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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2
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44 views

linear algebra formulation help

I asked this on mathoverflow as well and apologies for cross-posting. I am trying to compute this so-called bending energy matrix. The bending energy of a thin plate in 3D is given by: $$ BE = ...
0
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2answers
33 views

Conceptual query for finding eigen values during change of basis

Consider an n x n matrix. Suppose i wish to find the eigen values of this matrix. Now, we know that row transformation is equivalent to a change of basis in the vector space. But, we also know that ...
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0answers
33 views

Find out the smallest disk like ($|z-1| < r$ ) in the complex plane containing the eigenvalues of the given matrix

Consider the given matrix $$\left[\begin{matrix}1& -2& 3& -2 \\1& 1& 0& 3\\-1& 1& 1& -1\\0& -3& 1& 1&\end{matrix}\right]$$ Find out the smallest ...
3
votes
1answer
59 views

Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
2
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1answer
45 views

Determinant of $M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$

Calculate the determinant of the following matrix: $M \in M_{2n}(\mathbb{C})$ such that $$M = \begin{pmatrix} I_n&iI_n \\iI_n&I_n \end{pmatrix}$$ I find that that $\det M = 2^n$ is that ...
0
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1answer
33 views

Computing the Permanent of a Matrix in Terms of Permanents of its Sub-matrices

Say I have a matrix \begin{align*} G &= \left( { \begin{array}{cc} G^{\prime} & \vec{u} \\ \vec{v}^T & d \end{array} } \right) \end{align*} where $\vec{v}^T$ and $\vec{u}$ are row and ...
2
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1answer
155 views

Determinant of block matrix with certain properties

So I have the following 2N $\times$ 2N block matrix $H=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ where each block in an N$\times$N matrix. Each block have the following ...
0
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2answers
55 views

2nd order Matrix differential equation

$\ddot{X}+W\dot{X}=X$, $W$ is n-dimensional skew symmetric matrix. $X$ is a column vector and $I$ is identity matrix of appropriate dimension. \begin{equation} \left(\begin{array} XX \\ \dot{X} ...
0
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0answers
56 views

Finding basis of null space for block matrix

I am trying to determine the full-rank basis $Z \in \mathbb R^{(n+m) \times }?$ for the null space of the matrix $$M= \left( \begin{array}{yy} -A & B \\ x & y \end{array} \right)$$ with ...
3
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2answers
69 views

Basis of Kernel of a matrix

Given $\theta>0$. Let $H$ be $5 \times 6$ matrix $$\left[\begin{matrix} 1 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 ...
1
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1answer
28 views

Equivalence involving diagonalization of block matrix

Given $T^\top L T = diag(A, B, C)$ being a block diagonalization with $L$ symmetric $A,B$ positive definite (p.d.) $T = \left( \begin{array}{ccc}I & 0 & T_2 \\ T_1 & I & T_3 \\ 0 ...
2
votes
1answer
83 views

Congruence of a matrix

Let $X=\begin{bmatrix}a&b&0&0\\ c&d&0&0\\0&0&-a&-b\\0&0&-c&-d\end{bmatrix}$ where $a,b,c,d\in \mathbb{Z}$. For a such given $X$, is there a $4\times ...
2
votes
2answers
97 views

How to express a matrix as a product of two symmetric matrices?

Let $A$ be a matrix and $J$ its Jordan canonical form. How can one express $A$ as a product of two symmetric matrices? I expressed $J$ as a product of two symmetric matrices: block by block in the ...
8
votes
1answer
209 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 $X$: ...
1
vote
1answer
71 views

Given the inverse of a block matrix…

Given the inverse of a block matrix $X^{-1}$, where $$ X=\left(\begin{array}{cc} A & B \end{array}\right). $$ A is $m\times n$ and B is $m\times(n-m)$. Can I obtain the pseudo-inverse of A ...
1
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0answers
233 views

Proof that if a matrix is invertible, its rank is maximum

I have to prove that if a square matrix $A \in \mathfrak{M}_n (\mathbb{K})$ is invertible, then $rg(A) = n$. The thing is I cannot use vector spaces, subspaces, etc... to prove this, only matrix ...
1
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1answer
46 views

Proving that $M_p(M_q (K)) \cong M_{pq} (K)$.

My textbook finishes the proof of one of the theorems with the following fact: $$ M_p(M_q (K)) \cong M_{pq} (K), $$ where $K$ is a field, and it says that it is true by "block multiplication ...
2
votes
3answers
98 views

Name of this matrix product?

Suppose $A$ and $B$ have columns as follows, $$A = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix},$$ $$B = \begin{bmatrix} b_1 & b_2 & \dots & b_n \end{bmatrix}.$$ Is ...
1
vote
1answer
77 views

Rank of a partitioned matrix

Let $p$, $q$ and $n$ be positive integers such that $p+q \geq n$ and $p \leq n$. Denote by $\mathbf{I}$ the identity matrix of size $p$, by $\mathbf{0}$ the zero matrix of size $(p\times (n-p))$ and ...
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 ...
1
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0answers
59 views

Are all columns or rows in a block toeplitz convolution matrix linearly independent?

My question specifically relates to the case where the vector that the matrix blocks were formed from have lower orders than the dimensions of the sub matrices. Consider a vector (filter) $a[k]$ of ...
1
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0answers
28 views

DCT and split-radix algorithms

I am studying a paper which describes split-radix algorithms by making matrix factorizations, so that e.g. DCT 8x8 can be computed via 4 DCT4x4. I must apologize, but the question is related to an ...
8
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1answer
337 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 ...
1
vote
1answer
237 views

Jordan Canonical form 2x2 matrix

Compute the Jordan Canonical form of A = $\begin{bmatrix}i & 1\\1 & -1\end{bmatrix}$. My (feeble) attempt: After I compute the characteristic polynomial, which gives me $x^2=0$, the ...
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vote
2answers
233 views

Determinant of block matrices with non square matrices

Let $A$ be $m \times n$ matrix, and B be $n \times m$ matrix, then Show that $\det\begin{bmatrix}I_{n} & B\\ A & I_{m} \end{bmatrix}=\det\begin{bmatrix}I_{m} & A\\ B & I_{n} ...
3
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2answers
73 views

Completely baffled by this question involving putting matrices in matrices

This is homework, so only hints please. Let $A\in M_{m\times m}(\mathbb{R})$ , $B\in M_{n\times n}(\mathbb{R})$ . Suppose there exist orthogonal matrices $P$ and $Q$ such that ...
0
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0answers
110 views

Row & Column Removal and Rank Reduction

I have a problem involving a n x n square, real matrix $K$ which is initially full rank and is not positive definite. In each iteration of my program, I have to remove a row and the corresponding ...
3
votes
2answers
253 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 ...
0
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1answer
60 views

How many involutary matrices of order 2013 and trace = 2013 exist having one of it's non diagonal elements as -1?

How many involutary matrices of order 2013 and trace = 2013 exist having a57 = -1? My Attempt: An involutary matrix is the one which satisfies : $A^2$ = I . Now, if k is an eigen value of A, then it ...
13
votes
4answers
4k 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 ...
4
votes
2answers
108 views

Determinants of Block Matricies

I read on wikipedia that $Det \begin{pmatrix} A & B\\B& A\end{pmatrix}$ is equal to $ Det(A+B)Det(A-B) $ if $A$ and $B$ commute. Does this hold true even if $ A $ and $ B$ are not ...
0
votes
1answer
124 views

limit of an expression involving a matrix as a parameter approaches infinity

X is a symmetric positive definite n by n matrix. This also means it is invertible of course. Consider the matrix Y, which is X, but with an extra row and column at the end: the first n rows of column ...
0
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1answer
84 views

Inverse of a certain block matrix

So I'm trying to compute the inverse of a block matrix that's a subset of a larger consideration I was attempting (this particular matrix comes from the normal and orthogonal equations for least ...
0
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0answers
51 views

Can we perform this operation on block matrices?

We have a block matrix: $$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$ Here $A$, $B$ and $C$ are all permutation ...
3
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1answer
152 views

Applications/Motivations of matrix decomposition techniques

Matrix decomposition is one area of matrices that has always intrigued me. Every time I open a matrix book, I can interestingly follow it till Eigen values and Eigen vectors because they are well ...
6
votes
3answers
159 views

Eigenvalues of block matricies

If the eigenvalues of a matrix $A$ are $\lambda_1,\lambda_2,\dots,\lambda_n$, what are the eigenvalues of the matrix? $\begin{bmatrix}0 &A\\A&0\end{bmatrix}$ From some numerical examples I ...
4
votes
1answer
393 views

Determinant of a $2 \times 2$ block matrix

$\textbf{Problem}$: Let a $2n \times 2n$ matrix be given in the form $M=\left[ {\begin{array}{cc} A & B \\ C & D \\ \end{array} } \right]$, where ...
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1answer
93 views

If $A^m=I$ then A is Diagonalizable

Let $A$ be an $n\times n$ complex matrix. If $A^m=I_n$ for some positive integer $n$. How to show that $A$ is diagonalizable?
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0answers
78 views

Finding correct equivalence matrix for all group representations

This question is linked to a previous one that I asked: Attempting to find a specific similarity (equivalence) matrix I have a group of 24 elements, with two generators. I need to find an ...
0
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0answers
91 views

Pseudo-inverse of something similar to Kronecker product

I have a matrix $C$ that looks like a Kronecker product of two matrices $A$ and $B$, but not quite. $A$ is a $p\times q$-matrix and $B$ is a $r\times q$-matrix, and $C$ looks like this: $$C = ...
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0answers
156 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) ...
7
votes
3answers
103 views

A rational “fifth root” of the scalar matrix $2I$

I am working on the following problem from a past exam. Find a necessary and sufficient condition for there to exist a square matrix $A$ of order $n$ whose entries are all rational, such that $A^5 ...
2
votes
1answer
285 views

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 = ...
3
votes
3answers
411 views

Prove that the rank of the block matrix is rank $A$ + rank $B$

Assume $A,B$ are $n \times n$ matrices. Prove that the rank of the block matrix that is defined as the following $$ \pmatrix{A& AB\\B&B+B^2} $$ is equal to $ \mbox{rank} A +\mbox{rank} B$. ...
0
votes
2answers
87 views

Jordan block idempotent then it is a number?

in the book matrix theory but Fuzhen Zhang it is said on page 94, that if $J$ is a Jordan block and if $J^2=J$ then, J must be of size 1, that is, J is a number. I do not get it, take the matrix ...
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0answers
45 views

Is there any simple way to write down permutations set-theoretically?

Let $\{J_{n_1}(\lambda_1),...,J_{n_m}(\lambda_m)\}$ and $\{J_{l_1}(\mu_1),...,J_{l_k}(\mu_k)\}$ be finite sequences of Jordan blocks with entries in a field $F$. Let $A\triangleq ...
0
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1answer
54 views

Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
0
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1answer
127 views

Matrix of T, a linear transformation when Im T = Ker T

Let $V$ be a finite dimension vector space, $T:\ V \to V$ a linear transformation, and assume that $\operatorname{\rm Ker} T = \operatorname{\rm Im} T$. Prove that there is a basis $B$ of $V$, so that ...
0
votes
2answers
185 views

Minpoly and Charpoly of block diagonal matrix

I am currently struggling with an exercise where I have to treat a Block diagonal matrix (so it is a square matrix, where square block matrices are down the diagonal). Now I was wondering whether we ...
0
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1answer
140 views

Block matrix and invariant subspaces

I was wondering what the exact relationship between invariant subspaces and a block matrix is? Is it correct to say: Each diagonal block matrix "creates a vector space decomposition" and vice versa? ...