votes
2answers
17 views

How to find the corresponding matrix of a dot product over a polynomial ring to a specific basis

Let $V= \mathbb R[x]_{\leq 2}$ be the vector-space of real polynomials with degree $\leq 2$. We define a dot product on the $V$ as follows: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$ ...
0
votes
0answers
25 views

What is a transformation that can't have shearing called?

What is a transformation called when it can have separate scaling for x and y, rotation, and translation, but it cannot have shearing or scaling AFTER rotation? Basically if this transformation is ...
0
votes
1answer
11 views

matrix transformation help [on hold]

i'm really unsure of how to tackle the following questions (further maths gcse): the transformation A is represented by the matrix $$A = \begin{pmatrix} 2 & 0 \\ -1 & 3 \end{pmatrix}$$ ...
0
votes
2answers
36 views

Inequality $\sqrt[4]{x^TA^{-2}x}\sqrt{x^TAx}\leq 1$ for symmetric positive definite matrices

Assume that $x\in \mathbb{R}^{n}$ is a unit vector and $A$ is a symmetric positive definite matrix. Prove that $$\sqrt[4]{x^TA^{-2}x}\sqrt{x^TAx}\leq 1.$$ Progress Since A is spd, it is ...
0
votes
2answers
12 views

Rank of a matrix from a 5 X 7 matrix with a basis of 3 vectors

The question in my book is as follows: If the subspace of all solutions of Ax=0 has a basis consisting of thee vectors and if A is a 5 x 7 matrix, what is the rank of A? Now i thought because ...
0
votes
1answer
17 views

Given $\det(A)$ and $\det(B)$, is my calculation of $\det(-2B^T B A)$ correct?

Suppose $A$ and $B$ are $3 \times 3$ matrices with $\det(A) = -2$ and $\det(B) = -1$. What is the determinant of $C = -2 B^T B A$? I know that $$\det(A^T) = \det(A) \qquad \det(AB) = \det(A) ...
1
vote
1answer
16 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
-3
votes
0answers
14 views

Do those two expressions have the same eigenvalues?

I am encountering the following eigenproblem \begin{eqnarray} \text{min} ~ \epsilon' Z' \Omega^{-1} Z \epsilon, \end{eqnarray} where $\epsilon$ is N by one, Z is N by K, $\Omega$ is K by K and real, ...
0
votes
0answers
5 views

Variation of linear matrix inequality

When reading "Convex optimization, S. Boyd" p.76, Example 3.4, it says The last condition is a linear matrix inequality (LMI) in $(x,Y,t)$. Therefore, epi($f$) is convex. I am confused about ...
1
vote
0answers
12 views

Rectangular Orthogonal Matrix

Consider a overcomplete matrix $D$ of dimension $m\times n$ where $n>m$. I want to know under what conditions i can say $D$ has orthogonal columns or rows. More specifically when $D$ will be close ...
2
votes
0answers
41 views

Does this family of special matrices have a name?

These are the bisymmetric matrices that are "pyramid" shaped as follows: $$f(14) =\begin{bmatrix}1&1&1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\1& ...
0
votes
2answers
71 views

Can we find the inverse for a vector

Can we inverse a vector like we do with matrices, and why ? I didn't see in any linear algebra course such a concept of vector inverse and I was wondering if there is any such thing and if not, why.
0
votes
2answers
28 views

Generating a random binary matrix with fixed number of nonzeros

I want an algorithm (just the idea, not the actual code) to generate a random $n$ by $n$ matrix with binary entries, but with the condition that the number of nonzeros must be a fixed number $c$. Any ...
1
vote
1answer
16 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
2
votes
2answers
34 views

Properties of the matrix square root

In a paper I am reading, it is claimed that if $A, B \in \mathbb{R}^{n \times n}$ are positive definite, then $$ A^{1/2} (A^{−1/2} B A^{−1/2})^{1/2} A^{1/2} = A (A^{-1}B)^{1/2} $$ because of the ...
4
votes
3answers
208 views

All Two by Two Matrices Satisfy a Certain Property Problem

Show that if $A$, $B$ are $2 \times 2$ matrices over $\mathbb{R}$ then there exists a real number $\lambda$ so that $$ (AB-BA)^2 = \lambda I $$ I can do this problem using brute force (i.e. looking ...
1
vote
0answers
39 views

$\text{Ker}A=\text{span}(u) \implies A=mat_C\left( u\wedge . \right)$

i found this equality and i wonder how can i find the right term $$\dfrac{1}{2}\left(\begin{matrix}0&1&1 \\ -1&0&1\\ -1&-1&0 ...
2
votes
1answer
22 views

Commutativity of the square root of matrices

Let $A, B \in \mathbb{R}^{n \times n}$ two positive definite matrices such that $AB = BA$, that is $A$ commutes with $B$. It is easy to prove that $A^{1/2}$ commutes with $A$, indeed $AA^{1/2} = ...
1
vote
0answers
22 views

GMRES and Preconditioning

I am using GMRES to approximate the solution of a system of equations $Ax=b$, I am using a preconditioner $P$ to make GMRES converge faster. My question is how do I know if the preconditioner I am ...
2
votes
1answer
44 views

determinant of matrix $X$

Please hint me. ‎How I‎ ‎can ‎calculate ‎determinant ‎of ‎matrix ‎‎$‎X‎$‎?‎ \begin{equation*}‎ ‎\mathbf{X}=\left(‎ \begin{array}{ccc}‎ A&B&‎\cdots&B\\‎ B&A&‎\cdots& B\\‎ \vdots ...
1
vote
1answer
24 views

Integral defined on space of matrices

I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. ...
2
votes
1answer
37 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
3
votes
0answers
22 views

Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
1
vote
1answer
22 views

Commutativity of matrix square root

Let $A, B \in \mathbb{R}^{n \times n}$ and let us assume that $A^{1/2}$ exists. I have often seen people write something like $$ AB = A^{1/2}\, B\; A^{1/2} $$ when both $A$ and $B$ are symmetric, in ...
2
votes
2answers
79 views

Find smallest $n \in \mathbb{N}$ s.t $A^n=I$

Let $A$ be $2 \times 2$ matrix: $$ \left( \begin{matrix} \sin\frac{\pi}{18} \\ \sin\frac{4\pi}{9} \end{matrix} \begin{matrix} -\sin\frac{4\pi}{9} \\ \sin\frac{\pi}{18} \end{matrix} \right) $$ ...
0
votes
1answer
10 views

Show that rodriques formula is a linear transformation?

Can someone help me out on how to find the the matrix representation and show proof that it is a linear transformation? It is the rodrigues roation formula and the matrix representation should ...
0
votes
0answers
18 views

Where does this matrix rotation formula come from?

Im in a book and they use this rotation matrix formula in the picture. Where does it come from. I know that the c in the matrix is for Cos and the s is for Sin. Is there a proof?
2
votes
1answer
29 views

How to find the inverse of the matrix over $\mathbb Z_5$

How to find the inverse of the matrix over $\mathbb Z_5$ $$ \left( \begin{matrix} 1 & 2& 0\\ 0 &2& 4 \\ 0& 0& 3\\ \end {matrix} \right) $$
0
votes
1answer
20 views

Next step to show that these matrice expressions are equal?

This is a problem from Discrete Mathematics and its Applications I know invertible means it is possible to take the inverse of this matrix. This is definition of a power of a square matrix from my ...
0
votes
0answers
20 views

Linear Algebra - verification of my answer, basis for $ImT$

I'd like to verify this answer, because I think that the answer in my book is incorrect. I'll be very glad if someone could tell me, if the basis I found for $ImT$ is correct. Let : $T:R^3 ...
0
votes
0answers
28 views

Determinant of matrix n x n [duplicate]

How to calculate $det\begin{bmatrix}1 & x_1 & x_1^2 \dots x_1^{n-1} \\ 1 & x_2 & x_2^2 \dots x_2^{n-1} \\ \\ 1 & x_n & x_n^2 \dots x_n^{n-1}\end{bmatrix}$?
0
votes
0answers
13 views

Positive/Negative Definite Bordered Hessian?

I understand how to check a function for concavity and convexity using the Hessian matrix and the rules for the determinants of the leading principal minors. I understand if these rules are violated, ...
2
votes
1answer
16 views

$B - A \in S^n_{++}$ and $I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}$ equivalent?

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Now suppose that $A,B \in S^n_{++}$ are two ...
1
vote
0answers
29 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
2
votes
1answer
48 views

Finding orthogonal matrix that maps one vector to another

Let $w, v \in \mathbb{R}^k$ be two known vectors such that $||w|| = ||v||$ ($|| . ||$ is the usual Euclidean norm). My questions are related with the problem of finding $Q$ orthogonal such that $v = Q ...
0
votes
1answer
17 views

Rewrite an expression in terms of basis vectors

Given any vector k $\epsilon$ $R^{3}$ consider k= $\sum_{j=1}^{3}$ $c_{j}u_{j}$ where $u_{1}$,$u_{2}$,$u_{3}$ are the orthonormal basis vectors (I don't know how to make them bold sorry about that, ...
2
votes
1answer
132 views

How to solve 29 coupled quadratic equations?

I have a set of 29 coupled quadratic equations, with 29 unknown variables. Can anyone offer any advice on how I could go about solving this? 3 days of staring at a wall has so far given me no ...
0
votes
1answer
19 views

Matrix Differentiation using Kronecker operator issue

Let X an $n\times n$ variable matrix and given vectors and matrices $p_1$ ($1\times n$), $p_2$ ($n\times 1$), $\Omega$ ($n\times n$). What is the derivative of the function $f(X)=p_{1}X^{-1}\Omega ...
24
votes
3answers
2k views

The set of block matrices of order $(2m+1)\times (2m+1)$ with four constant blocks of size $m\times m$

Ok so what I found was a square matrix of order $n×n$ where $n$ follows $2m+1$ and $m$ is a natural number the pattern these matrices follow is as follows: for a $3×3$ matrix: $$ A = \left( ...
1
vote
2answers
22 views

limit of a function with a matrix exponential

I spent too many time trying to solve this problem...and finals are coming. Please help me! I just can't see a method to do this demonstration: "For an $A_{n \times n}$ matrix, demonstrate that a ...
0
votes
1answer
23 views

One eigenvalue and eigensystem

Matrix $A \in \mathbb{K}^{n,n}$ has one engenvalue $\lambda \in \mathbb{K}$ and its engensystem $V_{\lambda}$ has dimension that equals to $n$. How to show that $A = \lambda I_{n}$?
2
votes
1answer
44 views

proof on similarity of matrices

Could you please help me with the following problem? Let $A$ be an $n$$\times$$n$ complex matrix. Prove that $A$ is similar to $B$, which is an $n$ $\times$ $n$ real matrix, if and only if $A$ is ...
0
votes
1answer
27 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
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vote
0answers
12 views

Applications of Matrix in simplifying algebra [on hold]

Inversions (and the Mobius Transformation, though it belongs to complex numbers) are pretty good tools in simplifying an algebric mess. What other tools exist apart from this and how may we use them? ...
2
votes
2answers
29 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
0
votes
1answer
18 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
2
votes
1answer
53 views

Example of a non singular square matrix such that $A+A^{-1} = 0$

Is there any example of a non singular square matrix $A$ such that $A+A^{-1} = 0$? Are they any specific type of matrices or can these be found under any category of matrices (such as symmetric, ...
1
vote
0answers
20 views

Problems in metric space including matrices. [on hold]

Let $M(n, \Bbb R)$ denote the set of a real $n \times n$ matrices. We can always define a linear isomorphism between $M(n, \Bbb R)$ and $\Bbb R^{n^2}$....where the isomorphism is defined as for any ...
1
vote
1answer
30 views

Matrix multiplication computation

Any tips how to solve this? $$ \left[ \begin{matrix}1 & 2 & 0 \\ -2 & -5 & 1 \\ 11&15&5 \end{matrix}\right] \times \mathbf{X} \times \left[ \begin{matrix} -4&5&1\\ ...