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
votes
2answers
39 views

Use matrix algebra to show $A(B^{-1}(A+B)A^{-1})B = A+B$

I've got a super simple linear algebra question for an intro college course I can't seem to figure out. Using matrix algebra and matrix identities, show that: $$ A(B^{-1}(A+B)A^{-1})B = A+B $$ ...
2
votes
2answers
37 views

Proving a Trick to More Quickly Calculate N-Step Transition Probabilities

So, I have been working on a homework problem all day that asks me to prove that: $P^n= \Pi +Q^n$ where P is the transition matrix of a finite-state regular Markov Chain, $\Pi$ is a matrix whose rows ...
1
vote
1answer
23 views

I don't understand how Kirchhoff's Theorem can be true

Kirchhoff's Matrix-Tree theorem states that the number of spanning trees of a graph G is equal to any cofactor of its Laplacian matrix. Wouldn't this imply that all cofactors of a Laplacian matrix ...
0
votes
0answers
1 view

Lower bound on the sum of singular values for a sum of Hermitian matrices

Suppose $\mathbf{A}$ is a Hermitian $n\times n$ matrix with eigenvalues $\lambda_i(\mathbf{A})$, $i=1,\ldots,n$. Suppose $\mathbf{B}$ is an $n \times n$ complex-valued matrix and $b\neq 0$ is a ...
0
votes
0answers
24 views

Under What Intervals Is A Matrix Positive Definite, Positive Semi-Definite, Indefinite, Negative Definite and Negative Semi-Definite? [on hold]

Suppose we have a matrix which represents a quadratic form. $$ \begin{matrix} a & -a & -3a \\ -a & 2a & 2a \\ -3a & 2a & (9a+2) \\ ...
0
votes
3answers
15 views

Show that a matrix with (I) a row of zeros and (II) a column of zeros cannot be invertible (respectively)

Show that a matrix with a row of zeros cannot be invertible. Show that a matrix with a column of zeros cannot be invertible. What I tried: I tried to show that a matrix $A \in M_n (\mathbb{R})$ such ...
1
vote
1answer
37 views

Show that $(I − Q)^{−1} $= $Q^2 + Q+ I$.

Consider $Q\in M_n (\mathbb{R})$ Assume that $Q^3 = [0] $ show that $ (I − Q)^{−1} = Q^2 + Q + I$. What I tried: I tried to use $(I-Q)(I-Q)^{-1} = I$ and use that to manipulate the left side of the ...
5
votes
1answer
39 views

Factor the matrix (scalar $\times A$) into permutations of $A$

Here's an example of $A . B = scalar \times C$, done with magic squares. The last square does not have a consecutive range of digits. Drop the magic square requirement. In $2\times2$ matrices we ...
0
votes
0answers
10 views

What is the name of the sub-matrix?

Given a matrix $\mathbf{A}$ of size $n\times n$. Let $I=\{i_1,\ldots,i_k\}\subseteq\{1,\ldots,n\}$ for some $k\leqslant n$. How to call the sub-matrix of $\mathbf{A}$ that has its indices in $I$? (I ...
0
votes
1answer
12 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
4
votes
4answers
5k views

Raising a square matrix to a negative half power

I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations: $$S = (\textrm{diag } R^{-1})^{-1/2}$$ I understand the diagonal and inverse ...
4
votes
0answers
27 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
2
votes
1answer
48 views

The root system of $sl(3,\mathbb C)$

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a ...
0
votes
1answer
13 views

$0 * \Lambda ^{1/2} \neq 0$ (?) Problems with matrix multiplication

Suppose $\Lambda$ is a diagonal matrix of size $n > 1$ and rank $1$, let's denote the sole element on the diagonal as $\lambda$. Consider the following equation: $\Lambda ^ {-1/2} = 1/\lambda * ...
1
vote
0answers
9 views

Stationary points with matrix

I have an exercise but I do not even know where I should start: Consider the normalised quadratic form $(x^T Ax)/(x^T x)$ where $x∈R^2$, $A$ is a general 2x2 matrix. Find the vectors that make this ...
0
votes
0answers
8 views

Find a basis for the column space - why not reduce to RREF first?

Related to Understanding how to find a basis for the row space/column space of some matrix A. . When asked to find the basis for the column space of a matrix, can I first reduce to RREF, and then use ...
0
votes
2answers
43 views

Find $\det(A)$ of Matrix and condition on a and b

Let $$ A=\begin{bmatrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{bmatrix} $$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
3
votes
3answers
28 views

$A$ has more columns than rows and has full row rank, show there exist infinitely many $B$ s.t. $AB=I$

If A $\in M_{m\times n}(R)$ such that $n>m$. Prove that if $\text{rank} (A) = m$ then there are infinitely many matrices $B \in \ M_{n\times m} (R)$ such that $ AB = I_m$ So the question is ...
0
votes
1answer
16 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of ...
0
votes
1answer
56 views

Different results when multiplying matrices with a calculator

I'm having a big trouble when I have to multiply 2 matrices. I think I have a problem with my calculator (HP 50g) because I get a correct answer but not the one my professor has. For example, I have ...
0
votes
1answer
1k views

Difference between rotation and pure rotation

Hi i am trying to understand my teacher's assignment. I have 2 write 2 Matlab functions ...
0
votes
0answers
54 views

Prove that $A+2I$ is invertible [duplicate]

Given $A$ is a square matrix such that $A^{3} = 2I$ Prove that $A+2I$ is invertible and find its inverse. How do I prove that $A+2I$ is invertible? For proving $A-I$ is invertible, I use ...
0
votes
1answer
30 views

Operation count, LU-decomposition

I'm having trouble with an assignment question. The question is as follows: Determine the total number of multiplications and divisions (as a function of $n$) required to compute the LU-decomposition ...
0
votes
1answer
30 views

How many subsets of unknowns whose sum can be determined by the underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$

Consider a underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$ (i.e. being a binary matrix), $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. I want find a set $S$, $e \in S$ if and only if ...
0
votes
1answer
47 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
0
votes
1answer
25 views

Differentation of vector with respect the another vector [on hold]

$y$ is $m \times 1$ vector $y=Ax$. $A$ is $m \times n$ matrix in function of $z$. $x$ is $n \times 1$ vector in function of $z$. And $z$ is vector $r \times 1$. How can i find $\frac{dy}{dz}$? I ...
1
vote
1answer
22 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal

Let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to ...
3
votes
3answers
35 views

Intuitive understanding of vector / matrix calculcation

I am currently dealing with calculations done on vectors and matrices. For some parts I have gained an intuitive understanding, for others I didn't. E.g., when we are adding two vectors, you can ...
0
votes
2answers
1k views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
5
votes
2answers
1k views

$A$ is skew hermitian, prove $(I-A)^{-1} (I+A)$ is unitary

Given $A$ is a skew-hermitian, (i.e $A^H=−A$), the Cayley transform of $A$ is defined as: $W=(I-A)^{-1} (I+A)$. How can be proved that $W$ is unitary (i.e. $W^H W = W W^H = I$)?
6
votes
4answers
196 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
2
votes
1answer
33 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
4
votes
1answer
63 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ ...
0
votes
0answers
5 views

Difference between the SCM converging to the Marcenko-Pastur distribution and Johnstone's result about the top eigenvalue

I have a confusion which I suppose must be rather basic. As I understand, in the 60s/70s it was known that the empirical eigenvalue distribution of the sample covariance (of $n$ i.i.d. standard ...
0
votes
0answers
79 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix?
0
votes
1answer
42 views

Prove that a real matrix is a matroid

Problem $A$ real matrix, size $m\times n$ $M$ some structure, possible matroid $E(M)$ set of all columns of $A$ (we're considering them vectors) $I(M)$ set of all linearly independent columns of $A$ ...
1
vote
1answer
1k views

Does a Symmetric Matrix with main diagonal zero is classified into a separate type of its own? And does it have a particular name?

For example, I have a Matrix as shown below. Does this Matrix belong to a particular type. I am CS student and not familiar with types of Matrices. I am researching to know the particular Matrix type ...
11
votes
0answers
179 views
+200

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
2
votes
0answers
27 views

Find (linear) transformation matrix using the fact that the diagonals of a parallelogram bisect each other.

This is the first time I post something on this website. I'm on this question already for hours. I'm clearly not asking the community to do my homework, I'm hoping someone can explain me how I should ...
1
vote
1answer
23 views

L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a ...
1
vote
1answer
48 views

Proving matrix is invertible using the Banach Lemma

I have an assignment question that goes like this: Consider the $n \times n$ matrix $$ \begin{pmatrix} 2 & 1 & 2^{-1} & 2^{-2} & 2^{-3} & 2^{-4} & \cdots & ...
0
votes
2answers
20 views

Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
2
votes
1answer
18 views

Is it true that for all matrices $A$ and all traceless matrices $T$, there exists a traceless matrix $T'$ such that $AT = T'A$?

Fix a real number $n$. By a "matrix", I mean an $n \times n$ real matrix. Now let $A$ denote a matrix. Is it true that for all traceless matrices $T$, there exists a traceless matrix $T'$ such that ...
1
vote
0answers
33 views

Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
-1
votes
0answers
41 views

XOR binary matrix multiplication $AX=B$? [on hold]

Let $A$, $B$, and $X$ be binary matrices (in F2 ), where $A$ and $B$ are of size $n \times m$ with $n > m $. $X$ is an $m \times m$ matrix. Compute $X$ such that $AX=B$. ps: $A$ is not a ...
0
votes
0answers
11 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
0
votes
1answer
15 views

Prove of identity: $(Av) × (Aw) = CofA (v × w)$ [on hold]

How can I prove that for each $A \in M^{3×3}$ and $v, w ∈ \mathbb R^3$ $(Av) × (Aw) = CofA (v × w)$
0
votes
1answer
35 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{max}(B^{-1}A)}{\lambda_{min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
0
votes
2answers
14 views

Prove a covariance matrix is positive semidefinite

Given a random vector c with zero mean, the covariance matrix $\Sigma = E[cc^T]$. The following steps were given to prove that it is positive semidefinite. $u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = ...
3
votes
1answer
16 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...