Questions tagged [block-matrices]

For questions about matrices which are defined block wise, like $\pmatrix{A&B\\ C&D}$ where $A,B,C$ and $D$ are themselves matrices. Use this tag with (matrices), and often with (linear-algebra).

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3
votes
1answer
50 views

How to find the eigenvalues of the following block matrix

Suppose $n \times n$ matrix $A$ is symmetric, where $n \in \mathbb N$. If the eigenvalues of $A$ are denoted by $\lambda_1\le \lambda_2\le \dots \le \lambda_n$, find the eigenvalues of the following $(...
0
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1answer
27 views

Block diagonalization of an orthogonal matrix

Consider an orthogonal matrix $A\in \mathbb{R}^{3\times 3}$. Find an orthogonal matrix $T\in O(3)$ s.t. \begin{equation} T^\top A T=\begin{pmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&...
0
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0answers
8 views

Help in understanding the use of Schur's complement and the results

I would love if you can help me understand the following result I found in article I read (https://www.jmlr.org/papers/volume6/chechik05a/chechik05a.pdf).(All variables are multivariate gaussian) $\ ...
1
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0answers
27 views

How do you write a block diagonal matrix in components?

I'm doing some calculations with diagonal block matrices and I'd like to write the components of the block matrix explicitly with respect to the blocks. For diagonal matrices this is easy using the ...
0
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1answer
14 views

Eliminating Equations in a Block Matrix

I have the following linear system: $$ \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{0} & -\mathbf{Z}^{T} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{Y}^{T} \\ \...
0
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0answers
33 views

Question about determinant of a block matrix [duplicate]

I was studying block matrices and suddenly this question came to my mind. Let $A, B \in \Bbb R^{n \times n}$. From this Wikipedia page, $$\det \begin{pmatrix} A & B\\ B & A\end{pmatrix} = \det(...
0
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2answers
59 views

Show a matrix is non-singular and symmetric indefinite

Consider a matrix $A\in \mathbb{R}^{m\times n }$ with $m>n$ is of full column rank. Show that $$B=\begin{bmatrix} I & A \\ A^\top & 0 \end{bmatrix}$$ is non-singular and symmetric ...
2
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1answer
33 views

Inverting a Block-Toeplitz matrix with the Sherman-Morrison formula

Suppose we are given the following Block-Toeplitz matrix: \begin{eqnarray} T=\left(\begin{matrix} A & 0 & ... & 0\\ B & A & ... & \vdots\\ \vdots & \ddots & \ddots &...
0
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2answers
56 views

Eigenvalues of arrowhead-like Hermitian matrix [closed]

Let $\lambda \in \mathbb C, \ell \in \mathbb R, x \in \mathbb C^{n},$ and $$\mathcal H=\begin{pmatrix}\lambda I_n & x\\ x^{\ast} &\ell\end{pmatrix}.$$ Prove that $\lambda$ is an eigenvalue of ...
2
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0answers
16 views

$QR$ decomposition of block matrix

Given a square block matrix $$ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb R^{2d \times 2d}, $$ where $A, B, C, D \in \mathbb R^{d \times d}$. Is there some kind of a block $...
5
votes
2answers
70 views

Using a block matrix to show that $A$ and $B$ commute

I'm studying for a PhD qualifying exam in linear algebra and I wanted to ask about the following problem: Let $A$ and $B$ be invertible $n\times n$ matrices. Let $M$ be the matrix \begin{bmatrix} ...
5
votes
1answer
52 views

Nullspace of a block matrix

Suppose I have $M \in \mathbb{R}^{2n\times 2n}$ such that $$M = \begin{bmatrix} A &B\\0 & C\end{bmatrix}$$ for some $A,B,C \in \mathbb{R}^{n \times n}$. I wish to: (i) Find the nullspace of $...
0
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0answers
40 views

Eigendecomposition of symmetric block tridiagonal matrix with symmetric tridiagonal blocks

I have a symmetric block tridiagonal matrix of the form: $$ K = \begin{bmatrix} C_1 & D_1 & 0 & 0 & 0 & \cdots \\ D_1 & C_2 & C_2 & 0 & 0 & \cdots \\ 0 & ...
1
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0answers
40 views

Show that positive definite matrix $S$ can be factorised in particular way

I would like to show that every positive definite matrix $S \in \mathbb R^{2d \times 2d}$ can be written as: $$ S = M M^T, $$ where $M$ has the following form: $$ M = \Lambda \begin{bmatrix} I ...
0
votes
2answers
29 views

Eigenvalues of block matrix comprising diagonal matrices

This seems simple, but I can't seem to solve it. Let $$A = \begin{pmatrix} D_{11} & \dots & D_{1n} \\ D_{21} & \dots & D_{2n} \\ \vdots & \vdots & \vdots\\ D_{n1} & \dots &...
2
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0answers
31 views

Sufficient conditions for symmetric ($m \times m$) block matrices to be negative definite if diagonal blocks are negative definite

Consider a symmetric block matrix $$A = \begin{bmatrix} A_{11} & \dots & A_{1m}\\ \vdots & \ddots & \vdots\\ A_{m1} & \dots & A_{mm} \end{bmatrix}~,$$ where each diagonal ...
1
vote
1answer
50 views

Partitioned positive definite matrix property

I am interested in the following problem: Let $X$ be a real symmetric positive definite matrix partitioned into four submatrices as follows: $$ X = \begin{pmatrix} A&B\\ B^T & C \...
0
votes
1answer
39 views

How to find the inverse of this $3 \times 3$ block matrix?

Given $$E = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ define the following block matrix $$A = \begin{bmatrix} I & 0 & 0 \\ 0 & E & I \\...
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0answers
11 views

Solving bordered system

Given $m$ points $v_i$ in $\mathbb{R}^n$, $m\le n +1$, (think triangles in 2D, 3D, 4D space), I need to solve bordered equation systems of the form $$ \begin{pmatrix} V^T V & e\\ e^T & 0 \end{...
0
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1answer
72 views

Determinant of a block matrix using row operations [duplicate]

Consider an $n \times n$ matrix of the form $$ A = \begin{pmatrix} M & P \\ 0 & N \end{pmatrix} $$ with $M \in \Bbb K^{r \times r}$, $P \in \Bbb K^{r \times s}$, and $0 \in \Bbb K^{s \times r}$...
0
votes
3answers
48 views

matrix and eigenvalues

Please help me solve this question. This is not a homework assignment. If $\lambda_M$ is the eigenvalue of the matrix M, please prove that $max|\lambda_M|<1$ $$M_{2n\times2n}=\begin{vmatrix} A_{n\...
0
votes
1answer
27 views

Rayleigh Quotient with $2n \times 2n$ Matrices.

I need help understanding a problem. For a bonus question in our problem set, we are told that $\bf{A}$ is a $2n \times 2n$ symmetric matrix of form $$\bf{A} = \begin{pmatrix} A_1 & A_3 \\ A_3 &...
1
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0answers
38 views

Relation Between Eigenvalues of Certain Block Matrices

Let $A=\begin{bmatrix} cI & P & Q \\ P^\top & cI & R\\ Q^\top & R^\top & cI\\ \end{bmatrix}$ be a block matrix where $P$, $Q$ and $R$ are $n\times n$ matrices with real ...
0
votes
0answers
14 views

Efficient computation of eigen value decomposition for a block-diagonal plus a rank-1 matrix

I have a block-diagonal matrix A of size pq where each block of A is of dimension p<q. I also have another rank-1 matrix B. Is there an efficient method to compute the eigenvalue decomposition of ...
1
vote
1answer
30 views

Direct Determinant of Block Matrix

Prove $|C+\overrightarrow{Y}\overrightarrow{Y}'|=(1+\overrightarrow{Y}'C^{-1}\overrightarrow{Y})|C|$ where $Y$ is a $p \times 1$ vector and $C$ is a non-singular $p \times p$ matrix. See Exercise $4....
0
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0answers
18 views

How to write and simplify the OLS estimator from a block matrix?

The OLS estimator can be written has: $(X'X)^{-1}X'Y$ I am interested in analyzing this expression by having $X$ has a block matrix: $X = [\begin{matrix} X_a & X_b \end{matrix}]$ So that the OLS ...
1
vote
1answer
46 views

Spectral radius of a block circulant matrix

Let $A$ the block matrix given by the blocks: $$\tilde{A}=\begin{pmatrix} 0&-\mu&0&...&0&-\mu\\ -\mu&0&-\mu&...&0&0\\ 0&-\mu&0&...&0&0\\ ...&...
1
vote
1answer
34 views

Reformulation of LMI

In a paper I have read, the authors reformulated the following LMI $$X_{t} \succeq \left[\begin{array}{cc}\alpha_{i}\left(B_{t} U+C_{t}\right)^{\top} E_{i}\left(B_{t} U+C_{t}\right) & \left( B_{t} ...
1
vote
0answers
65 views

characteristic polynomial of matrix with several blocks

$\mathbf{M}=\begin{bmatrix} \mathbf{y}&0&0&\cdots&0&0&\mathbf{x}\\ \mathbf{I}&0&0&\cdots&0&0&0\\ 0&\mathbf{I}&0&0&0&0&0\\ 0&...
0
votes
1answer
48 views

Square root of a block matrix

Given the $2n \times 2n $ matrix $$A=\begin{pmatrix} \bar{A} & A^* \\ A^* & \bar{A} \end{pmatrix}$$ let $X$ the matrix such that $XX=A$, that is $X=\sqrt{A}.$ So $$X=\begin{pmatrix} X_{1,1}&...
0
votes
0answers
15 views

Inequality about Eigenvalues of negative semi-definite Block Matrix

Let $M = \left(\begin{array}{cc} A & B \\ B^T & C \end{array} \right)$ be a real symmetric matrix. Suppose that $M, A, B, C$ are all negative semi-definite matrices. I want to find some ...
0
votes
1answer
31 views

Inversibility of a block matrix in schur decomposition

Let $A'$ be a given, n × n, real, positive definite matrix partitioned as follows: \begin{pmatrix} A & B \\ B^T & C \end{pmatrix} show that $C − B^TA^{−1}B$ is positive definite. I know that I ...
2
votes
2answers
182 views

Simplifying matrix equation $A^{-1}B^T(BA^{-1}B^T)^{-1}BA^{-1}$

I want to simplify the inverse of a $2\times 2$ block matrix: $$ \begin{bmatrix} A_{n\times n} & B^T_{n\times m}\\ B_{m\times n} & 0_{m\times m} \end{bmatrix} $$ $A_{n\times n}$ is square and ...
0
votes
0answers
29 views

What are the eigenvalues of this block-tridiagonal matrix?

I have a block-tridiagonal matrix of the form \begin{equation} M = \begin{pmatrix} A & I &&&&\\ I & B & I &&&\\ & I & A & I &&\\ & ...
0
votes
1answer
35 views

Finding eigenvalues of block type matrix

If I know the eigenvalues of matrix $A$, is there any way to calculate eigenvalues of $B$ using some tricks? Here $A$ is an $m\times m$, $O$ is the zero matrix and $I$ denotes $m \times m$ identity ...
3
votes
2answers
52 views

Why do block matrices behave so similarly to regular matrices?

I've been using block matrices a bit in my numerical analysis course. They have many identities that mimic the identities for regular matrices. I understand the proofs, which involve boiling things ...
0
votes
0answers
28 views

Infer spectral radius inequality from 2-norm inequality in block matrices

Say $\mathbf{B}$ is a $N\times N$ block matrix \begin{equation} \mathbf{B} = \begin{bmatrix} \mathbf{B_1} & \mathbf{B_2} \\ \mathbf{B_3} & \mathbf{B_4} \\ \end{bmatrix}, \end{equation} where ...
0
votes
1answer
32 views

Trace and Jordan normal form of Block matrix

Given the Matrix $A \in \mathbb{R}^{2\times2}$ with $Spec(A) = \{300,333\}$ I have the Block matrix $M = \begin{bmatrix}A&A\\A&A\end{bmatrix} \in \mathbb{R}^{4\times4}$ for which I have to ...
0
votes
1answer
36 views

Eigenvectors of a block matrix

I am trying to find the eigenvectors and eigenvalues of the following block matrix $$M = \begin{bmatrix} I & A\\A^T &I\end{bmatrix}$$ where $A \in \mathbb{R}^{n\times n}$. I know I must use $A$...
1
vote
1answer
68 views

Understanding the set $M_1 \cap M_2$

Let $ M_1= \left\lbrace \left( \begin{matrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & f \end{matrix} \right), a,d,f \in i\mathbb{R}, b = \overline{c} \right\rbrace $ be a subspace ...
0
votes
0answers
16 views

How to construct a matrix with one dimension nullspace?

Suppose we have a fat matrix $\boldsymbol{E} \in \mathbb{R}^{p \times t}$ and we conduct SVD as follows \begin{eqnarray} \boldsymbol{E} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^\top. \end{...
0
votes
1answer
24 views

$\text{diag}(M^t,\ldots,M^t)=[\text{diag}(M,\ldots,M)]^t$ with $t$ is real and $M$ is a positive operator

Let $M$ be positive operator on a complex Hilbert space $E$. Since the square root of a positive operator is unique, then $$\begin{pmatrix} M^{1/2} & \\ & \ddots & \\ & & M^{1/2}...
2
votes
1answer
41 views

Diagonalizing a block diagonal matrix

I have a matrix of $2 \times 2$ blocks which looks like \begin{equation} M = \begin{pmatrix} A & B & B & \cdots \\ B & A & B & \cdots \\ B & B & A & \cdots \\ \...
2
votes
1answer
69 views

Eigenvalues of a 2x2 block matrix with invertible diagonal blocks

There is something I do not understand for eigenvalues of block matrices. If we consider a matrix \begin{equation} M = \begin{pmatrix} A & B \\ C & D\end{pmatrix}. \end{equation} Here each ...
2
votes
3answers
67 views

Determinant of associated real matrix from a complex one

Given $X\in \operatorname{GL}(n, \mathbb{C}) $, let $X_r\in \operatorname{GL}(2n, \mathbb{R}) $ be the real matrix obtained by substituting to each complex entry $a+ib$ the matrix $\begin{pmatrix} a &...
0
votes
0answers
18 views

Finding invertible matrices P,Q such that PCQ=D (C and D being certain block matrices)

Two matrices $A\in \mathbb F ^{m\times n}$ and $B\in \mathbb F ^{n\times p}$ are given. Then, if the matrix $C$ is defined as $C=\begin{bmatrix} B & I_n \\ 0 & A \end{bmatrix}$, I have to ...
1
vote
1answer
26 views

Spectral radius of a block matrix with a block that has spectral radius larger than 1

Suppose we have a $n\times n$ block matrix $$ X= \begin{bmatrix} A_{11} & \ldots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{n1} & \ldots & A_{nn} \end{bmatrix}, $$ where each ...
0
votes
1answer
31 views

SVD decomposition of diagonal matrix $A = \begin{pmatrix}0 & C^T \\ C & 0 \end{pmatrix}$

Assume I have a matrix $A = \begin{pmatrix}0 & C^T \\ C & 0 \end{pmatrix}$ I want to do sigular value decomposition of $A$. Can I take the benefit of the diagonal property if $A$? like only ...
1
vote
1answer
26 views

Block Diagonal Polynomial and Inverse

Suppose that you have a block diagonal matrix $B$, which has blocks $B_1 \dots B_r$ along the diagonal. Show how you can express the polynomial function $f(B)$ and the inverse of $B$ in terms of ...
0
votes
1answer
20 views

Relating the Entries of a Block Operator Matrix to Compositions of Projections

I am having some difficulties proving the following statement: If $V$ and $W$ are vector spaces over a field $F$ such that $V=V_1\oplus V_2$ and $W=W_1\oplus W_2$ for some subspaces $V_i\subseteq V$ ...

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