1
vote
0answers
11 views

generalisation of Knonecker matrix product

In the Kronecker matrix product $C = A\otimes B$ we have that $C(i,j)=A(i,j)*B$ where the elements $A(i,j)$ are just numeric scalar values. What if the $A(i,j)$ are matrix operators which act on ...
2
votes
2answers
38 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
1
vote
1answer
17 views

Solving Ax = b where A is composed of diagonal blocks

I would like to solve the equation $Ax=b$ where $x\in\mathbb{R}^n$ and $A$ is of the form: $$A= \begin{bmatrix} D_1 & D_2 &D_3 \\ D_2 & D_4 & D_5 \\ D_3 & D_5 & D_6 ...
1
vote
0answers
52 views

Use Frobenius inequality to prove $\mathrm{rank}(A)+\mathrm{rank}(B)\geq\mathrm{rank}(AB)+ \mathrm{rank}(A+B)$

If $A, B$ are $n\times n$ matrices, $AB=BA$, then $$\mathrm{rank}(A)+\mathrm{rank}(B)\geq\mathrm{rank}(AB)+ \mathrm{rank}(A+B)$$ Can one try Frobenius inequality to prove this? we must find three ...
1
vote
0answers
34 views

Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where, $J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$ $A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$ $B \in ...
3
votes
1answer
148 views

Eigenvalues of a submatrix

A self-adjoint matrix $A$ in a complex linear space has real eigenvalues bounded between $0$ and $1$. If $A$ is projected onto an arbitrary two-dimensional subspace, what would be the bounds on the ...
3
votes
1answer
64 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 ...
2
votes
1answer
58 views

The Matrix Inversion Lemma: the General Case

I find it is hard to understand the application senario of the Matrix Inversion Lemma in non-special cases. Suppose I already computed $A^{-1}$ and want to find $\left(A+UCV \right)^{-1}$. The Matrix ...
0
votes
1answer
31 views

Extending a set of vectors to a basis by picking from a given basis

I have a linear independent set ${\cal K}=\{v_1,\dots,v_{k-d}\}\subset\mathbb{R}^k$. I'd like to find $\cal W=\{w_1,\dots,w_d\}$ such that $\cal K\cup W$ is a basis for $\mathbb{R}^k$. To do this, ...
6
votes
0answers
122 views

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 ...
2
votes
1answer
16 views

Factorizing a block column matrix with element-wise factors

Is it possible to factor this matrix $$\begin{bmatrix} x_{11} a_{11} & x_{11} a_{12} & x_{12} a_{11} & x_{12} a_{12} & \\ x_{21} a_{21} & x_{21} a_{22} & x_{22} a_{21} ...
0
votes
1answer
53 views

Zeros diagonal element of a semidefinite matrix leads to zeros row/column. Why?

I have a similar problem as in this question. In short words: Assume a square, positive semidefinite matrix $A\in\mathbb R^{n\times n}$. Show that if a diagonal element of $A$ is zeros then the ...
0
votes
0answers
33 views

Can we reduce this matrix to the identity, which contains binomial elements?

We are given a function: $$f(a,b,m) = \binom{n}{b}\binom{n-b}{a}\binom{n-a-b}{m-a}$$ We can suppose we have the following $(n/2)^2 \times (n+1)$ matrix (form), that we wish to find the value for the ...
0
votes
1answer
24 views

Calculating Partitioned Matrices from subs

Say you have a matrix $A$ which is of size $P\times P$ and a number $Q < P$ can be used to take a partition of said matrix, where: $A_1$ is the upper-left sub matrix, with dimension $Q\times Q$, ...
1
vote
0answers
25 views

Can we find a reduced row-echelon form for these matrices?

Starting with a Vandermonde matrix: $$V = \begin{bmatrix} 1^1 & 1^2 & 1^3 & \dots & 1^n \\ 2^1 & 2^2 & 2^3 & \dots & 2^n \\ 3^1 & ...
1
vote
0answers
32 views

Understanding of a formula with matrix summation

We are a quite a few students at in the class struggling to compute this would anyone be able to help? Also note --> ' means transpose. Sorry for the misunderstanding, when I said that it is not ...
0
votes
1answer
22 views

Proving the equality: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$ where $P=(A-BD^{-1}C)^{-1}$

I am trying to prove the following equality that I need to use as an intermediate step to solve one of my problems. The equality is the following: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$, where ...
2
votes
0answers
40 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
votes
2answers
23 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 ...
0
votes
0answers
29 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 ...
2
votes
1answer
36 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
votes
1answer
22 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
votes
1answer
103 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
votes
0answers
48 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
votes
2answers
55 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
vote
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
80 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 ...
1
vote
0answers
171 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
vote
1answer
44 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
93 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
44 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
98 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
vote
0answers
20 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
votes
1answer
326 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
169 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 ...
1
vote
2answers
167 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} ...
0
votes
0answers
103 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 ...
0
votes
1answer
57 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 ...
12
votes
4answers
2k 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
91 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
113 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
votes
1answer
70 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
votes
0answers
50 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
votes
1answer
135 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
145 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
354 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 ...
1
vote
1answer
86 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?
1
vote
0answers
141 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
100 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 ...