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
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1answer
207 views

The Matrix Inversion Lemma: the General Case

I find it is hard to understand the application senario of the Matrix Inversion Lemma in non-special cases. Suppose I already computed $A^{-1}$ and want to find $\left(A+UCV \right)^{-1}$. The Matrix ...
2
votes
1answer
116 views

N-th power of matrix

Find the formula for the n-th power of this matrix. $$ \pmatrix{1&1\\1&0} $$ Well $f^2 = \pmatrix{2&1\\1&1}$ and $f^3 = \pmatrix{3&2\\2&1}$ and $f^4 = ...
2
votes
1answer
63 views

Best algorithm for computing eigenvalue decomposition of a $3 \times 3$ symmetrix matrix

In one of my applications, I need to compute the eigenvalue decomposition of a $3 \times 3$ symmetric matrix. What algorithms can I use? Which is the most efficient one? More specifically, the ...
2
votes
1answer
75 views

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. ...
2
votes
2answers
61 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
2
votes
1answer
98 views

intuition for matrix multiplication not being commutative

I want to have an intuition for why A*B in matrix multiplication is not same as B*A. It's clear from definition that they are not and there are arguments here (Fast(est) and intuitive ways to look at ...
2
votes
1answer
382 views

Eigenvalues of 3x3 Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
2
votes
1answer
33 views

Matrix of linear maps

I need a bit of clarification for an assignment question that I have. Let T: *F*$[t]_n$$\to$*F*$^2$ (where *F*$[t]_n$ represents polynomials of degree n) given by $T(f) = (f(1) , f(2))$. I am asked ...
2
votes
1answer
68 views

Parameter Transformation with the Jacobian

If $\phi:U\rightarrow V$ and $\tilde{\phi}:\tilde{U}\rightarrow\tilde{V}$ are parametrizations of a regular surface $S$ with $V\cap\tilde{V}≠0$ and $V, \tilde{V}\subset S$. Let $E,F,G$ and ...
2
votes
2answers
45 views

Merging Linear Regression

If I have built two linear regression models over sets $A$ and $B$, and now want a linear regression over set $A\cup{}B$. Is there a way to reuse what I already have?
2
votes
2answers
527 views

Product rule when differentiate matrix products

I want to differentiate the following expression with respect to $b$ $(Y-Xb)'(Y-Xb)$ Where $Y$ is $n\times1$ and $X$ is $n\times k$ and $b$ is $k\times1$, ' denotes transpose. If i do it term by ...
2
votes
2answers
146 views

The sum of a positive definite matrix and small symmetric matrix

Assume $\bf A$ is a (real) positive definite matrix. Let $\varepsilon \neq 0$ be any real (not necessarily positive) number of your choice and $\bf B$ be a fixed (real) symmetric matrix. Is $\bf A + ...
2
votes
1answer
143 views

Triangular Matrices and Simple Modules

Let $\Bbb{T}_n(k)=\{n \times n \text{ upper triangular matrices (which includes the diagonal entries)}\}$ I want to first express $\Bbb{T}_{n}(k)$ (as a $\Bbb{T}(k)$-module) as a direct sum ...
2
votes
1answer
130 views

bound on trace-norm of product of matrices

Is it true that $$ \|ABA^\dagger\|_1\leq \|A\|^2\|B\|_1, $$ where $\|A\|_1$ is the trace norm, $\|A\|$ is the spectral norm, and $A$ and $B$ are square matrices?
2
votes
1answer
129 views

The dimension of the subvariety of matrices of rank 3 in M(n, m)

Consider the space $M(m, n)$ of matrices of size $m \times n$ over field $K$. Let $X \subset M(m, n)$ be the subset of matrices of rank $3$. Show that $X$ is an algebraic subvariety of $M(m, n)$. ...
2
votes
1answer
738 views

Invertibility, eigenvalues and singular values

I am confused about the relationship between the invertibility of a matrix and its eigenvalues. What do the eigenvalues of a matrix tell you about whether a matrix is invertible or not? Also, what ...
2
votes
1answer
32 views

Let $f\colon V \rightarrow W$ and $A$ be the matrix of $f$ for a certain bases, find $\dim(V), \dim(W), \dim(\text{null}(f))$ and $\dim(\text{im}(f))$

Let $f\colon V \rightarrow W$ and A be the matrix of $f$ for certain bases, find $\dim(V), \dim(W), \dim(\text{null}(f))$ and $\dim(\text{im}(f))$. $A$ is given by the following matrix: ...
2
votes
3answers
320 views

Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
2
votes
1answer
302 views

Gauss-Jordan Elimination to solve for variables

I have the following linear system: $$x + 2y - 3z = 4$$ $$3x - y + 5z = 2$$ $$4x + y + (s^2 - 14)z = s+2$$ Im trying to solve for $s$ to figure out how many solutions it has (if any). I know how to ...
2
votes
1answer
91 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
2
votes
1answer
139 views

Can we classify commuting pairs of matrices up to conjugacy?

Recall that two $n\times n$ matrices over $\mathbb{C}$ are conjugate if and only if they have the same Jordan canonical form. Question. Is there a similar classification for commuting pairs of ...
2
votes
1answer
325 views

Determinant of block matrix with certain properties

So I have the following 2N $\times$ 2N block matrix $H=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ where each block in an N$\times$N matrix. Each block have the following ...
2
votes
2answers
57 views

Calculate the eigenvectors

We calculate the eigenvectors for the matrix $$ \begin{equation*} \mathbf{A} = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & -1 & 3 \\ ...
2
votes
1answer
277 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
2
votes
1answer
54 views

Given 9 positive numbers taking $N$ distinct values, what is minimum $N$ so that they can be arranged into an invertible square matrix?

Assume that $(a_1, \dotsc, a_9)$ are different positive numbers. Let us make a $3\times 3$ matrix $A_S$ by putting them arbitrarily into 9 positions available. Show that there is always ...
2
votes
1answer
119 views

A canonical form for this equivalence relation on matrices

This question is inspired by http://cs.stackexchange.com/q/19250/755. Define the equivalence relation $\sim$ as follows: If $M,N$ are two $8\times 8$ (or $n\times n$ if you prefer generality) $(0,1)$ ...
2
votes
1answer
69 views

Nondiagonalizable Matrix and Polynomials

I got the following problem: If $A$ is a nondiagonalizable square matrix of order $n$ over field $\mathbb{F}$ then there exists a polynomial $P$ of degree $n-1$ over $\mathbb{F}$ such that ...
2
votes
1answer
65 views

Vector $p$-norm for square matrices is submultiplicative for $1 \le p \le 2$

I'm trying to prove that the vector $p$-norm for square matrices is submultiplicative for values of $p$ between $1$ and $2$. The vector $p$-norm for a square matrix $A$ is defined as $\displaystyle ...
2
votes
1answer
260 views

Logarithm and tensor products

We define Von Neumann Entropy for a density matrix $\rho$ (hermitian, positively defined, with trace 1) as : $S(\rho)=-tr(\rho \ln(\rho))$ Considering $\rho = \rho_1 \bigotimes \rho_2$, I want to ...
2
votes
2answers
86 views

Equation of plane without cross product

We know that vectors $(3,3,4)$ and $(-1,-1,5)$ span a plane in $\mathbb{R}^3$. Can we somehow readily infer that the plane's equation is $x_1 - x_2 = 0$? Cross-products have not yet been introduced ...
2
votes
1answer
92 views

Coordinate System Rotation and Cross Term

If I have a conic equation $$ 5x^2 - 4xy + 8y^2 = 36 $$ and $ \left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right] $ in matrix form, whose eigenvalues are 4 and 9, how would I rotate ...
2
votes
1answer
272 views

Linear Algebra: Finding the matrix representation with respect to standard basis

I would appreciate some help with a linear algebra practice question, I'm studying for my final and I am stuck, this is a screenshot of the question: Are my answers correct? a) $P_{2}$: $ ...
2
votes
1answer
50 views

If $A=LL^T$, is $A\otimes I_3 = (L \otimes I_3)(L \otimes I_3)^T$?

$A$ is a symmetric positive definite matrix and $LL^T$ its Cholesky factorization. $A \otimes I_3$ is the Kronecker product of $A$ with the 3x3 identity matrix. Is the relation $A\otimes I_3 = (L ...
2
votes
2answers
73 views

When does an element of $\mathrm{Sl}_n(\mathbb{R})$ preserve a scalar product?

I was asking myself the following question : when does an element of $\mathrm{Sl}_n(\mathbb{R})$ preserve a certain scalar product ? A simple necessary condition is that it's complex eigenvalues are ...
2
votes
1answer
75 views

Equivalent Conditions of Projection Map

I got a problem in doing the following: Let $A_1,\dots,A_k$ be linear operators on a vector space $V$ with dimension $n<+\infty$ such that $$A_1+\cdots+A_k=I.$$ Prove that the following conditions ...
2
votes
1answer
92 views

Sylvester matrix and GCD degree

How to prove that the degree of a $\gcd$ of two polynomials is equal to the dimension of the null space of the Sylvester matrix? I know that any linear combination of the rows of $S(u,v)$ is a linear ...
2
votes
1answer
49 views

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$.

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$. How to construct this example? I have obtained that of ...
2
votes
2answers
222 views

Find the Norm of Matrix using Cauchy-Schwarz inequality

Let $A$ be $n \times n$ matrix and such that all of its entries are uniformly $O(1)$. Using Cauchy-Schwarz inequality, show that the operator norm of matrix $A$, which is $\|A\|_{op}=\sup_{x\in R^n: ...
2
votes
1answer
73 views

Invariant under the choice basis

Suppose $H$ be a full rank $m \times n$ matrix with $m<n$ and $A$ is any invertible $n \times n$ matrix. Consider the subspace $S=\{x\in \mathbb{R}^n:Hx=0\}$; It is well-known that this subspace ...
2
votes
2answers
667 views

Not commuting exponential matrices

Reading this book I came across the following formula : $$ e^A e^B = e^{A+B}e^{\frac{1}{2}[A,B]} $$ where $A$ and $B$ are two matrices and $[A,B] = AB-BA$. I tried to find a demonstration without ...
2
votes
1answer
311 views

rank one update for Cholesky factor

I have covariance matrix known to be $$K = \sum_{i=1}^Nx_ix_i^T$$ where the dimension of $x$ is big (like 50000) so I don't want to really compute any outer-product to expand it as a full matrix. ...
2
votes
2answers
77 views

What's the insight for a 3x3 matrix with orthogonormal columns,the rows are also orthogonormal?

I know this can be easily proved with simple matrix tricks, But I don't know the insight for this, and just feels it amazing that if I pick up 3 orthogonormal vectors in 3d space, their corresponding ...
2
votes
1answer
83 views

Orthogonally diagonalize a matrix with variables as elements?

So, I understand how to orthogonally diagonalise a basic matrix with numbers in it. However, I have reached a question asking me to do so for a matrice involving only variables. Orthogonally ...
2
votes
1answer
79 views

Question on Matrix Derivatives

When differentiating with respect to a matrix, is it possible to rewrite the derivative operator via some transformation so that you're differentiating with respect to the diagonal eigenvalue matrix ...
2
votes
1answer
104 views

Getting rotation matrix from a vector

I have a vector pointing in some direction and I'm trying to find a matrix $M$ that rotates the vector $v_1=(1,0,0)$ to $v_2=(x,y,z)$, i.e., $M v_1 = v_2$. What is $M$ if $v_1$ and $v_2$ are known? ...
2
votes
1answer
117 views

Update SVD for an added diagonal

I have a positive definite matrix $K$ which has an SVD of $UDU^T$. Is there a way of finding the SVD of $K+\operatorname{diag}(\sigma_1,\sigma_2,\dots,\sigma_n)$ efficiently by the knowledge of the ...
2
votes
2answers
67 views

Given $A, B\in R^{n\times n}$ diagonal matrices, there exist $p,q \in R[x]$ and $X\in R^{n\times n}$ such that $A = p(X),B=q(X)$

(1) We are given $A,B \in R^{n\times n}$ diagonal matrices of n rows and n columns with real values. Show that there are $X \in R^{n\times n}$ and polynomials $q$ and $p$ such that: ...
2
votes
1answer
264 views

Show that ${\bf x} \cdot A^t {\bf y} = {\bf y} \cdot A{\bf x}$

Let $A \in \mathcal M_n (R)$ and ${\bf x}, {\bf y} \in R^n$. How can I show that: $${\bf x} \cdot A^t {\bf y} = {\bf y} \cdot A{\bf x} \, ?$$ Thanks for any help.
2
votes
1answer
71 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
2
votes
2answers
58 views

Singular Value Decomposition-noisy data

I have a system of the form $$Ay=f,$$ where $A$ is a $N\times4$ matrix, $y$ a 4-element array of unknows and $f$ an $N$-element array. I add Gaussian noise in my data. I tested the following ...