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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2
votes
0answers
101 views

Inverse of two matrices multiplied [on hold]

I've been asked to find the inverse of $AB$ where $A$ and $B$ are: $$A=\begin{bmatrix}5 & 3 \\4 & 2\end{bmatrix}$$ $$B=\begin{bmatrix}2 & -3 \\1 & 3\end{bmatrix}$$ My answer: What I ...
2
votes
0answers
10 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$. I'd like a ...
2
votes
0answers
22 views

bilinear forms on $M_{n, n}(K)$

Let $K$ be a field and $V = M_{n, n}(K)$ the ring of $n \times n$ matrices over $K$. For any $f \in V^*$ (the dual space of $V$), we set: $\gamma_f: V \times V \to K, (A, B) \mapsto f(A B^t)$. I now ...
2
votes
1answer
57 views

Absolute value of eigenvalues of a $3 \times 3$ matrix

Let $$A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 1 & w^2& w \\ 1 & w & w^2 \end{array} } \right] $$ where $w$ is a cube root of unity (other than 1). Let ...
2
votes
0answers
26 views

How to prove the following determinant identity?

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
2
votes
0answers
14 views

Cartan matrices: motivation and intuitive examples?

could anyone provide me with a sketch of the motivation that gave rise to Cartan matrices in abstract (homological) algebra, Lie algebrae and so on? Which was the trigger or the need for them? It ...
2
votes
0answers
15 views

Constrained zero diagonal low rank approximation of a matrix with zero diagonal

Suppose that you have a $n\times n$ matrix $A$ that is symmetric and has zero diagonal, such as for example $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0 & 1\\ 2 & 1 & 0}, $$ and you want ...
2
votes
0answers
18 views

Change of coordinates matrix

My linear algebra textbook has an example that I don't really understand. We're trying to find the change of coordinates matrix from $B$ to $B'$ where $B = \{(1, 1), (1, -1)\}$ and $B' = \{(2, 4), (3, ...
2
votes
1answer
59 views

Eigenvalue and eigenvector implies that matrix $A$ satisfies $a_{ik}=a_{ij}a_{jk}$

Consider the following theorem (2.2). The author says the 'if' part is obvious so the proof was not given. Theorem 2.1 above says that every positive $n \times n$ matrix whose elements satisfy the ...
2
votes
2answers
16 views

special relation between the expression of a matrix by block and its rank

Given two matrices $A$ and $C$ of order $n\times n$ and $m\times n$ respectively. We define the following matrix by block: $$ D=\left( \begin{matrix} C\\ CA\\ :\\ :\\ CA^{q-1} \end{matrix}\right) $$ ...
2
votes
0answers
22 views

Trick to rewrite operator in terms of another?

In the book Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems by Bill Sutherland, I would like to understand the trick done in (4), see the excerpt from p 29 shown below I ...
2
votes
0answers
37 views

Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
2
votes
0answers
42 views

How to find the eigenvectors of two closely related hermitian tridiagonal matrices

Given two tridiagonal hermitian matrices A,B with $a_i\in \mathbb{R}$ and $b_i\in \mathbb{C}$ as follows \begin{align} A= \begin{pmatrix} a_{1} & |b_1| & \cdots & 0 \\ |b_1| & ...
2
votes
2answers
50 views

The positive determinant of one special matrix

I try to prove the positive value of determinant for matrix ($n\times n$ for any $n$): \begin{equation*} ||a_{ij}|| = ||f(x_i - y_j)|| , \text{where}~f = f(\lambda(x - y)) = \exp(-\lambda(x - ...
2
votes
0answers
27 views

Smallest bound for convex combination of columns of non-negative matrix

The problem can be formulated as following linear program: $\min_{\mathbf{x},y}\;\;y$ subject to: $\mathbf{Ax}\le y\mathbf{1}$ $\mathbf{x}^T\mathbf{1}=1$ and $x_i \ge 0,\;\forall i$ Here, ...
2
votes
2answers
44 views

Let $A$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
2
votes
1answer
37 views

Similar matrices NOT over the complex numbers [duplicate]

We say that two matrices $A,B$ with complex entries are similar if and only if there exists an invertible complex matrix $P$ so that $A = P^{-1} B P$. Does $P$ always have to be a complex matrix? ...
2
votes
1answer
56 views

Let $\text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}$. Why are $A$ and $B$ similar?

Let $A$ and $B \in M_n$ be two matrices such that $$\forall k=1,2,\dots,n,\ \forall \lambda\ \text{eigenvalue of $A$},\ \text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}.$$ Why are $A$ ...
2
votes
0answers
45 views

How to calculate the eigenvalue of the following general matrix [duplicate]

Let the $n\times n$ matrix $Z$ with $(i,j)$-element defined by $Z_{i,j}=i+j$. How to calculate the eigenvalue of $Z.$? I have used Matlab to calculate it. I find no matter how bigger n is, there are ...
2
votes
0answers
14 views

Eigenvalues and positivity of Hermitian Toeplitz matrices

I want to check the eigenvalues (and also the positivity) of the $n \times n$ complex Toeplitz matrix \begin{equation} T = \begin{bmatrix} r & z_1 & z_2 & z_3 &\cdots & z_{n-1}\\ ...
2
votes
0answers
53 views

Efficient computation of matrix determinant in finite ring

I am trying to implement generalization of Hill cipher. My idea is very simple: the size of key matrix should be arbitrary number not only three. All steps of this cipher is trivial except computation ...
2
votes
0answers
34 views

$trc(A)=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices? [duplicate]

Let $trc(A)=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?($A \in {M_n}$)
2
votes
1answer
31 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
2
votes
0answers
20 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
2
votes
1answer
35 views

Which matrices are covariances matrices?

Let $V$ be a matrix. What conditions should we require so that we can find a random vector $X = (X_1, \dots, X_n)$ so that $V = Var(X)$? Of course necessary conditions are: All the elements on ...
2
votes
2answers
49 views

Matrix Differential Equations

I am working on a practice problem with the following equation: $$ \frac{d^3 x}{dt^3} + (k + 1)\frac{d^2x}{dt^2} + (k+1)\frac{dx}{dt} + kx = 0 $$ I understand the first part which is to convert to a ...
2
votes
0answers
46 views

Is the matrix with these coefficients invertible?

Let $0 \leq x_{i-1} < x_i < x_{i+1} \leq 1$. Let $p, q$ be functions that depend on that such that $p$ is positive and $q$ is non-negative. Let $c_i = a_{i+1,i} = a_{i,i+1}$. Let all other ...
2
votes
0answers
33 views

Matrix Point-Wise Vector Multiplication

I have the following equation, where $\mathbf{M}$ denotes a singular Square Matrix (dim= $n$ x $n$), $\mathbf{x}$ and $\mathbf{y}$ denote vectors (with dimension $n$, too). The operator $\odot$ ...
2
votes
2answers
33 views

how to combine angle rotations along different axes into one rotation along a single vector [duplicate]

So, lets say I have some rotation a about the x-axis(vector:$(1, 0 ,0)$) and some other rotation about y-axis(vector $(0, 1, 0)$) and a rotation about the z-axis(vector: $(0,0,1)$). How would I ...
2
votes
0answers
34 views

How to compute determinant (or eigenvalues) of this matrix?

Let us have the $n \times n$ circulant matrix given by \begin{equation} C(c_0,c_1,\cdots, c_{n-1}) =\begin{bmatrix} c_0 & c_1 & c_2 &\cdots & c_{n-1}\\ c_{n-1} & c_0 & c_1 ...
2
votes
0answers
24 views

Change of basis and similarity

Consider the transformation $T$ in the standard basis: $$[T]_B\begin{bmatrix} 0&3&1 \\ -1&3&1 \\ 0&1&1 \end{bmatrix}$$ Also consider the two matrices: $$A_1 = ...
2
votes
2answers
22 views

Incremental algorithm for matrix eigenvalues

I try to solve the following problem: Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of ...
2
votes
1answer
39 views

how to prove if [T]b is diagonal then there is a scalar “a” such that T(v)=av

hey i was trying to prove the next proposition: given T:V->V for every Basis B, if the matrix [T]B is diagonal, then there is a scalar "a" for every v in V such that T(v)=av this is what i managed ...
2
votes
1answer
30 views

Symmetricity and orthogonality

Can a 3 or more dimensional orthogonal matrix be symmetric ? I am learning linear algebra and I couldn't seem to figure it out. I understand an Identity matrix or any column matrix with either 1 or ...
2
votes
0answers
40 views

Inequality with eigenvalues

Let matrix $ X $ is Hermitian and denote $ \lambda_1(X) \ge \lambda_2(X) \ge \ldots \ge \lambda_n(X) $ eigenvalues of matrix $ X $. Prove that $ \lambda_i(A + B) \le \lambda_i(A) + \lambda_1(B) $ I ...
2
votes
1answer
49 views

Find orthogonal Q given eigenvalue and eigenvector?

Given some upper Hessenberg matrix $H \in R^{n \text{x} n}$, i know how to find an orthogonal matrix which is a product of Givens rotations such that $P^THP$ is also upper Hessenberg, but I'm not sure ...
2
votes
0answers
81 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
2
votes
0answers
43 views

Let A be an m × n matrix, and b an m × 1 vector, both with integer entries.

Let $A$ be an $m \times n$ matrix, and $b$ an $m \times 1$ vector, both with integer entries. If $Ax = b$ has a solution over $ \mathbb Z/p \mathbb Z $ for every prime $p,$ is a real solution ...
2
votes
2answers
107 views

Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their ...
2
votes
0answers
57 views

Restoring Bidiagonality to a Matrix in SVD Algorithms

Good Afternoon, I am implementing the Golub-Reinsch SVD algorithm and am having difficulty with a boundary case Given a bidiagonal matrix of the form: $$ \begin{bmatrix} b11 & ...
2
votes
1answer
40 views

Question about matrices?

I have been learning about matrices in my math class and I am confused as to how exactly they work. Take this example: $\left(\begin{array}{c c c c c | c} 1 & 4 & 1 & 0 & 0 & ...
2
votes
0answers
43 views

Value of determinant using given conditions.

Let $A$ be a $2$ x $2$ matrix with real entries and $det(A)$ is equal to $d$ which is non-zero. It is given that $det(A +d(adjA))=0$ where $adj$ stands for the adjoint of the matrix. We have to find ...
2
votes
1answer
119 views

Volume of a parallelepiped, given 8 vertices

Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped. I'm having trouble finding the 1 vertex ...
2
votes
0answers
34 views

Does this matrix normal form have a name and has it been used?

In a research paper in Theoretical Computer Science, we are using a certain matrix normal form, which I was not able to find in the literature (I have to admit that my Linear Algebra got a bit rusty, ...
2
votes
2answers
30 views

Relation Between Eigenvalues of Block Matrices

Is there any relation between eigenvalues, or spectral radii, of $M$, $M_1$, and $M_2$ block matrices? \begin{equation} M= \begin{bmatrix} A&B\\B^T&C \end{bmatrix} \end{equation} ...
2
votes
0answers
27 views

How to solve the equation $Au+Bv=C$

How do I solve $Au+Bv=C$ Where $A$ and $B$ are constant known matrices that are nxn, $C$ is a constant known nx1 vector while $u$ and $v$ are unknown nx1 vectors with the condition given that $u_i = ...
2
votes
0answers
44 views

algebraic equation with trace

I have a problem which I don't know how to attack. Actually I am not even sure there is a way to do it. Is it possible to solve an equation of this form $$A²-\frac{tr(A)²}{4}Id_{4\times 4}=B$$ where ...
2
votes
0answers
96 views

Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is equal to $\dim(\text{null}(T-\lambda I)^{\dim V})$

Without using induction, prove that the the algebraic multiplicity of an eigenvalue $\lambda$ is $$\dim (\text{null} (T-\lambda I)^{\dim V});$$ here, the algebraic multiplicity of an eigenvalue ...
2
votes
0answers
24 views

Some maximization over stochastic matrix

I am writing some applied assignment which leads me to the following problem. I will be very grateful if anyone can provide a solution or even some thoughts. Thanks a lot! Consider a (row-)stochastic ...
2
votes
0answers
51 views

Are there Latin squares with no repeats on the diagonal not of the form 2y+x+1(mod n)?

I am looking for a certain kind of latin square (nxn). Rules: No repeats in any column or row (Definition of Latin Square) No repeats in any diagonal including others than the main diagonal. So ...