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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1
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1answer
32 views

Are the spaces of real orthogonal, complex unitary, hermitian or symmetric matrices connected?

I want to know which of these are connected and which are not. I think I've to take some continuous map from the set of matrices to $\mathbb R$ or $\mathbb C$ and interpret these matrix sets as ...
1
vote
0answers
32 views

Fast Cholesky Factrorization for Tree Laplacians

Suppose $T_1$ and $T_2$ represent two Laplacian matrices of two spanning trees of $n$ vertices. Since the Cholesky factorization needs $O(n)$ time for each $T_i\ (i=1,2)$ due to the tree structure, ...
1
vote
1answer
50 views

Proof that if $A$ is similar to $B$, then $B$ is similar to $A$

$A$ is similar to $B$ if there is an invertible matrix $S$ such that $B = S^{-1}AS$. Prove that if $A$ is similar to $B$, then $B$ is similar to $A$. So if $A$ is similar to $B$ then $B = ...
0
votes
0answers
5 views

Matrix with highly correlated adjacency entries

I am learning about SVD from this book. One of the exercise questions asks me to create matrix with highly correlated adjacency entries and then conduct some experiments to discover the nature of the ...
2
votes
3answers
66 views

What is inverse of $I+A$ given that $A^2=2\mathbb{I}$?

I have the next problem: Let $A$ be a real square matrix such that $A ^ 2 = 2\mathbb{I}$. Prove that $A +\mathbb{I}$ is an invertible matrix and find its inverse. I tried with the answers given ...
0
votes
0answers
12 views

Eigenvalues of Hankel matrices

Let $\mathbf{A}$ be a $4-$ dimensional symmetric matrix with real entries, whose elements are given as \begin{equation} \mathbf{A} = \left( \begin{array}{cccc} a & b & c & d \\ b & c ...
3
votes
2answers
32 views

Eigenvalues of 3D rotation matrix

I'm having some trouble calculating the eigenvalues for this rotation matrix, I know that you subtract a $\lambda$ from each diagonal term and take the determinant and solve the equation for $\lambda$ ...
1
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3answers
38 views

Eigenvectors of $\left( \begin{array}{ccc} 0 & -b \\ a & 0 \end{array} \right)$

This is similar to my previous question in that I when I form a system of simultaneous equations and solve them all the terms cancel and I don't get any information on the eigenvectors. The matrix in ...
1
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3answers
67 views

Eigenvectors of $\left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$

I calculated the eigenvalues of the following matrix to be $a$ and $-b$. $J = \left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$ But when I use the formula $(J - \lambda I)v = 0$ ...
0
votes
2answers
40 views

Does $TS$ being an isomorphism imply that $S$ is an isomorphism?

Let $V, W, U$ be vector spaces over a field $\Bbb F$, and suppose that $S : V → W$ and $T : W → U$ are linear. If $TS$ is an isomorphism, is it true that $S$ must be an isomorphism? If it's not ...
1
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0answers
35 views

Linear Algebra - Find basis for $ImT$ and $KerT$.

$B=(u1,u2,u3) \in R^3$ $u1=(1,-1,0)$ $u2=(1,1,1)$ $u3=(1,2,3)$ $T : R^3 \rightarrow R^3$ This is expression matrix (not sure if that the right term in English) on basis B: $[T]_B ...
0
votes
1answer
24 views

Eigen values of a positive semidefinite matrix and its transpose

$A\in M_n(\mathbb{C})$ is positive semi-definite so there there exists unitary matrix $U$ such that $A=U^*DU$ where $D$ is the real diagonal matrix consisting of eigen values $(\ge 0)$ of $A$, now I ...
-1
votes
1answer
29 views

How Can I define the derivative of matrix?

If I have a matrix: $$F(x) = \begin{pmatrix}f_1(x)& f_2(x) \\ g_1(x) & g_2(x) \end{pmatrix} $$ where $f_1,f_2,g_1$ and $g_2$ are differentiable functions. What would be the derivative of ...
0
votes
1answer
48 views

Expressing determinant as a linear combination of minors of fixed dimension

Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of ...
1
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0answers
10 views

Non square inversion

I need to find the matching conditions for an adaptive system in terms of the $k_x$ and $k_r$ that satisfy: $A+B K_x=A_m$ $B k_r=B_m$ Where: $A=\begin{pmatrix}-1 & 1\\1 & 0\end{pmatrix}$ ...
-1
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0answers
37 views

Linear Alegbra - The $U + W$, $U \cap W$, $U \cup W$, $U-W$ of subspaces and not subspaces

Lets assume $U,W \subseteq \mathbb{R}^4$ $U=\{u_1 = (0,0,0,1), u_2=(1,0,0,0)\}$ $W=\{w_1=(0,0,1,0),w_2=(1,0,0,0),w_3=(0,1,0,0)\}$ I understand that in case $U$ and $W$ are not subspaces: Case $U ...
0
votes
0answers
11 views

Equivalence of condition number from equivalence of vector norms

I must show that the equivalence of vector norms implies the equivalence of the condition number of its induced matrix norm. That is, given that for two arbitrary vector norms (+ and *) and an ...
0
votes
2answers
14 views

Outer Product of Two Matrices?

How would I go about calculating the outer product of two matrices of 2 dimensions each? From what I can find, outer product seems to be the product of two vectors, $u$ and the transpose of another ...
8
votes
3answers
495 views

Proving or disproving A+B is invertible

Given two matrices $A,B\in M_n (F)$, where $A$ is $k$ -nilpotent and $B$ is invertible, is it true that $A+B$ is also invertible? I was having trouble on how to prove this, and then I thought maybe ...
0
votes
1answer
59 views

Linearly dependent eigenvectors of a matrix

I read a theorem that says squared matrix $A_{n\times n}$ is diagonalizable iff there is a set of $n$ linearly independent vectors ,each of which is an eigen-vector of $A_{n\times n}$ . I understand ...
0
votes
0answers
14 views

Prove this relation between truncated SVD and eigen decomposition?

For a real matrix $M$, we have a full SVD $M=USV^T$ and a truncated SVD $M_{k}=U_kS_kV_k^T$. The truncated SVD means (matlab grammar): $U_k=U(:,1:k), S_k=S(1:k,1:k), V_k=V(:,1:k)$. Based on the ...
0
votes
0answers
9 views

Sherman Morrison Formula for hermitian updates

I have a problem in which, in principle I can apply twice Sherman-Morrison formula but it seems to me that for this case, there should be a simpler solution so my question is "May the process ...
0
votes
4answers
84 views

If matrix $AB=A$, does it mean B must be an identity matrix?

If matrix multiplication $AB=A$, does it mean $B$ must be an identity matrix? If not, why? What conditions? $A$ is not a zero matrix.
0
votes
0answers
40 views

What is the maximum absolute value in RREF form of a matrix

what is the maximal absolute value that one can get in a RREF form of a matrix $A\in\mathbb{R^{n\times m}}$, where $A_{ij}$ is always an integer from range $<-r;r>$ for some given $r$? I'm able ...
1
vote
1answer
27 views

Construct an $Q$ orthogonal using Givens matrices that for all unitary vectors $x$ and $y$ we have $Q^Tx=y$

Find a method to use given matrices to create an orthogonal matrix $Q \in R^{n \times n}$, that for unitary vectors $x,y$ $\in R^{n}$, $$Q^Tx=y$$ The ideia that i have is: take a sucession of Givens ...
3
votes
2answers
63 views

$P(AB=BA)$ , $A,B\in M_{3x3}(\mathbb Z/p\mathbb Z)$

Let $A,B\in M_{3x3}(\mathbb Z/p\mathbb Z)$ ($p$ a prime number). Find the probability $P$ that $AB=BA$ that is $P(AB=BA)$ $$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} ...
0
votes
2answers
32 views

Show that the inverse of a strictly diagonally dominant matrix is monotone

I have been struggling with this problem for awhile. Given that $A$ is a strictly diagonally dominant matrix with positive diagonal entries and non-positive off-diagonal entries, show that $A$ is ...
1
vote
1answer
28 views

Constructing a matrix of order $3\times 3$ such that the limiting matrix also exists in which all the rows are not the same.

Let $$ A = \left[\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right] $$ be a matrix where $a_{ij}\in{[0,1]}$, ...
3
votes
1answer
52 views

Real vs. complex conjugate

Suppose I have matrices $A,B\in\mathrm{Mat}_n(\mathbf{R})$ which are conjugate in $\mathrm{Mat}_n(\mathbf{C})$ in the sense that there is $S\in\mathrm{GL}_n(\mathbf{C})$ with $A=SBS^{-1}$. Then is it ...
4
votes
2answers
58 views

How to find $(Ker(A^{*}))^{\perp}$

Let $$A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & -2 & -1 \\ 1 & 2 & 4 & 3 \end{pmatrix}$$ Find a basis for $(Ker(A^{*}))^{\perp}$. Find vectors $b_i$ such that $ ...
2
votes
0answers
38 views

Changing the basis of a matrix of the linear mapping

Let $$A= \begin{pmatrix} 1 & 3 & 1 & 4 \\ 2 & 3 & 4 & 5 \end{pmatrix}$$ be the matrix of the linear mapping $F: \mathbb{R}^4 \to \mathbb{R}^2$ in the usual bases of ...
0
votes
1answer
24 views

Controllable and observable

The square matrices $A$ is invertible, $Q$ and $G$ symmetric positive semidefinite. Moreover, $(A,G)$ is controllable, and $(Q,A)$ is observable. I have the following question Is $(-A,-G)$ ...
0
votes
1answer
60 views

Im(A) = Im(A*A)

How does one prove that $Im(A^*) = Im(A^*A)$ and that the $Im(A) = Im(AA^*)$? I have found that $Ker(A) = Ker(A^*A)$ and that $Ker(A)=Ker(AA^{*})$. Also, the $Rank(A) = Rank(AA^{*}) = Rank(A^{*}A)$. ...
0
votes
1answer
53 views

Linear Alegbra - Find Base for ImT and KerT

Find basis for $ImT$ and basis for $kerT$. $v_1=(1,1,1)$ $v_2=(0,0,1)$ $v_3=(0,1,1)$ $B=(v_1,v_2,v_3) \in R^3$ My solution $[T]_B=\left[\begin{array}{cccc} 1 & 2 & 1 \\ -2 & 0 & ...
0
votes
1answer
20 views

What will $A^+A$ and $A^gA$ actually or exactly get if $A$ is not invertible?

I know if $A$ is invertible then $A^{-1}$ is the inverse of $A$, and $AA^{-1}=A^{-1}A=I$. I just learnt the concept of Generalized inverses and Moore–Penrose pseudoinverse. For a matrix $A$ that is ...
0
votes
0answers
17 views

Composition of functions in vector form

Is H equal to the matrix multiplication of G*F How do I use the chain rule to calculate H'? G'(F(x))*F'(x)? How does this work in practice for matrices? For (b) I will compute the matrix of ...
0
votes
1answer
31 views

Prove a condition from properties of matrices

Suppose that $A$ is an $n\times n$ real symmetric matrix of rank $n-1$. Suppose also that $Au=0$, where $u=(1,1,...,1)^T$. Show that $Ax=y$ has a solution if and only if $\sum_{j=1}^ny_j=0$. ...
1
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2answers
20 views

How do you solve a general second order matrix differential equation?

How do you find solutions to the equation of the form : \begin{equation}A\frac{d^2X}{dt^2} + B\frac{dX}{dt} + CX = 0 \end{equation} where A,B,C are 3X3 positive definite and symmetric matrices with ...
-1
votes
2answers
74 views

Linear Algebra - $n\times n$ matrices [closed]

Let $A$ and $B$ be any $n \times n$ defined over the real numbers. Let assume that $I+AB$ invertible matrix. Prove : $I+BA$ invertible matrix $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ Any help will be ...
0
votes
1answer
40 views

Matrices and bases

Can you please verify my argument: Let $M = \begin{pmatrix} a & b\\ c& d\end{pmatrix}$, where $a,b,c,d$ are all real. $$AM=\begin{pmatrix} c & d\\ a& b\end{pmatrix}$$ Let $B$ be ...
1
vote
1answer
55 views

Rational solution to AX=0

Let $\mathcal{M}_{n,p}(\mathbb{K})$ be the set of matrices $n\times p$ with coefficients in $\mathbb{K}$. Let $A\in\mathcal{M}_{n,p}(\mathbb{Q})$. We suppose there exists a non zero solution ...
1
vote
1answer
28 views

Why are the spectral norm of $A^{*}A$, $AA^{*}$ and $A$ equal?

I'm learning matrix norm now, but i don't have learned Hermitian before. Is there any theorem about hermitian i can use to prove that three matrices norm are equal?? Thanks a lot.
2
votes
2answers
48 views

if matrix multiplication $B*A=C*A$, does it mean $B=C$?

If matrix multiplication $B*A=C*A$, does it mean $B=C$? If A is invertible, then I guess this should work. If not, then?
7
votes
1answer
113 views

Matrix exponential converse. Baker-Campbell-Hausdorff

I am currently reading about the Baker-Campbell-Hausdorff formula and in a textbook on Lie Algebras, he shows that if $$[X,[X,Y]] = 0 \quad \text{ and } [Y,[X,Y]] = 0$$ then $$e^{Xt}e^{Yt} = e^{Xt ...
0
votes
1answer
27 views

Are following statements about matrices true?

1) If $AB+BA=0$, then $A^2B^3=B^3A^2$ 2) If $A$ and $B$ are non-singular, then $AB$ is non-singular 3) If $A^3=0$, then $A-I$ is non-singular $A$ and $B$ are $n \times n$ matrices and $I$ is ...
1
vote
2answers
49 views

Show that $V$ is a vector space

If we let $$V = \{ x \mid x = \begin{bmatrix} x_1 \\[0.3em] x_2 \end{bmatrix},\text{ where }x_2 > 0 \} $$ and define addition and scalar multiplication by $$u + v = ...
2
votes
1answer
32 views

Linear Algebra - Invertible matrice

I have this problem and I'm not sure my solution is correct. Let $A$ be any $n \times n$ matrix, defined over the real numbers, A is not invertible matrix. Proof that there's is $B \neq 0$ and $C ...
1
vote
2answers
29 views

Linear Algebra - Invertible matrices and determinants

Let $A$ be any $n \times n$ invertible matrix, defined over the integer numbers. Let assume that $A^{-1}$ (Inverse of A) is also defined over the integer numbers. Prove that $\det A\in\{-1,+1\}$. ...
1
vote
0answers
15 views

Can anyone help me prove block diagonal matrix? [duplicate]

If I have Block matrix A 0 B C , where A,B,C are square matrices I was trying to prove why determinant of this matrix is equal to product of determinant of A and C. ...
0
votes
1answer
26 views

When does a square matrix have an eigen-decomposition? When is a matrix defective? [duplicate]

Some square matrices, like $ \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$, don't have a complete set of eigenvectors. By complete I mean that the eigenvectors span the entire ...