0
votes
1answer
37 views

most general form of $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution $X = f(A,B,C)$?

What is the most general form of the cubic matrix equation $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution of the form $X = f(A,B,C)$, where $A,B$ and $C$ are positive definite ...
0
votes
0answers
7 views

Solving Toeplitz plus Diagonal System, how should I make use of the structure?

I learned that a Toeplitz system, $Ax = B$ where $A$ is Teoplitz, can be solved in $O(n \log n)$ time using Superfast method. or approximate $A$ similar to Approximation method. I am keep ...
7
votes
5answers
118 views

Positive semi-definite of a matrix composed of semi-definite blocks

Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or $$\begin{pmatrix} A ...
0
votes
0answers
30 views

Computing characteristic polynomial of tridiagonal block matrices

I want to compute the characteristic polynomial of symmetric matrices of the form \begin{bmatrix} A & U & & & 0\\ U & B & V & &\\ & V& C& W &\\ ...
0
votes
2answers
19 views

Problem solving for linear algebra's matrix R

i found "U" but failed to find "R". What I did is add row3 to row2, then divide by pivots, which gave me my "R". i thought that;s how its done, why did i fail it. How is it done?
0
votes
0answers
17 views

Block Circulant matrix

Let \begin{align} A=\left(\begin{array}{cc} B & C \\ -C & B \end{array}\right), \end{align} where $B=\left(\begin{array}{ccc} \alpha^R & \beta^R & \gamma^R \\ \gamma^R &\alpha^R ...
0
votes
1answer
35 views

Find Eigenvalues of Circulant Matrices

$$ A=\begin{pmatrix} \alpha^R&\beta^R &\gamma^R&-\alpha^I&-\beta^I &-\gamma^I\\ \gamma^R&\alpha^R&\beta^R &-\gamma^I&-\alpha^I&-\beta^I \\ \beta^R ...
1
vote
1answer
20 views

The similarity of the block matrices

Let $\mathbb{F}$ be a field,and let $A,B,C$ be matrices over $\mathbb{F}$ of respective sizes $n\times n , k\times k, $and $n\times k$. put $M=\begin{bmatrix} A&0 \\ 0&B ...
0
votes
0answers
19 views

Block Diagonalization related to Direct Sum and Single Eigenvalue?

I'm just a beginner in Linear Algebra, and I've proved myself the following: A matrix $A^{n \times n}$ is block diagonalizable if and only if the base field $F^n$ can be divided into at least two ...
4
votes
3answers
73 views

Prove or disprove that trace of matrix $X$ is zero

I was trying to solve a question from a competitive exam paper. This is a part of that question. Let $I_n$ and $O_n$ be $n\times n$ identity and null matrices respectively.Let $S$ be $2n\times ...
1
vote
1answer
15 views

determinant of special structure block matrix

How do you compute the determinant of the block matrix: $$ M = \begin{bmatrix} A+B & A &A &A &... \\ A & A+B &A &A &... \\ A & A &A+B &A &... ...
1
vote
1answer
69 views

Does there exist $B$ for which $BB^T=I$?

My question is Does there exist a real matrix $B_{n\times m}$ with $m<n$ for which $BB^T=I_n$? Why do I need this? Suppose we are given a real matrix $Q_{m\times n}$ (again, with ...
3
votes
4answers
112 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 ...
0
votes
1answer
27 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 ...
0
votes
2answers
60 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 ...
1
vote
0answers
13 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 ...
3
votes
2answers
47 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
32 views

Nonsingular block matrix

Let us consider a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ and the block partitioning $$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & ...
1
vote
1answer
25 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
1answer
42 views

solving matrix equation by matrix rearranging

I have a matrix equation like following: $$p=KTm $$ where $$T: {4}\times{4} \text{ homogeneous transformation matrix}$$ $$K: {3}\times{4} \text{ matrix}$$ $$p: {3}\times{1} \text{ column ...
1
vote
0answers
40 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 ...
1
vote
0answers
47 views

Condition for a block matrix to be positive semi-definite

Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? ...
0
votes
0answers
109 views

Block Diagonalisation of 4x4 Matrix

I'm attempting to find a 4x4 matrix, P, that will convert my matrices, $A = \begin{bmatrix}1&1&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$ and, ...
0
votes
1answer
39 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
151 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
18 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
0answers
77 views

Block Circulant Matrix Inverse

I have a nearly block-tridiagonal "block circulant" matrix of the form $$G=\left(\begin{array}{ccccccc} B & C & & & & & A\\ A & B & C\\ & A & B & C\\ ...
1
vote
3answers
141 views

eigenvalue of block matrix in terms of original matrix

A is a $4*4$ matrix with eigenvalues $\lambda_A$. Consider a block matrix $B = \left( \begin{array}{ccc} A & I \\ I & A \end{array} \right) $. Then how can we find eigenvalue $\lambda_B$ of ...
0
votes
1answer
25 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
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
votes
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 ...
0
votes
0answers
34 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 ...
0
votes
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 ...
3
votes
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 ...
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
98 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
234 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
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
78 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
vote
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 ...
8
votes
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
238 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
234 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
votes
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
votes
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 ...
0
votes
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 ...