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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1answer
21 views

Orthogonally diagonalizing a matrix

Can anybody explain how to orthogonally diagonalize the following matrix: $$ \begin{pmatrix} 9 & \sqrt10 \\ \sqrt10 & 0 \\ \end{pmatrix} $$ Am I correct in ...
0
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1answer
16 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
1
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0answers
16 views

Following problem on topology $(N.B.H.M - 2015)$

let $X = \{ f \in C[-5 , 5] : f(-5) = f(5) = 0 \}$ . Then Which of the following statement are true : (a) There exist $f \in X$ such that $f \equiv 2$ on $[-1 ,0 ]$ and $f \equiv 3$ on ...
1
vote
2answers
26 views

Similarity in two 2x2 Matrices and finding the S in A=SBS-1

I am doing something wrong here and I am not sure what. The object of the exercise is to find the S for similar matrices $A$ and $B$. $A=SBS^{-1}$ with $B=\begin{pmatrix}4& 1\\1& ...
2
votes
2answers
37 views

Help me to solve this matrix determinant problem, please.

Show that LHS = $$\begin{vmatrix}a_1+b_1t & a_2+b_2t & a_3+b_3t \\ a_1t+b_1 & a_2t+b_2 & a_3t+b_3 \\c_1 & c_2 & c_3 \\\end{vmatrix}$$ RHS = (1-t^2) ...
0
votes
1answer
11 views

When does the Singular Value Decomposition fail?

Does the singular value decomposition ever not work? The statement of the associated theorem, here from wikipedia: http://en.wikipedia.org/wiki/Singular_value_decomposition#Statement_of_the_theorem is ...
2
votes
1answer
58 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
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0answers
19 views

Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...
1
vote
1answer
35 views

Determinant and matrix power

I was wondering if there is a relation between the determinant of a matrix and the determinant of its powers. I mean I am looking for something like $$ \det (A^k) = f(\det(A), k). $$ A few check I ...
0
votes
2answers
32 views

given- $BA+B^2=I-BA^{2}$ what can be said about A ,B matrices

let A,B be $n\times n$ matrices such tha $BA+B^2=I-BA^{2}$ where $I$ is the identity matrix.which of the following is true 1.$A$ is nonsingular 2.$B$ is nonsingular 3.$A+B$ is nonsingular 4.$AB$ ...
1
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0answers
41 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
0
votes
1answer
30 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
19 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
39 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
13 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
20 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) ...
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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
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0answers
15 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, ...
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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 ...
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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
42 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.
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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
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1answer
17 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
35 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
209 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
41 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
23 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
55 views

determinant of matrix $X$

Please hint me. ‎How ‎can I ‎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
23 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
19 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
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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
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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
31 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 ...