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
36 views

Proof related to matrix with if and only if condition

Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined. Prove that $$(A+B)^2=A^2+B^2+2AB\quad\text{if and only if}\quad AB=BA.$$ Someone help me with this.
1
vote
1answer
17 views

How to approach specific dimensionality reductions?

I'm having difficulty in rationalizing dimensionality reduction (I've used other sources), and I would appreciate it if someone could help me out with a specific example. Given an $M \times M$ PCA ...
1
vote
1answer
52 views

Making a matrix diagonal with its eigenvectors

I'm trying to make my matrix diagonal. this is my matrix (for matlab and octave) ...
1
vote
2answers
41 views

Prove these matrix-vector products are linearly dependent/independent.

I have two statements that I wanto to prove or disprove. Let $A \in \Bbb R^{n \times n}$ and $b_1,\ldots,b_n \in \Bbb R^n$ be linearly independant. Then, $Ab_1,\ldots,Ab_n$ are also linearly ...
1
vote
0answers
43 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
1
vote
1answer
77 views

How to prove Cholesky decomposition for positive-semidefinite matrices?

According to Cholesky decomposition $A$ is a Hermitian positive-definite matrix if and only if $A=T^*T$ for some upper triangular matrix $T$. When $A$ is positive-semidefinite we have such ...
2
votes
2answers
89 views

matrix inequality proof [completion of squares]

Can someone help me to prove this? $\begin{bmatrix} 0 & B^\top W^\top \\ WB & 0 \end{bmatrix} \leq \begin{bmatrix} B^\top Q B & 0 \\0 & W^\top Q^{-1}W \end{bmatrix}$ with $Q$ ...
2
votes
2answers
77 views

Proving generalized form of Laplace expansion along a row - determinant

Definition: Let $A$ be an ($n \times n$)-matrix. Let $M_{ij}$ denote the matrix acquired from $A$ by deleting row $i$ and column $j$. For $n \geq 2$ we define the determinant of $A$ inductively as \...
4
votes
2answers
120 views

Prove that $\dim(V)$ is even

Let $V$ be a finite dimensional vector space. Let $A_1,A_2: V\rightarrow V$ be commuting linear operators such that $A_1+A_2=-I$ where $I$ is the identity operator. Also $A_1,A_2$ have no negative ...
0
votes
0answers
29 views

Matrix Calculation .

$$P\pi=\pi$$ $$ \begin{bmatrix} 0&0&0&0&1 \\ 1&0&0&0&0 \\ \frac{1}{2}&\frac{1}{2}&0&0&0 \\ \frac{1}{3}&\frac{...
0
votes
1answer
44 views

Such a matrix: Except the main diagonal, the rest are Hermitian

In a matrix, all the entries are complex numbers. If we set the main diagonal entries to be zero, then the matrix will be Hermitian. Does this matrix has a name or some nice properties, especially ...
0
votes
2answers
81 views

Explain how the proof is done

A solution of matrix problem appears to be as follows some one explain the following in the solution why is A cube is eliminated and fourth power of A is obtained? In the seventh line In the ...
1
vote
1answer
68 views

Understanding representation of permutation matrix as vector

I hope this question is relevant here: I'm using some external software that does an LU decomposition of a square $(n\times n)$ matrix; the result is returned as three matrices L, U and P where P is ...
6
votes
2answers
263 views

Determinant of a Certain Block Structured Positive Definite Matrix

PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED VERSION....
1
vote
1answer
54 views

Tightness of inequalities for various matrix norms

For a general inequality involving matrix norms, does the choice of the norm influence the tightness of the inequality? Eg. In $\|AB\| \leq \|A\| \|B\|$, Does the choice of the norm affect the ...
1
vote
0answers
47 views

How do we find the eigenvalues of the matrix?

We have the $m \times m$ matrix $$A=\begin{bmatrix} -2 & 1 & & & 0 \\ 1 & -2 & 1 & & \\ & \dots & \dots & \dots & \\ & & 1 & -2 &...
0
votes
3answers
45 views

What is $[M_1,M_2]$ equal to? ($M_1$ and $M_2$ are matrices)

This is an old exercise that I had a year ago: $$M_1 = \dfrac{1}{\sqrt{2}} \begin{bmatrix}0 & 1 &0\\1 & 0 & 1\\0 & 1 & 0\end{bmatrix}$$ $$M_2 = \dfrac{1}{\sqrt{2}} \begin{...
1
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0answers
24 views

Product expression maximization given 4 matrices as function of 2 column vectors

Hypothesis: - we are given 4 complex matrices denoted with $H_1, H_2, G_1$ and $G_2$. - the 4 matrices are not necessarily square so their size is $N$ by $M$. - we denote with $w_1$ and $w_2$ two ...
0
votes
0answers
51 views

Help understanding a theorem about diagonalizable matrices

So while studying for my Linear Algebra test, I'm required to study some theorems and their proofs, and I have trouble understanding a particular part of the proof for the following (I'm translating ...
0
votes
1answer
41 views

Write the equation in cartesian form

I can't remember what method to use, help would be appreciated. The question: A line is given by $r = 2i +3J -k + \lambda$. Write the equation in cartesian form. Thanks
0
votes
1answer
51 views

Problem on singular value and trace of matrix

Let $A,B\in \mathbb{R}^{n\times n}$ show that there exists (1) Orthogonal matrix U,V satisfying $|trace(AB)|\leq\sum_{i=1}^{n}\sum_{j=1}^{n}(\sigma_{i}(A)\sigma_{j}(B)|u_{ij}v_{ij}|)$ (2) Double ...
0
votes
1answer
57 views

Help me proving a property of the determinant

I'm trying to prove the following property using cofactor expansion along the first row. Not sure if my proof is correct (don't really know what to do with the induction hypothesis), and I got trouble ...
1
vote
0answers
102 views

Is my proof true or not ? $rk(A) + rk(B) \ge rk(A+B)$

I know that my question has already an answer here, but I have proved it another way and I want to see whether my proof is true or not? If we assume $A$ , $B$ and $A+B$ are respectively the matrices ...
0
votes
1answer
25 views

Matrix entropy measure

I have a matrix (its dimension is $n$ x $m$) where each cell can be $0$ or $1$. I would like to calculate an "entropy" measure on it that tells me how close are the ones together or how spread they ...
1
vote
1answer
181 views

Finding minimal polynomial of big blocks diagonal matrix [duplicate]

Consider the following matrix: \begin{bmatrix} -3 & 1 & -1 & & & & & \\ -7 & 5 & -1 & & 0 & & & 0\\ -6 & 6 & -2 & & ...
1
vote
1answer
30 views

Question about the signature of a matrix

Let $A$ and $B$ be real symmetric $n \times n$ matrices with the same rank such that $B$ differs from $A$ only by two sufficiently small nondiagonal entries. Can we say that $B$ has the same signature ...
0
votes
1answer
59 views

Strict inequality of vector norms

Given a non orthogonal projection $p$ and non zero vector $x$. I am going to prove that $$\|Px\|<c\|x\|$$ for some $c<1$, where $\|\cdot\|$ is the usual Euclidean norm. I can only have the ...
0
votes
1answer
38 views

How to solve for matrix $X$ in $Y=X(X^TDX)^{-1/2}$

Let $Y \in \mathbb{R}^{n \times n}$ be any matrix such that $Y^T D Y = I$ for some positive diagonal matrix $D$ and $I$ the identity matrix. Further it is known that $Y=X(X^TDX)^{-1/2}$ for some ...
1
vote
0answers
58 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
1
vote
1answer
385 views

Jacobian of matrix product

I understand from Wikipedia (https://en.wikipedia.org/wiki/Matrix_calculus#Matrix-by-scalar) that if two Matrices $M \in \mathbb{R}^{m+k}$ and ${N \in \mathbb{R}^{k+t}}$ whose elements are real ...
1
vote
1answer
82 views

If $A$ is a $12 \times 12$ real matrix such that $A^{17}=I$ , is $A$ diagonalizable ? Are all eigenvalues of $A$ real ?

If $A$ is a $12 \times 12$ real matrix such that $A^{17}=I$ , is $A$ diagonalizable ? Are all eigenvalues of $A$ real ?
1
vote
3answers
200 views

The rows of an orthogonal matrix form an orthonormal basis

A matrix $A \in \operatorname{Mat}(n \times n, \Bbb R)$ is said to be orthogonal if its columns are orthonormal relative to the dot product on $\Bbb R^n$. By considering $A^TA$, show that $A$ ...
0
votes
1answer
39 views

Algorithm to find positive definite matrix given conditions.

I want an algorithm that always find one solution for the given problem below: Given a positive n length vector, b, a n vector of 1 values, u: I want to determine B, matrix n * n, positive definite ...
2
votes
1answer
55 views

finding two conditions so that for two polynomials, there exists exactly one matrix

Let $K$ be a field, and $f, g \in K[t]$, the ring of polynomials over $K$. I want to find a necessary and a sufficient condition for $f$ and $g$, so that there exists an (except for similar matrices) ...
1
vote
2answers
47 views

Matrix inequality with maximum singular value

Let $P$ be a positive definite matrix: $P = P^\top \succ 0$. For any square matrix $A$, show that the matrix inequality $$ A^\top P A \preccurlyeq \sigma^2 P $$ holds if $\sigma$ is the maximum ...
3
votes
1answer
58 views

$ (x x^T)^{-1}$, efficient matrix inversion for matrix composed as product of a vector with itself?

Given a vector $x$, is there an efficient way of computing $(x x^T)^{-1}$? I mean without first computing the matrix $(x x^T)$ and then applying matrix inversion techniques to it?
1
vote
3answers
52 views

Contraction maps with $3\times 3$ matrices, choice for $x$?

$f:\Bbb R^2\to \Bbb R^2$ given by $$f(x)=\begin{bmatrix}\frac12&0\\0&\frac13\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$ We can determine if this is a contraction map by showing that $\...
0
votes
0answers
27 views

Reading a basis for $\operatorname{Null}(A)$ and $\operatorname{Null}(A^T)$ from the reduced row-echelon form of $A$

I know that it is possible to read the basis for $\operatorname{Null}(A)$ and $\operatorname{Null}(A^T)$ by simply looking at the reduced row-echelon form (RREF) of the matrix $A$. I have only an ...
3
votes
0answers
40 views

What do we know about inverses of matrices which are “like” Laplacians of graphs?

Consider the Laplacian $L$ of a bipartite graph. Is there any generic understanding we have about what $1/(z-L)$ looks like? [say $z > \lambda_\max(L)$)] You can consider variations of $L$ like ...
0
votes
0answers
107 views

Inverse of a toeplitz matrix with fft based methods

I have a covariance matrix, Q and I need to find out Q^-1. Here, Q is a Toeplitz matrix. Now, I want to calculate the inverse of the matrix with fft based methods rather than the conventional ones ...
1
vote
0answers
35 views

Finding a path to calculate a tangent space in a matrix manifold

Let $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$ be in $\mathbb{R}^{m \times n}$ such that $\mathrm{rank}(A) = k$ and $A_{11}$ is a $k \times k$ invertible matrix. ...
1
vote
1answer
43 views

What relation holds between these two sets?

Question: $\sigma (X)$ is the set of all eigenvalues of $X$. Let $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{m \times m}$. For which condition(s) what relation holds between these ...
3
votes
1answer
47 views

Is having $G^N = I$ enough to a matrix $G$ orthogonal?

I have a matrix $G$ that is cyclic ($G^N = I$) and has determinant $\pm 1 $; is this enough to show that it is orthogonal? If not, what more could I add to make it so?
6
votes
3answers
99 views

The set of all matrix with rank $n-1$ is a hypersurface.

Prove that the set $M$ of $n\times n$ matrices with rank $n-1$ is a hypersurface in $\mathbb{R}^{n²}$ and find the tangent space at $A=(a_{ij})$ where $a_{ij}=\begin{cases} \delta_{ij} \ \text{if} \ (...
3
votes
1answer
82 views

Is $A+uv^T$ invertible?

Let $u, v \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n\times n}$. For which condition(s) this matrix is invertible? $$A + uv^T$$ and find the inverse of this matrix. I tried to take elementary ...
-1
votes
1answer
122 views

Does $A^2B^2=I$ imply $AB=I$? [closed]

If $A$ and $B$ are square matrices and $A^2B^2=I$, does it mean that $AB=I$? If that is true i would like to see the proof to it. Thanks.
3
votes
1answer
25 views

Understanding a matrix bound/inequality

I came across the following statements; For a positive matrix $A$, that is bounded $0 \leq A \leq I,$ where $I$ is the identity matrix and a statement like $Y \leq X$ means that $X-Y$ is positive ...
1
vote
0answers
17 views

Inverse of Lumped Markov Process

Suppose we have an (ergodic, with dumping factor $\alpha$) Markov chain $P$ on a graph $G$, and we lump the transition matrix (it's a sort of reduction of the matrix using a equivalence relation of ...
1
vote
0answers
27 views

Inverse Condition

Given a matrix $H \in \mathbb{R}^{M \times K}$, the new matrix $W$ is defined as $$ W = H^T H,$$ where $T$ denotes the transpose operator. I know that $W$ is invertible if $M \geq K$, but how can I ...
0
votes
0answers
19 views

Three diagonal matrix inverse condition [duplicate]

Let $a$ is a constant. For which condition(s) this matrix has an inverse? $$\left[ \begin{array}{cccccc} a & 1 & 0 & \cdots & & 0 \\ 1 & a & 1 & \cdots & & ...