2
votes
3answers
92 views

Skew Symmetric Matrix Properties

We have a theorem says that "ODD-SIZED SKEW-SYMMETRIC MATRICES ARE SINGULAR" . Proof link is given here if needed. Now let us assume we have a $3\times 3$ skew symmetric matrices of the form $ ...
2
votes
2answers
57 views

Determinant of the sum of an identity matrix and a rank-two-symmetric matrix

Suppose $I$ is an $n \times n$ identity matrix, and $S$ is the $n \times n$ symmetric matrix with rank equals two. I was reading something saying that: $$\det(I-S)=(1-\lambda_1)(1-\lambda_2)$$ where ...
2
votes
2answers
28 views

Positive Semi Definite Matrix

If $A$ is a positive semi definite matrix, is $\left[ \begin{matrix}c_1A & c_2A \\ c_3A & c_4A\end{matrix} \right]$ positive semi definite? ($c_1, c_2, c_3, c_4 > 0)$ In general, what ...
3
votes
1answer
21 views

Partition of a Matrix

In Linear Algebra, we have been taught that the partition of a matrix $A$ consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to ...
4
votes
1answer
199 views

Null space basis

Let $V\in\mathbb{R}^{a\times b}$ be a matrix such that it is not a full column rank. Then there will be a nonsingular matrix $H$ such that $$VH=\left[\begin{array}{cc} V_{1} & 0_{a\times ...
1
vote
2answers
54 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
0
votes
1answer
30 views

Transpose of higher dimension matrices

We all know transpose of 2D matrix A Old $A_{ij}$ will be replaced by $A_{ji}$ in the transpose matrix and vice versa Question If A is 3D matrices of $3\times 3 \times 8$ then what is old ...
1
vote
0answers
81 views

Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
0
votes
1answer
31 views

Is this symmetric, block-diagonal matrix positive semi-definite?

I have a matrix of the following form, where $a,b,c>0$ \begin{align*} A = \left[ \begin{array}{cccccc} aM_{12}^2 & aM_{12}M_{13} & 0 & 0 & 0 & 0 & 0 \\ aM_{13}M_{12} ...
3
votes
2answers
227 views

Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
1
vote
1answer
87 views

Finding eigenvalues of a block matrix

I have a block matrix of size $2N \times 2N$ of the form $$B = \begin{bmatrix} A_N & C_N \\ C_N & A_N \end{bmatrix}$$ where $A_N$ and $C_N$ are both $N \times N$ matrices. Specifically, $$A_N ...
1
vote
0answers
44 views

Is there a name for the following type of block matrices?

Is there a name for the following type of block matrices? A matrix $A$ is [insert name here] if it can be decomposed into non-zero non-scalar submatrices such that each sub-matrix $B$, with $B$ ...
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 ...
0
votes
0answers
55 views

show that the determinants are equal

Prove that the determinants are equal $$ \begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \\ \end{vmatrix}= ...
1
vote
0answers
13 views

Expanding a product of matrices with tensor product and transpose

I'm trying to expand the following product of $\pm1$ matrices $H_1, H_2, K_1, K_2$: $(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 ...
7
votes
5answers
122 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
2answers
20 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
19 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 ...
1
vote
1answer
33 views

Block partitioned symmetric orthogonal matrix

Let $A\in\mathcal{M}_n(\mathbb{R}),\ n\in\mathbb{N}$ and $X\in\mathbb{R}^n$. For any $\lambda\in\mathbb{R}$, define $M(\lambda)=\left( \begin{array} {c|c}A & X\\ \hline\phantom{}^tX & \lambda ...
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 ...
1
vote
1answer
40 views

Anyone worked with this particular orthogonal matrix

In my recent studies of quaternions, the following orthogonal matrix has come up. For example, it is related to the matrix representation of quaternion multiplication. Has anyone seen it come up in ...
1
vote
0answers
26 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
77 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
70 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 ...
2
votes
0answers
37 views

Canonical Form of Nilpotent Matrices

Given the matrix $$\hat{S}=\begin{bmatrix} S & *& *&* \\ 0& S &* &* \\ 0& 0& S &* \\ 0&0&0&S\\ \end{bmatrix} $$ where $S$ is an $n \times n$ ...
0
votes
0answers
23 views

Design feedback control law to make the whole matrix Hurwitz

Suppose $(A_1, B_1)$ and $(A_2, B_2)$ are both stabilizable. Then we know that we can find some $K_1$ and $K_2$ to make $A_1+B_1K_1$ and $A_2+B_2K_2$ Hurwitz, respectively. Now, for non-zero constant ...
0
votes
1answer
40 views

Solve Matrix: How many trips can be made?

We have the matrix: $$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$ The matrix ...
3
votes
4answers
121 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
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 ...
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 ...
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
51 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
26 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
44 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
168 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
86 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
107 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
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
164 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
20 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
150 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
34 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
26 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
30 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
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
35 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
35 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
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 ...