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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5
votes
3answers
1k views

Derivative of matrix involving trace and log

I'm stuck on this problem. Let $X\in\mathbb{R}^{n\times n}$, compute the following matrix derivatives $$\frac{\partial}{\partial X}\mathrm{tr}(\log(XA)\log(XA)^\top),$$ $$\frac{\partial}{\partial ...
2
votes
1answer
252 views

Adjoint matrix eigenvalues and eigenvectors

I just wanted to make sure that the following statement is true: Let $A$ be a normal matrix with eigenvalues $\lambda_1,...,\lambda_n$ and eigenvectors $v_1,...,v_n$. Then $A^*$ has the same ...
1
vote
1answer
67 views

Incomplete Circulant matrix

The eigenvectors and eigenvalues of a Circulant matrix are well-known to be related to the discrete Fourier transform of entries of one row (the exact terms are given here). My question: is there any ...
0
votes
1answer
30 views

A basis such that $A$ is of a certain type.

Let $A$ be a $2\times 2$ real matrix without eigenvalues, and the roots of its characteristic polynomial be $\alpha+i \beta$ and $\alpha - i \beta$. Show that there exists a basis of $\mathbb{R^2}$ ...
0
votes
0answers
45 views

When random walk Markov matrix of the graph is normal?

I'm considering random walk on undirected graph $G$. At every time step, walker moves to a random neighbour, with all neighbours being equally likely. With adjacency matrix $A$ the random walk Markov ...
2
votes
2answers
109 views

Spectrum of Lyapunov exponents of a linear system

Question: How to show that the eigenvalues of matrices $\mathbf{A}$ and $ \mathbf{L} = \log \lim_{t \to \infty} \left((e^{\mathbf{A}t}e^{\mathbf{A^T}t})^{\frac{1}{2t}}\right) $ have equal real parts? ...
0
votes
1answer
308 views

Determining Jordan form from ranks of matrix powers.

Suppose you're working over an algebraically closed field. If $J$ is a Jordan matrix, then one can determine the number of Jordan blocks and their sizes for any eigenvalue $\lambda$ by looking at the ...
3
votes
2answers
884 views

If $AA^T$ is the zero matrix, then $A$ is the zero matrix

Let $A$ be a $4 \times 4$ matrix. Show that if $A^TA$ or $AA^T$ is the zero matrix, then $A$ is the zero matrix. I feel very close to solving the problem so far. I have said that ...
6
votes
3answers
656 views

Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
0
votes
2answers
212 views

A B matrices and Av Bv dependent vectors

A,B nxn complex matrices : Prove that exist a vector v(not 0), that A(v) and B(v) are dependent. Extra question: What if A,B are real matrices?
1
vote
1answer
316 views

Reverse engineer transition matrix from steady state?

I had this open ended question. Is it possible to reverse engineer a transition matrix from the steady state. Is there a unique solution or could there be many? Basically given $P^{\infty}$ is there ...
0
votes
1answer
60 views

Smart way $2\times 2$ JNF

I wanted to find a fast way to construct the JNF (with basis transformation) of a $2\times 2$ Matrix which is not diagonalizable, which means that we need to have one eigenvalue with algebraic ...
0
votes
1answer
94 views

anomalous minus sign in commutator of vector fields

Define $$X_A(x):=A^i_{\ j}x^j$$ where $A$ is a matrix. Why is there a minus sign in the following formula? $$[X_A,X_B]=-X_{[A,B]}$$ Edit: perhaps the question is not well posed, since what I really ...
1
vote
2answers
84 views

behavior of scalar product defined by trace under commutator

Define $$\langle X,Y \rangle := \operatorname{tr}XY^t,$$ where $X,Y$ are square matrices with real entries and $t$ denotes transpose. I have some troubles in proving that $$ \langle [X,Y],Z \rangle = ...
1
vote
2answers
109 views

Moore-Penrose Inverse and Standard Inverse

I have read that the Moore-Penrose inverse $A^+$ of a matrix $A$ is the same as the standard inverse $A^{-1}$ in the case $A$ is a square, invertible matrix. Is there any relation between the ...
1
vote
1answer
45 views

Is the largest number of zero eigenvalues at least 2

For any $x\le3$, is the largest (or maximal) number of zero eigenvalues of $diag\left(1,2,3\right)-U\cdot diag\left(4,5,x\right)\cdot U^{T}$ for orthogonal matrix $U$ at least $2$?
3
votes
1answer
60 views

Scaling of eigenvalues

Suppose $A_N$ is a positive definite matrix of size $N$ with eigenvalues $\Lambda=\{\lambda_1,\ldots,\lambda_N\}$. Let $D = \text{diag}\{d_1,\ldots,d_N\},\ d_i>0$ be a diagonal matrix. Can the ...
2
votes
1answer
75 views

Does this normalization of a positive definite matrix alter its positive definiteness?

I have a matrix $A$ that is positive definite. Denoting the elements of $A$ by $a_{ij}$, let $A'$ be a new matrix formed as: $$A'_{ij} = \frac{a_{ij}}{\sqrt{a_{ii}a_{jj}}}$$ Is $A'$ also positive ...
2
votes
1answer
462 views

convex hull of orthogonal matrices

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm non greater than one? It is easy to show that a convex combination of ...
0
votes
1answer
27 views

convulotion associative between vectors

We learnt that convolution is commutative, meaning that: $$xh = hx.$$ However if I take: $$h=[-1,0,1] \mbox{ and } x=[1,1,1]^T $$ ($T$ is transpose) I get that $xh$ is not equal to $hx$. Could ...
2
votes
2answers
95 views

Matrix $I + 2 A A^T$ is nonsingular for any A

Suppose A is $m\times n$ matrix with real entries. Could you prove that $\det (I + 2 A A^T) \neq 0$
3
votes
2answers
82 views

Matrix determinant $\neq 0$

I have problem with this task: Given a square matrix $A = [a_{ij}]^n_{i,j=1} \in M_{n \times n}(\mathbb{R})$ and $t \geq 0$ satisfies conditions: $\forall i \neq j : a_{ij} = t,$ $\forall i : ...
0
votes
1answer
1k views

The upper triangular matrix problem.

Hello guys please help me whit this problem. Hints is fine but if you can explain that would be great. For $1.$ I think is just the definition but I can't figure out how to combine the two conditions ...
1
vote
1answer
220 views

Trace, Kronecker and vec relations

I'm reading a paper and got stuck on one of the simplifications that was done without any elaboration. I've taken a course on Linear Algebra, but this is a little out of reach for me... The ...
1
vote
1answer
82 views

When A and B are similar matrices,what conditions hold A = C.B

When A and B are similar matrices,In which cases,A = C.B I'm not sure whether this is a valid mathematical question... $A=D^{-1}B D=C B$
0
votes
1answer
35 views

Verifying statements for non-zero matrix

Let $N$ be non-zero $3 \times 3$ matrix with the property $N^2=0$. Which of the following is true? (A) $N$ is not similar to a diagonal matrix. (B) $N$ is similar to diagonal matrix. (C) $N$ has ...
2
votes
2answers
368 views

How to prove that normal matrix with property $A^2=A$ is Hermitian?

I am given a matrix $A\in M(n\times n, \mathbb{C})$ normal (in matrix form $AA^*=A^*A$) and $A^2=A$. The task is to prove that the matrix is Hermitian. But when I try something like $A^*=\,\,...$ , ...
1
vote
2answers
3k views

Find the determinant by using elementary row operations

I'm having a problem finding the determinant of the following matrix using elementary row operations. I know the determinant is -15 but confused on how to do it using the elementary row operations. ...
0
votes
2answers
605 views

Under what conditions is $(A+B)(A-B) = (A^2 - B^2)$ for two $n \times n$ matrices $A$ and $B$?

so my approach to this problem was to view $A$ as a matrix of the form $\begin{bmatrix}a_1 & a_2 & a_3 & \dots & a_n\end{bmatrix}$ and $B$ as $\begin{bmatrix}b_1 & b_2 & b_3 ...
1
vote
0answers
36 views

Can we always solve these linear algebra equations given rows of multiples?

We can find $n$ elements of multiplicative order $(n+1)$ modulo some large prime $p$, according to this question. Now I'm wondering if we can always perform linear algebra on the elements, as ...
2
votes
2answers
193 views

$A$ be a complex $3\times 3$ matrix such that $A^3=-I$

Let $A$ be a complex $3\times 3$ matrix such that $A^3=-I$, then we need to find out which of the following statements are correct? $A$ has three distinct eigenvalues; $A$ is diagonalizable over ...
4
votes
3answers
422 views

Solving for X in a simple matrix equation system.

I am trying to solve for X in this simple matrix equation system: $$\begin{bmatrix}7 & 7\\2 & 4\\\end{bmatrix} - X\begin{bmatrix}5 & -1\\6 & -4\\\end{bmatrix} = E $$ where $E$ is the ...
4
votes
1answer
284 views

Is Householder orthogonalization/QR practicable for non-Euclidean inner products?

The question Is there a variant of the Householder QR algorithm to orthonormalize a set of vectors with respect to an inner product if no orthonormal basis is known a priori? Background Let's ...
1
vote
2answers
136 views

more than n eigenvectors

I am learning diagonalization of matrices. We are given the following theorem: If $A$ is an $n\times n $ matrix with $n$ distinct eigenvalues, then $a$ is diagonalizable Now the proof is: ...
0
votes
2answers
203 views

square root of a symetric matrix

I have a symmetric matrix which positive-definite, but it contains zero as eigen value. So the method of Cholesky does not work, could someone give another method to do this? I do not want an ...
2
votes
2answers
262 views

I am going to learn these math topics , please suggest me where to start?

I always did poor in mathematics and i even quit my mathematics from 10th grade but since I was good in programming ( C++ and Java) I took course related to computers in my college where I am going to ...
0
votes
0answers
268 views

General form of Matrix family

Find the general form of matrices $^1U,^2U,^3U \in \mathbb{R}^{3\times3}$such that $^iU$ is nonsingular and for every permutation $\pi : \left[{3} \right] \rightarrow \left[{3} \right]$ ...
3
votes
0answers
52 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [2] [duplicate]

Let $A$ be a $n \times n$ matrix with entries on the set $\{0,1\}$, with exactly two ones on each column and two ones on each row. Give necessary and sufficient conditions for rank$(A)$ to be $n$. I ...
4
votes
1answer
247 views

Singular Value Decomposition to predict a missing value from a FULLY POPULATED matrix

Let's say I have a matrix of data that is fully populated except for one value. For example, Column 1 is Height, Column 2 is Weight, Column 3 is Years Weight Lifting, and Column 4 is Bench Press ...
0
votes
2answers
380 views

Minpoly and Charpoly of block diagonal matrix

I am currently struggling with an exercise where I have to treat a Block diagonal matrix (so it is a square matrix, where square block matrices are down the diagonal). Now I was wondering whether we ...
1
vote
1answer
88 views

Complement of invariant subspace

Assuming that I have a given vector space $V$ and a subspace $U$, which is invariant under an endomorphism $A\in End(V)$. I want to prove that $U^\perp$ is also invariant on $A$ .
0
votes
1answer
240 views

Block matrix and invariant subspaces

I was wondering what the exact relationship between invariant subspaces and a block matrix is? Is it correct to say: Each diagonal block matrix "creates a vector space decomposition" and vice versa? ...
2
votes
0answers
36 views

The dimension of birkoff polytope

Let $P_m$ be a subset for R^mxm be the polytope given by: $x_i,_j \ge 0$ $x_i,_1 + ... + x_i,_m \le 1$ $x_1,_j + ... + x_m,_j \le 1$ $\sum_{1 \le i,j \le m } \ x_i,_j \ge m-1$ Contruct a ...
3
votes
2answers
146 views

Set of permutation matrices

I'm stuck in this problem. Prove the set $P$ of $n×n$ permutation matrices spans a subspace of dimension $(n−1)^2+1$
1
vote
1answer
72 views

Is this function involving matrices convex?

Let $X\in \mathbb{R}^{n \times n}$. Then, is the function $$ \text{Tr}\left( (X^T X )^{-1} \right)$$ convex in $X$? ($\text{Tr}$ denotes the trace operator)
1
vote
1answer
42 views

question about a $7\times 5$ matrix

I have a matrix $A$ of order $7\times5$ and of rank $4$. Let $P$ and $Q$ be the projection matrices that project vectors in $\mathbb R^7$ onto $R(A)$ and $N(A^T)$, respectively. I have to show that: ...
4
votes
1answer
519 views

constructive canonical form of orthogonal matrix

For every orthogonal matrix $Q$ over the reals there is an orthogonal matrix $P$ and a block diagonal matrix $D$ such that $D=PQP^{t}$. Each block in D is either $(1)$, $(-1)$ or a two dimensional ...
2
votes
2answers
139 views

matrix representations of linear transformation

I have a indexing problem about the matrix representation of linear transformation. Let $V$ be a $3$ dimensional vector space over a field $F$ and fix $(\mathbf{v_1},\mathbf{v_2},\mathbf{v_3})$ as a ...
0
votes
0answers
41 views

Finding the inverse of a matrix involving reciprocals of positive integers [duplicate]

The result of Example 16 suggests that perhaps the matrix $A=\begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} \ \ \cdots \ & \frac{1}{n} \\ \frac{1}{2}& \frac{1}{3} & \cdots ...
2
votes
1answer
42 views

Question about projection

If $B^T AB$ is not a projection, then either $B$ isn't orthogonal, or $A$ isn't a projection. I understand that orthogonal $B$ and projection $A$ help transform the following: $B^T ABB^T AB = B^T AAB ...