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), ...

learn more… | top users | synonyms (1)

1
vote
1answer
38 views

Matrices - Find matrix E

Suppose $A = \begin{bmatrix}1&2&-1\\1&1&1\\1&-1&0\end{bmatrix}$ and $D = \begin{bmatrix}1&2&-1\\-3&-1&3\\2&1&-1\end{bmatrix}$. I need to find the matrix ...
1
vote
1answer
99 views

Alternating multilinear map and products

I was reviewing some school notes from many semesters ago and I came across a point which I wish to prove but can't. Let $F$ be a field (real or complex for example), and we say $\delta : ...
1
vote
3answers
97 views

Matrices - Understanding row echelon form and reduced echelon form

I have the following two matrices: 1) $$\begin{bmatrix}1&0&0\\ 0&1&1\\ 0&0&0\\0&0&0 \end{bmatrix}$$ I believe this matrix is in the form of reduced row echelon form ...
1
vote
2answers
69 views

Given a matrix of basis transformation what is the algorithm to find $ker(T)$ and $im(T)$?

I'm given the following transformation matrix of the linear map $T:\mathbb R^4\to\mathbb R^3$: Find $\mathrm{ker}(T)$ and $\mathrm{im}(T)$ So I should probably get this matrix to $\mathrm{rref}$: ...
1
vote
1answer
89 views

Increase the diagonal entries of a positive definite matrix

Assume that we have a positive definite matrix $C$, and a positive definite diagonal matrix $\Lambda$. Are all the diagonal entries of $(C + \Lambda)^{-1}$ smaller than those of $C^{-1}$? In other ...
1
vote
3answers
111 views

Generalization of permutation matrix

For integers $n$ and $k$, I am interested in $n\times n$ matrices with exactly $k$ non-zero entries in each row and each column. The case $k=1$ corresponds to (generalized) permutation matrices. In ...
1
vote
2answers
109 views

Mirror Matrix Multiplication

Usual matrix multiplication is done from left to right and top to bottom. Does there exist an application or a theory that does matrix multiplication from right to left and top to bottom? EXAMPLE: ...
1
vote
2answers
161 views

Direct summand of skew-symmetric and symmetric matrices

Let $W_1$ be the subspace of $\mathcal{M}_{n \times n}$ that consists of all $n \times n$ skew-symmetric matrices with entries from $\mathbb{F}$, and let $W_2$ be the subspace of $\mathcal{M}_{n ...
1
vote
1answer
93 views

Finding minimum of the trace of the matrix equals finding maximum of the trace of the inverse matrix?

Let $K$ be a positive definite, symmetric matrix. Let $C$ be a nondegenerate matrix of the same order. Elements of $K$ and $C$ depend on some parameter $a.$ Is it true to say that $$ ...
1
vote
3answers
45 views

Why is the map bewteen a matrix and its characteristic polynomial continous?

One may define the following map : $\begin{array}{l|rcl} f : & M_n(\mathbb R) & \longrightarrow & \mathbb R_n[X] \\ & A & \longmapsto & p_A \end{array}$ Why is $f$ ...
1
vote
2answers
48 views

How to solve equation using inverse matrix method?

I'm trying to solve the following equation: $$ AX=B $$ where $A=\begin{bmatrix}1 & 5\\2 & 3\\-4 & 1\end{bmatrix}$, $B=\begin{bmatrix}1 & 2 & 5\\2 & 4 & 3\\-4 & -8 ...
1
vote
2answers
527 views

What are the properties of symmetric, anti-symmetric, and diagonal matrices

I know the definition of each one but I don't know how to answers questions about them, or what their properties are and how I can use them to prove/disprove statements about them. If P, Q, and D are ...
1
vote
1answer
34 views

linear algebra : matrix decomposition

Let $X \in \mathrm{Mat}_{n \times p}(\mathbb{R})$ a matrix such that $\mathrm{rank}(X)=p$. Let $S = \mathrm{I}_{n} - X \big( X^{\top} X \big)^{-1} X^{\top}$ be the orthogonal projection on $\big( ...
1
vote
1answer
60 views

confusion about rank and nullity of matrix and rank-nullity theorem

I can see that rank + nullity = number of columns of the matrix. Does that mean the dimension of matrix is the number of columns? Isn't dimension of a $m\times n$ matrix $mn$? Thanks.
1
vote
1answer
32 views

$A$ is singular and has nonzero row sums that are the same for every row. then $A+\lambda 11^{\prime}$ is singular

$A$ is singular and has nonzero row sums that are the same for every row. then $A+\lambda 11^{\prime}$ is singular, where $1$ is a vector of one's. Let $A=\{a_{i1} a_{i2}\ldots a_{i(n-1)}\space ...
1
vote
1answer
39 views

Is there a smarter way to solve? Matrix multiplication

So i have this Problem: I know how to solve it basically finding the inverse matrices and so on, but i was wondering if there isn't a quicker and smarted way, because the matrix on the right is ...
1
vote
2answers
111 views

Matrix diagonalizable or not [duplicate]

Let $A$ is in $M_3(\mathbb R^3)$ which is not a diagonal matrix. Pick out the cases when $A $ is diagonalizable over $\mathbb R$: a. when $A^2=A$; b. when $(A-3I)^2=0$; c. when $A^2+I=0$. My ...
1
vote
1answer
100 views

calculating an incoherence property

With respect to Matrix Completion and Compressive Sampling (CS) I'm trying to understand how to calculate an incoherence property μ between two bases Φ and Ψ. Getting this incoherence is important ...
1
vote
1answer
25 views

Incoherence property in Matrix Completion

I'm not a math major and I find the statement below confusing (from a paper by Candes and Recht on Matrix Completion). Can someone clarify this? I'm sure it's painfully simple/obvious. "For ...
1
vote
2answers
34 views

Can this condition infer that the matrix is Hermite?

$\boldsymbol{A^H A=AA^H}$ does this imply that $\boldsymbol{A}$ is Hermite matrix? Why? $\boldsymbol{A^H}$ is the conjugate transpose of $\boldsymbol{A}$
1
vote
1answer
68 views

Schur's Lemma and division algebras

Let $A$ be an abelian subgroup of the unimodular group of degree $n$ (i.e. $GL(n,\mathbb Z)$). $A$ can be regarded as a group of automorphisms of a free abelian group of rank $n$ ($\mathbb Z^n$), and ...
1
vote
1answer
29 views

Finding a left- and right-multiplied matrix A given the product

I am working my way through this book, in an attempt to teach myself matrix algebra. In the first chapter, the author asks the student to find a matrix $A$ such that: $\begin{bmatrix}1 & 0 & ...
1
vote
2answers
47 views

From positive semi-definiteness to positive definiteness

One of my lectures includes the following quote from my professor (on a part of a chapter about compatible systems): $A^TA$ is semi-positive-definite. If columns of $A$ are linear independent, ...
1
vote
1answer
39 views

Solving for trivial solutions of a matrix

My friend and I came across this problem while looking through some homework. Say you had a $3 \times 4$ matrix that reduced down to something like this: $$ \begin{pmatrix} 1 & 2 & 0 ...
1
vote
1answer
36 views

The eigenvalue or the two norm of a matrix

Let $M\in \mathbb{R}^{n\times n}$. And $$B=\begin{bmatrix} -\theta M-M+c_1 I^{n\times n}& \theta M-c_2I^{n\times n}\\ I^{n\times n}& 0^{n\times n} \end{bmatrix};\quad c_1, c_2,\theta \in ...
1
vote
1answer
45 views

Decide if the vector (1,1,1) is in the row space of the matrix

Decide if the vector $(1,1,1)$ is in the row space of the matrix $$ \begin{bmatrix} 1& 1& 3\\-1&0&1\\-1&2&7 \end{bmatrix}$$ Yes. To see if there are $c_1$ and $c_2$ such ...
1
vote
1answer
86 views

Repeated Iteration of a 2x2 matrix

Suppose I am given a $2$x$2$ matrix $A=$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} And an initial vector $x_n$ = \begin{pmatrix} x_0 \\ y_0\end{pmatrix}. Under repeated iteration $x_{n+1} ...
1
vote
1answer
40 views

Can this equation have more than one solution?

Consider the following equation: $\left[\array{1 & 0.1353 & 1 \\0.3678 & 0.3678 & 1 \\ 0.1353 & 1 & 1 \\ 0.3678 & 0.3678 & ...
1
vote
1answer
121 views

Calculating a basis of vector space $U \cap V$

So I have two vector spaces: $ U := \langle(1,2,1,2), (1,2,3,3), (1,2,2,3)\rangle $ and $ V := \langle(2,0,2,1), (3,2,3,2), (0,4,0,1)\rangle $ I was able to calculate the base of both $U$ and $V$: $ ...
1
vote
1answer
70 views

If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable

Suppose that $A \in M_{n\times n}(\Bbb F)$ has two distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ and that $\dim (E_{\lambda_1})=n-1$ show that $A$ is diagnolizable.
1
vote
2answers
43 views

Matrix and eigenvectors

$\quad$The matrix $\mathbf A=\frac19\begin{bmatrix} 7 & -2 & 0 \\ -2 & 6 & 3 \\ 0 & 2 & 5 \\ \end{bmatrix}$ has eigenvalues $1$, $\frac23$ and $\frac13$n ...
1
vote
1answer
180 views

Minors of a positive definite matrix are positive definite

All main minors of a positive definite matrix are positive definite as well and therefore $A$ is strictly invertible. All I know about positive-definiteness is that for the symmetric matrix ...
1
vote
3answers
56 views

When does a square matrix is equivalent to the identity matrix?

Why does a square matrix A with an order of $n \times n$ and $rank(A)=n$ can be reduced to Identity matrix (or by multiplying a sequence of elementary matrices)
1
vote
1answer
116 views

Spectral radius, second induced norm

In my textbook there are few facts left without any sign of a proof, which really bugs me, and I was thinking maybe someone can help me: $A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = ...
1
vote
2answers
90 views

Independent components of trace?

A $3 \times 3$ symmetric matrix has $6$ independent components: \begin{equation} \{ S_{ij} \} = \begin{pmatrix} S_{11} & S_{12} & S_{13} \\ S_{12} & S_{22} & S_{23} \\ S_{13} & ...
1
vote
4answers
103 views

Proof $\langle Ax,y\rangle = \langle x,A^*y\rangle$ when $A$ Hermitian

I was trying to understand a proof of why a Hermitian $A$ matrix has its eigenvectors orthogonal. As part of the proof they state $$\langle Ax,y\rangle = \langle x,A^*y\rangle$$ From which property ...
1
vote
2answers
69 views

Representations of the set of Natural Numbers [closed]

Is there a finite number of ways that the entire set of natural numbers can be represented by an infinite matrix? eg (excluding the number $1$): $$ \left[\begin{array}{cccccc} 2 & 4 & 8 ...
1
vote
1answer
168 views

Find transformation matrix with respect to a basis of an invariant subspace

Simple question but I've never encountered ones like that. $T$ is a linear transformation from $\mathbb R^3$ to $\mathbb R^3$ defined by: $T(v)=Av$ when $A=\begin{pmatrix} 1 & 2 & 2\\6 & ...
1
vote
2answers
71 views

Why do diagnolizable matrices have to be invertible?

My professor gave us this definition of a diagnolizable matrix. A matrix $A$ is diagnolizable if it's invertible and $$(Ax)_{\mathcal{B}} = Dx_{\mathcal{B}}$$ for some diagonal matrix $D$, basis ...
1
vote
1answer
93 views

Conjecture about some group semiring representations ( and roots of unity ).

Let $\Bbb R_+=[0,\infty)$ be a semiring. $\Bbb R_+[C_n]$ is the group semiring formed by the semiring $\Bbb R_+$ and the cylic group $C_n$. Let $\Bbb R_+[X_n]$ be the polynomial semiring. ...
1
vote
1answer
22 views

conditions for LU on a $2 \times 2$ matrix

I have $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ I want to do LU factorization on it, so I need to find the elementary matrix $E$ such that $EA=U$ ...
1
vote
2answers
33 views

Finding associated eigenvalue and eigenvector

Having troubles with this question Suppose that $\det(A) \not= 0$, and $A$ and $B$ both have eigenvector $v$, but the corresponding eigenvalue is $\lambda_{A}$ for $A$ and $\lambda_{B}$ for $B$. Show ...
1
vote
2answers
75 views

characterization of uniform ellipticity

Let $B$ be a $n\times n$ matrix over $\mathbb{R}$ and define $A:=BB^*$. I read in a paper that the following two statements are equivalent: (1) the matrix $A$ is uniformly elliptic; i.e. for all ...
1
vote
1answer
137 views

Prove scalar products are invariant under all orthogonal transormation

I wondering how to prove: That scalar products are invariant under all orthogonal transformation: $<\!x, y\!>\; =\;<\!Qx, Qy\!>$ which holds for all vector $x$,$y \in \Re^n$ and all ...
1
vote
2answers
54 views

Unitary map between sets of vectors

Suppose I have two sets of vectors, $E_1=\{v_i\}_{i=1}^{k}$ and $E_2=\{u_i\}_{i=1}^{k}$, with each vector belonging to $\mathbb{C}^k$. When is it possible to find a unitary matrix that maps $E_1$ to ...
1
vote
1answer
46 views

Resolvent recurrence relation

Let the resolvent matrix of $\mathbf{X}$, a symmetric matrix with real entries, be defined as \begin{align} R_{\mathbf{X}}(\lambda):=\bigl(\mathbf{X}-\lambda\mathbf{I}\bigr)^{-1}, \qquad \lambda ...
1
vote
2answers
213 views

How to diagonalize a matrix

So I am trying to diagonalize this matrix {2,0,-2} {1,3,2} {0,0,3} so that those are the rows of the matrix. I know the eigen values are 2 and 3. I don't think that this matrix can be ...
1
vote
1answer
16 views

Nullspace and different solutions

$\begin{pmatrix} R_{11} & \cdots & R_{1A} \\ \vdots & \ddots & \vdots \\ R_{S1} & \cdots & R_{SA} \\ 1 & \cdots & 1 \end{pmatrix} ...
1
vote
1answer
111 views

Jordan normal form for a characteristic polynomial $(x-a)^5$

Write down all the possible Jordan normal forms for matrices with characteristic polynomial $(x-a)^5$. In each case, calculate the minimal polynomial and the geometric multiplicity of the eigenvalue ...
1
vote
2answers
67 views

Find all matrices that satisfy $BA=I_2$

I am given the matrix $$A=\begin{pmatrix}1&8\\3&5\\2&2\\ \end{pmatrix}$$ and I need to find all $2 \times 3$ matrices in $B \in M_{2 \times 3}(\mathbb R)$ with $BA=I_2$. Here's what I ...