# 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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### QR decomposition subcases

Is the full QR decomposition the most general, which includes the reduced QR, i.e, is it alright to always compute the full QR Decomposition for a given matrix blindly? What's the point of having two ...
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### Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
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### Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
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### Rank of a lower triangular block matrix

For $$A= \begin{bmatrix}B&0\\C&D\end{bmatrix}$$ where $B, C, D$ are matrices that may be rectangular, is it true or false that $$\text{rank}(A)=\text{rank}(B)+\text{rank}(D)$$ I think that if ...
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### Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$AC=I \quad \mbox{and} \quad BA=I,$$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
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### A proof of the Continuity of the inverse matrix function

I would like to see a proof to this fact. If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there ...
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### How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular.

$\mathbf{K}$ is a Krylov matrix. \begin{align} \mathbf{K}&= \left[ \begin{array}{ccccc} \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{N-1}\...
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### Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
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### Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
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### Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
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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 ...
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 ...