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).

735 questions
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Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks

Let $A$ be a block upper triangular matrix: $$A = \begin{pmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{pmatrix}$$ where $A_{1,1} ∈ C^{p \times p}$, $A_{2,2} ∈ C^{(n-p) \times (n-p)}$. Show that the eigenvalues of $A$ are the combined eigenvalues of…
tsiki
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Determinant of a block lower triangular matrix

I'm trying to prove the following: Let $A$ be a $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then, $$\det\left(\begin{array}{cc} A&0\\ C&D \end{array}\right) = \det(A)\det(D).$$ Can I just say that $AD - 0C =…
Buddy Holly
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Determinant of a block upper triangular matrix

How prove the following equality for a block matrix? $$\det\left[\begin{array}[cc]\\A&C\\ 0&B\end{array}\right]=\det(A)\det(B)$$ I tried to use a proof by induction but I'm stuck. Is there a simpler method? Thanks for help.
user66407
23
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4 answers

Block diagonal matrix diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diagonalizable, if only if $A$ and $B$ are diagonalizable. If $A\in GL(\mathbb{C}^n)$ and $B\in GL(\mathbb{C}^m)$ are diagonalizable, then…
FASCH
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How to denote matrix concatenation?

Trivial question: Is there any standard notation for the concatenation of two or more matrices? Example: $$A = \left(\begin{array}[c c] - a_1 & a_2\\ a_3 & a_4 \end{array}\right),$$ $$B = \left(\begin{array}[c c] - b_1 & b_2\\ b_3 & b_4 …
Anibal Troilo
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Inverse of a block matrix with singular diagonal blocks

I have a special case where $$X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$$ and: $X$ is non-singular $A \in \Bbb R^{n \times n}$ is singular $B \in \Bbb R^{n \times m}$ is full column rank $C\in \Bbb R^{m \times n}$ is full row…
Shyam
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17
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Principal submatrices of a positive definite matrix

Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k
16
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Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have some group $G$ of matrices, where $G$ is one…
16
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6 answers

What is the codimension of matrices of rank $r$ as a manifold?

I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a submanifold of $\mathbb{R}^{m\times n}$ with codimension…
16
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2 answers

The determinant of block triangular matrix as product of determinants of diagonal blocks

I am given the following partitioned, upper-triangular matrix: $$ \begin{bmatrix} A_1 &* &* &* &* &* \\ 0& A_2 &* &* &* &* \\ .& 0& A_3 &* &* &* \\ .& 0& 0 &... &* &. \\ .& 0& 0& 0& ... &. \\ 0& .& ...& 0&0 &…
Dor Shalom
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16
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Bounding the minimum singular value of a block triangular matrix

Question: What is the sharpest known lower bound for the minimum singular value of the block triangular matrix $$M:=\begin{bmatrix} A & B \\ 0 & D \end{bmatrix}$$ in terms of the properties of its constituent matrices? Motivation: Block triangular…
Nick Alger
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14
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2 answers

How to find the eigenvalues of a block-diagonal matrix?

The matrix $A$ below is a block diagonal matrix where each block $A_i$ is a $4 \times 4$ matrix with known eigenvalues. $$A= \begin{pmatrix}A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_n …
cgo
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13
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4 answers

eigenvalues of certain block matrices

This question inquired about the determinant of this matrix: $$ \begin{bmatrix} -\lambda &1 &0 &1 &0 &1 \\ 1& -\lambda &1 &0 &1 &0 \\ 0& 1& -\lambda &1 &0 &1 \\ 1& 0& 1& -\lambda &1 &0 \\ 0& 1& 0& 1& -\lambda &1 …
13
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3 answers

Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?

I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these. It seems to be a well known result that when you take the eigenvectors of $A$ as a basis and diagonalize $B$ with it then you get a block…
13
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4 answers

General expression for determinant of a block-diagonal matrix

Consider having a matrix whose structure is the following: $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & 0 & 0 & 0 & 0 & 0 & 0\\ a_{2,1} & a_{2,2} & a_{2,3} & 0 & 0 & 0 & 0 & 0 & 0\\ a_{3,1} & a_{3,2} & a_{3,3} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0…
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