# Tagged Questions

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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### Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...
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### Cholesky decomposition of $I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

I need to compute the Cholesky decomposition of the following matrix: $\varPi=I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$ Here $n$ is the dimension of the matrix and $x>0$. $\iota_{n}$ is ...
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### Projectors solution of a matrix equation

If I have two $n \times n$ complex matrices $A$ and $B$, where $A$ and $B$ are both projectors, i.e., $A^2=A$ and $B^2=B$. If $A B = A$ and $A \neq I$, clearly if $B=A$, or $B=I$ then the equality ...
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### How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Let $x \in X \subset \mathbb{R}^n$, then I define a set: $$A = \{x \in X| 1^Tx = 0\}$$ Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$ I concatenate $x,y$ in to a single vector ...
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### What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
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### Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
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### What if 1st pivot is missing but the 2nd one is there?

I have the following matrix : $$A= \begin{bmatrix} 0 &1 &2 &3 &4 \\ 0 &0 &0 &1 &2\\ 0 &0 &0 &0 &0\\ \end{bmatrix}$$ So here the 1st pivot is missing ...
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### CUR decomposition

In SVD, reconstruction of matrix X from its rank-2 approximation X2 ( i.e. using two PCs) is as follow: X_reconstructed = U(:,1:2) * S(1:2,1:2) * V(:,1:2)' How to reconstruct matrix X from its ...
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### Finding the matrix representation of a transformation

Question is : The vectors $(2,1)$ and $(1,1)$ form a basis for $R^2$. Let $T$ be a linear transformation satisfying $T(2,1)=(-2,6)$ and $T(1,1)=(0,5)$. Find the matrix of $T$ with respect to the ...
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### Matrices that represent rotations

So the question is What 3 by 3 matrices represent the transformations that a) rotate the x-y plane, then x-z, then y-z through 90? I believe this is the matrix that rotates the xy plane \begin{...
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### Minimize the inner product of this tensor function

Minimize the following function: $f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
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### Linear Algebra Eigenvalues and Eigenvectors [closed]

So I have a 2x2 matrix where equation 1(EQN1) is 1 and 2; equation2(EQN2) 2: 4 and 3 The determinant is det(A-λI)=0 When I first solve the eigenvalues I get λ=5, λ=-1 Now this is where I am lost,...
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### About the distributive property of matrices

So We all know that matrix operations are distributive, so here is my question.$A^2+AB\\$ and $BA+B^2$ is two matrix operations I have, I know we can do $A(A+B)$ in the first operation but I'm not ...
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### Matrix decomposition into square positive integer matrices

This is an attempt at an analogy with prime numbers. Let's consider only square matrices with positive integer entries. Which of them are 'prime' and how to decompose such a matrix in general? To ...
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Let $A=[a_{ij}]$ be a real $n\times n$ matrix. Prove that the following conditions are equivalent: $(1)$ for every $t\ge 0$, all elements of the matrix $\exp (tA)$ are nonnegative $(2)$ a_{ij}\ge ... 1answer 24 views ### How to find a scalar given a matrix equation with an unknown matrix? I am not expert in linear algebra. I couldn't come up with any solution for my problem with my limited knowledge. So the question may be even silly or have no solution, I don't know. But I appreciate ... 1answer 48 views ### Complex matrix decomposition If I have a block matrix of complex matrices $$\begin{bmatrix} P &Q\\ Q^T & P \end{bmatrix}$$ while Q being skew symmetric, the decomposition is \begin{bmatrix} I & -iI\\ . &... 0answers 18 views ### Transpose of matrix (block matrix form) [closed] Suppose A is a matrix 2N \times 2N which is made by a matrix a, b,c,d, a N\times N matrix. I want to know following holds \begin{align} &A= \begin{pmatrix} a & b \\ c& d ... 3answers 44 views ### Is W=\{A \in M_{n\times n}: \det(A)\neq0\} a subspace of M_{n\times n}(\mathbb{R})? How can I prove if two matrices of W, say w_1 ,w_2, are closed under addition and scalar multiplication. I know that under scalar multiplication w_1 is still in W but is there a way to prove \... 0answers 18 views ### How can I take a distance matrix and construct a coordinate representation from it Say I have a distance matrix M of rank n, where the distance between the ith and jth point is M[i,j]. the diagonal of such a matrix will be 0. How can I convert the distance matrix M to coordinates ... 0answers 33 views ### Dot product of two vectors as the eigenvalue of a special matrix [duplicate] I just noticed that for any two Cartesian vectors their dot product is precisely the only non-zero eigenvalue (if such exists) of the following matrix:\vec{a}=(a_1,a_2,a_3,\dots)\vec{b}=(b_1,... 0answers 25 views ### Do I calculate the determinant of a Jacobian Matrix the same way as a normal symmetrical Matrix? As the title says: Do I calculate the determinant of a Jacobian Matrix the same way as a normal symmetrical Matrix? Or is there another way to do it? 1answer 34 views ### Derivation of gradient for non negative matrix factorization I am looking at a paper for non-negative matrix factorization and can't seem to figure out the derivation for the gradient. The function is as follows:f(W,H) = \frac{1}{2}||V-WH ||^2_F$Where V ... 3answers 86 views ### Derivative of projection's norm squared with respect to a matrix Background: Let$M^{n\times k}(\mathbb{R})$denote the$n\times k$matrices with real entries. For any smooth function$f: M^{n\times k}(\mathbb{R}) \to \mathbb{R}$, define the derivative$\frac{\...
For a matrix $A$ the Singular Values Decomposition allows getting the closest low-rank approximation $$A_K=\sum_i^K\sigma_i \vec{v}_i \vec{u}_i^T$$ so that $\|A-A_k\|_F$ is minimal. I'd like to do ...