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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13
votes
4answers
5k views

Show that the determinant of $A$ is equal to the product of its eigenvalues.

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
5
votes
2answers
946 views

Are there any decompositions of a symmetric matrix that allow for the inversion of any submatrix?

I am given a $J \times J$ symmetric matrix $\Sigma$. For a subset of $\{1, ..., J\}$, $v$, let $\Sigma_{v}$ denote the associated square submatrix. I am in need of an efficient scheme for inverting ...
4
votes
3answers
295 views

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 ...
3
votes
1answer
85 views

eigenvalues of sum of matrices: $A$+block matrix are strictly less than eigenvalues of $A$+identity

Let $A$ whose sum for rows is 0, can I prove that the $ \lambda_i \left ( \begin{bmatrix} I & 0\\ 0 & 0 \end{bmatrix} +A\right ) $ are strictly less than $\lambda_i \left ( \begin{bmatrix} I ...
0
votes
2answers
67 views

Find the axis of rotation from the rotation matrix.

This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ...
8
votes
1answer
338 views

Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component ...
8
votes
1answer
214 views

Given the inverse of a block matrix - Complete problem

Given $X$ a block matrix $$\pmatrix{A&B}$$ where $A$ is $m \times n$ and $B$ is $m \times (n−m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: ...
5
votes
5answers
379 views

How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that ...
3
votes
4answers
120 views

Find the axis of rotation of a rotation matrix by $INSPECTION$ (NOT by solving $Kv=v$)

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$ by INSPECTION. This is from my other thread ...
2
votes
1answer
83 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
2
votes
1answer
320 views

Block diagonalizing a real matrix

I need to prove that for every real linear operator $T:\mathbb{R}^n\longrightarrow \mathbb{R}^n$, there exists an orthonormal basis of $\mathbb{R}^n$ such that the corresponding matrix is block ...
8
votes
1answer
2k views

Inverse of a block matrix

I have a special case where $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: $X$ is non-singular $A$ is singular $B$ is full column rank $C$ is full row rank How do you ...
3
votes
1answer
100 views

Proving matrix $A$ is similar to matrix $B$

Question: If the matrix $\begin{pmatrix} A & 0 \\ 0& A \end{pmatrix}$ is similar to $\begin{pmatrix} B & 0 \\ 0 & B \end{pmatrix}$ show that: the matrix $A$ is similar the matrix ...
3
votes
1answer
246 views

Checking if one “special” kind of block matrix is Hurwitz

I have the next block matrix $$ J = \begin{bmatrix}A & B \\ K &0\end{bmatrix} $$ all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...
2
votes
0answers
144 views

Properties about Matrices that can be proved by only using Block Multiplication of Matrices

I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove ...
1
vote
0answers
163 views

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked - I searched but couldn't find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) ...
1
vote
2answers
210 views

deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...
0
votes
1answer
32 views

Find the Axis of rotation of rotation matrix $K$ after solving $(K-I)v=0$

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$. This is from my previous thread click here ...