For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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3
votes
2answers
68 views

Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Let $X$ be the vector space of all complex $n \times n$ matrices and define $T \colon X \to X$ by $Tx \colon= bx$, where $b \in X$ is fixed and $bx$ denotes the usual product of matrices. I know that ...
0
votes
1answer
35 views

Linear maps that are matrices

If I have the linear map $A:\Bbb{R}^3\rightarrow \Bbb{R}^3$ where $A$ is a matrix. Is the matrix $A$ (along with the vectors it operates on) in a basis or not? I think it is not, since the vectors it ...
0
votes
1answer
18 views

iterated matrix multiplication in scilab\matlab [closed]

Please help me with my code. I can't write the iteration in codes ...
1
vote
0answers
33 views

Given similar matrices $A$ and $B$, how to find $M$ such that $B=M^{-1}AM$?

I am trying to teach myself linear algebra using Strang's Introduction to Linear Algebra. I would like to know what the most (or more) efficient way to solve this problem is by hand. The question: ...
4
votes
5answers
124 views

Find the determinant of a matrix definition [duplicate]

Let $A$ be a matrix that is defined like this: $$A_{ij}=\begin{cases} \alpha, & \text{if i=j} \\ \beta , & \text{if i $\ne$ j} \end{cases} $$ So I realized this matrix looks somehow like ...
2
votes
2answers
157 views

Easy way to calculate the determinant of a big matrix?

Given this matrix: \begin{matrix} 2 & 3 & 0 & 9 & 0 & 1 & 0 & 1 & 1 & 2 & 1 \\ 1 & 1 & 0 & 3 & 0 & 0 & 0 & 9 & 2 & 3 & ...
1
vote
3answers
42 views

Find the dimension of the space of $4\times 4$ real matrices with zero trace

I'm wondering if someone can help me to understand this problem. If $S$ is the subspace of $M_{4,4}(\mathbb{R})$ consisting of all matrices with trace $0$, what is $\dim(S)$? I've created a matrix ...
-1
votes
1answer
37 views

eigenvalue problem [closed]

Prove the following statement: $\max_i{\lambda_i(A^TA)}=\max_i{\left|\lambda_i(A^T)\right|^2}$ where matrix A is a N-by-N circulant matrix and $\lambda_i(X)$ denotes the $i$-th eigenvalue of matrix ...
1
vote
1answer
42 views

What is the linear space of Eigenvectors associated with a certain Eigenvalue?

The following matrix $A$ has $\lambda=2$ and $\lambda=8$ as its eigenvalues $$ A = \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{bmatrix}$$ let $P$ be the ...
-1
votes
2answers
60 views

Eigenvalues of a nilpotent matrix can only be $0$ [duplicate]

Prove that the eigenvalues for a square Nilpotent matrix A can only be $0$. Definition of nilpotent A $^n$=$0$ n is a positive whole integer
0
votes
2answers
65 views

Under what assumptions is it correct to say “a matrix is diagonalizable if and only if its eigenvalues are real”?

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real. Which statement is most correct: The proposition is true only if the eigenvalues are all greater than zero. The ...
0
votes
1answer
24 views

Algorithm for the Hill cipher (finding the inverse of the determinant of a $2 \times 2$ matrix modulo $26$)

I have a good understanding of how to do the Hill cipher on paper but putting it into program form is somewhat of a problem. Finding the the determinant is the thing I'm having problem with. On ...
0
votes
1answer
7 views

Converting nth order ODE with RHS into system of 1st order ODEs

I looked at these two questions, but they weren't directly relevant to my specific question: How to reduce higher order linear ODE to a system of first order ODE? Express differential equations as ...
0
votes
0answers
34 views

Matrix Proof relating Unitary and Hermitian Matrices

I am currently reading through my lecture notes and the following situation is given. Suppose that a ($n$ by $n$) unitary matrix $U$ can be written as $U=M+iN$ where $M$ and $N$ are Hermitian ...
0
votes
1answer
16 views

M22 → R Matrix Transformation Kernel

For a transformation such as this, how does one determine the form of the kernel? Is it simply making the right side equal to zero, solving for each individual variable, and then creating a matrix ...
0
votes
0answers
55 views

Finding complex eigenvectors of $n \times n$ matrix, $n\geq 3$

An example: $$ \begin{pmatrix} 1 & 2 & 0 \\ 2 & -3 & 4 \\ 4 & -8 & 7 \\ \end{pmatrix} $$ Has eigenvalues $3$, $1+2i$, $1-2i$. How ...
0
votes
1answer
28 views

If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable [duplicate]

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I ...
1
vote
1answer
42 views

Is my proof correct that function is a bijection iff matrix is invertible?

For given $B\in \mathbb{C}^{n\times n}$ let's define a function $$f: \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$$ so that $$f(A) = B^HAB$$ I have to prove that f is a bijection iff B is ...
4
votes
1answer
61 views

is this kind of symmetric matrix invertible?

Give a matrix $A=\begin{bmatrix}M&B\\ B^T&0\end{bmatrix}$, where $M\in\mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}, (m<n)$. If we know that $rank(B)=m$ and for any $v\neq 0$ and ...
0
votes
3answers
51 views

Can a matrix transformation ever make a linearly dependent matrix linearly independent?

I'm curious. Can ANY matrix transformation make some matrix with its columns linearly independent, or with an empty kernel, linearly independent? For example, if A is a linearly dependent matrix, and ...
0
votes
2answers
29 views

Inverse of Matrix?

What is the inverse of the following matrix ? Give a general formula for calculating the such matrix of dimension n-by-n. Grateful.
0
votes
1answer
20 views

Can anyone help me with “rotation matrix” and “Image of matrix”?

If A is a 3 by 3 matrix which gives a rotation about some line through the origin in R^3 , then columns of A form a basis of R^3 For any matrix A, the image of A^7 is contained in the image of A ...
4
votes
0answers
98 views

How does pointwise multiplication of two matrices affect their eigenvectors?

More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$. What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ ...
3
votes
2answers
34 views

Can we specify all row equivalent matrices of a given matrix?

Say we have a RREF matrix like $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}$$ From this matrix, is there some way of specifying ...
0
votes
2answers
28 views

Mutually orthogonal set of vectors

Show that the standard basis: $$..$$ $\mathscr{B} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \\ 0 ...
0
votes
1answer
46 views

Derivative of scalar function with respect to vector

Suppose I have three constant symmetric matrix $\mathbf{M}_{n\times n}$, $\mathbf{C}_{n\times n}$ and $\mathbf{D}_{n\times n}$ and two variable vectors $\mathbf{q}_{n\times 1}$ and ...
0
votes
2answers
54 views

Prove that $T,S$ are simultaneously diagonalizable iff $TS=ST$. [duplicate]

Definition: We say that $S,T$ are simultaneously diagonalizable if there's a basis, $B$ which composed by eigen-vectores of both $T$ and $S$ Show that $S,T$ are simultaneously diagonalizable iff ...
0
votes
0answers
48 views

Can anyone help me with these true and false questions about linear algebra?

1 A system of real linear equations can have exactly two solutions. 2 If U and W are subspace of V, V=U+W (Finite dimension), then dimV is less than or equal to dimU+dimW 3.Every inner product ...
2
votes
1answer
38 views

How to determine if a 3x3 matrix is diagonalizable?

The matrix is given as: $A=\begin{bmatrix} 0 & 1 & 1 \\0 & 0 & 4 \\ 0 & 0 & 3 \end{bmatrix}$ So the matrix has eigenvalues of $0$ ,$0$,and $3$. The matrix has a free ...
2
votes
0answers
27 views

Polynomial Matrix Eigenvalue problem. Conditions under which there are only two complex eigenvalues?

I'm solving the polynomial matrix eigenvalue problem $(A\lambda^2+B\lambda+C)v=0 $. This is what I want the eigenvalues to look like. Are there any conditions on the matrices A,B,C such that there are ...
1
vote
0answers
19 views

What is the algebra behind the Cuthill Mckee Bandwidth Reduction

From my understanding a Sparse matrix is converted to banded matrix and then the cuthill mckee is used to reduce the bandwidth. I have spent about three days browsing the web to find an example where ...
1
vote
3answers
41 views

Solving variables in a matrix for a specific determinant

The matrix is as follows: $$ A = \begin{pmatrix} 0 & x & 1 & 2 \\ x & 1 & 1 & x \\ 1 & x & x & 1 \\ 1 & x & 1 & x \end{pmatrix} $$ What I want to do ...
4
votes
3answers
44 views

Are $\vec{v}$ and $\vec{w}$ linearly independent?

Am I correct in saying that $\vec{v}=\begin{pmatrix}1\\2\\0\end{pmatrix}$ and $\vec{w}=\begin{pmatrix}2\\4a\\a-1\end{pmatrix}$ are linearly independent $\forall a\in\mathbb R$ /{1}? It seems to me ...
0
votes
1answer
12 views

Relationship between $U_{-\phi}A^nU_\phi$ and $(U_{-\phi}AU_\phi)^n$

If I have two matrices: $A=\begin{pmatrix} 3 &-1 \\ -1 & 1 \end{pmatrix}$ $U_\phi=\begin{pmatrix} \cos\phi &-\sin\phi \\ \sin\phi & \cos\phi \end{pmatrix}$ What is the ...
2
votes
3answers
36 views

MATLAB: matrix multiplication with exponentiation

Normal matrix multiplication computes $C=A*B$, such that $C_{ij}=\sum_k{A_{ik}*B_{kj}}$ I want to compute D, such that $D_{ij}=\sum_k{e^{A_{ik}*B_{kj}}}$ Basically I want to exponentiate each ...
1
vote
2answers
22 views

Co-ordinates of a vector in relation to the basis

Find the co-ordinates of the vector $u = (2,-1,4)$ of $\mathbb R^3$ in relation the basis $S = \{(1,1,1),(1,1,0),(1,0,0)\}$. Please could someone help/explain this to me, I'm doing revision for my ...
0
votes
1answer
29 views

Prove that A + B is invertible iff $I_n$ + $A^{-1}$B is invertible (matrices)

We are given that a matrix $A$ in $R^{n\times n}$ is invertible. We must show that $A + B$ (also in $R^{n\times n}$) is invertible if and only if $I_n$ + $A^{-1}$$B$ is invertible. I cannot figure ...
0
votes
1answer
15 views

Name for reversed main diagonal of a non-square matrix

Consider the matrix $ \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{pmatrix} $ Then the main diagonal elements are $1$ and $4$. I want something like the reversed main diagonal elements, ...
0
votes
2answers
33 views

Finding the rank of a matrix $A_{m \times n}$

I have a H.W question which I try to tackle but unsuccessfully. Given $m$ elements $a_1,a_2....a_m$ (not all elements equal $0$) and $n$ elements $b_1,b_2....,b_n$ (not all elements equal $0$) and ...
0
votes
1answer
15 views

Non-singularity of a block anti-diagonal matrix

Let us suppose we have an $(2n)\times(2n)$ symmetric matrix with the block structure: $$\begin{pmatrix} 0 & B \\ B^T & 0\\ \end{pmatrix}$$ where every sub-block is of ...
3
votes
4answers
117 views

Diagonalization and find matrix that corresponds to the given condition

Diagonalize the matrix $$ A= \begin{pmatrix} 1 & 2\\ 0 & 3 \end{pmatrix} $$ and find $B^3=A$. I derived $A \sim \text{diag}(1,3)$ but I have problem finding any $B$. I tried to solve it by ...
1
vote
1answer
51 views

Rank(AB) and Rank(BA)?

$A$ is a matrix of size $n\times r$. $B$ is a matrix of size $r\times n$. The rank of $A$ and $B$ are both equal to $r$. Assuming $r < n$. My question is: $\def\rank{\operatorname{rank}}\rank(AB) ...
1
vote
2answers
25 views

An exponential map of a matrix computation

Suppose the $n\times n$ matrices $A$ and $M$ satisfy $AM+MA^{T}=0.$ Show by direct computation that the product $\mathrm{exp}(At)~M~\mathrm{exp}(A^{T}t)=M$ for all $t\in \mathbb{R}.$ Note: By ...
1
vote
2answers
20 views

Finding Bases from polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? $$p_1 ( x ...
0
votes
1answer
50 views

The invertibility of matrix $(I - XX')$?

$I$ is an identity matrix of size $n \times n$. $X$ is a matrix of size $n \times k$(Assuming $k \leq n$). As we know, $(I+XX')$ is invertible. Because $(I+XX') = (I(blank)X)*(I(blank)X)'$, where $(I ...
0
votes
1answer
18 views

Matrix exponential of a non-nilpotent matrix?

An old exam problem asks to compute $e^{At}$ for $$ A=\begin{pmatrix} 2 & 6\\ -3 & -4\end{pmatrix} $$ However, I compute a few powers of $A$, and it doesn't seem like $A$ is nilpotent, so ...
1
vote
1answer
37 views

Find the diagonal matrix A that satisfies the equation

Find the diagonal matrix $A$ that that satisfies the equation: $$A^{-3}=\pmatrix{-27&0&0\cr0&8&0\cr0&0&-1}$$ Attempted solution: my intuition tells me that the inverse of ...
1
vote
1answer
29 views

How many square root matrices?

I would like to see a proof of the following statement: A positive-semidefinite matrix has precisely one positive-semidefinite square root, which can be called its principal square root. I ...
2
votes
2answers
31 views

Restrictive definition of diagonalizable matrix

There is a theorem that says that every matrix of rank $r$ can be transformed by means of a finite number of elementary row and column operations into the matrix $$D=\begin{pmatrix} I_r & O_1 \\ ...
0
votes
1answer
30 views

Can a non-zero element of $M_n(\Bbb{C})$ be nilpotent?

Let $A\in M_2(\Bbb{C})$. If $A$ is nilpotent, then $A^2=0$. If $A\in M(\Bbb{C})$, then $AA^*=I$, where $A^*$ is the conjugate transpose of $A$. How can a non-zero element of $M_2(\Bbb{C})$ ever ...