Articles of matrix decomposition

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 & C_{22} \end{pmatrix}\begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}$, where $\xi_1$ and $\xi_2$ have $p_1$ and $p_2$ dimensions, respectively. This problem has $p_1+p_2$ eigenvalues: $\{\rho_1, -\rho_1, …, \rho_p, -\rho_p, 0,…., 0\}$, where $p=min\{p_1, p_2\}$. Would somebody […]

On monomial matrix (generalized permutation matrix )

A matrix $a\in GL_{n}(F)$ is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices. I want to prove that following are equivalent $A\in N$ there exist a non singular $D$ (diagonal matrix ) and a permutation matrix $P$ such that $A=DP$ […]

Show the negative-definiteness of a squared Riemannian metric

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices. The geodesic distance between $A,B\in\Bbb{S}_{++}^n$ is given by the following Riemannian metric $$ d(A,B):= \Bigg(\operatorname{tr}\bigg(\ln^2\big(\sqrt{A^{-1}}B\sqrt{A^{-1}}\big)\bigg)\Bigg)^{\frac{1}{2}}. $$ EDIT It could also be defined as follows $$ d(A,B):= \lVert\log(A^{-1}B)\rVert_{F} $$ EDIT II I could show the negative-definiteness of $d^2$ if I could write […]

rank of block triangular matrix and its relation to the rank of its diagonal blocks

Prove that $$rank\begin{pmatrix}A & X \\ 0 & B \\ \end{pmatrix}\ge rank(A) + rank(B)$$ where $$A,B\in \mathbb C^{m \times m}$$. I know the intuition behind it (i.e. maximal independent rows, etc.), but I am looking for a formal proof. I have tried QR decomposition of A and B, then broke the block triangular matrix up […]

To what extent are the Jordan-Chevalley and Levi Decompositions compatible.

I know that the Jordan-Chevalley decomposition for real Lie algebras only applies to semisimple Lie algebras, but in general the addititive J-C decomposition says that for ANY operator, we can decompose it as $X = X_{SS} + X_{N}$, where $X_{SS}$ is semi-simple and $X_{N}$ is nilpotent. Similarly, the Levi decomposition of a Lie algebra $\mathfrak{g} […]

How to use LU decomposition to solve Ax = b

Using LU Decomposition how can I solve for vector $x$ in the system $Ax = b$, given $A$ and $b$. For simplicities sake where $A$ is a 3×3 matrix and $b$ is a vector of size 3. For example how to find x when: $$A= \begin{pmatrix} 3 & 1 & 2\\ 5 & 7 & […]

QR factorization of a special structured matrix

A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm I}$ is an identity matrix, both of order $n$, and $\xi \in \mathbb{R}$ is a scalar. Is there an efficient way to compute a QR factorization […]

Expressing a quadratic form, $\mathbf{x}^TA\mathbf{x}$ in terms of $\lVert\mathbf{x}\rVert^2$, $A$

EDIT: This question is actually an attempt to solve this. Please take a look. Let $A$ be a symmetric postive-definite $n\times n$ matrix, i.e. $A\in\mathbb{S}_{++}^{n}$ Also, let $\mathbf{x}\in\mathbb{R}^n$. Let $Q\colon\mathbf{R}^n\to\mathbb{R}^{*}_{+}$ be the following quadratic form $$ Q(\mathbf{x})=\mathbf{x}^TA\mathbf{x}. $$ If we apply SVD (Singular Value Decomposition) on $A$, we have $$ A=P\Lambda P^T, $$ where $P$ […]

Matrices that Differ only in Diagonal of Decomposition

Suppose that $\mathbf A_1$ and $\mathbf A_2$ are $n \times n$ matrices. Are there necessary and sufficient conditions such that there exists $n \times n$ matrices $\mathbf U$ and $\mathbf V$ and $n \times n$ diagonal matrices $\mathbf D_1$ and $\mathbf D_2$ that satisfy $$ \mathbf A_1 = \mathbf U \mathbf D_1 \mathbf V , […]

Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.

I have some questions regarding the following problem Let $ A + iB $ – hermitian and positive definite, where $A, B \in \mathbb R^{n\ \times\ n} $ show that the real matrix $$C =\begin{pmatrix} A & -B \\ B & A \end{pmatrix} $$ is symmetric and positive definite. How can the following system of […]