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 (2)

0
votes
1answer
20 views

Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
-4
votes
1answer
36 views

$A$ and $B$ have the same singular values.Why $A$ and $B$ are unitary equivalent? [closed]

Let $A,B \in {M_n}$ and they have the same singular values.Why $A$ and $B$ are unitary equivalent?(by SVD)
2
votes
2answers
40 views

What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices?

Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them?
-2
votes
0answers
34 views

Rotation quaternions and rotation matrix for electron spin [on hold]

Is it possible to construct a rotation quaternion and rotation matrix for the approximation of an electron spin?
-3
votes
0answers
20 views

If $AA^*$ and $BB^*$ are unitary equivalent. Is it true that $A$ and $B$ are unitary equivalent? [closed]

Let $AA^*$ and $BB^*$ are unitary equivalent. Is it true that $A$ and $B$ are unitary equivalent?
0
votes
1answer
43 views

Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$.

Let , $A_{6\times 6}$ diagonal matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$. From characteristic polynomial of $A$ , first ...
0
votes
1answer
26 views

What is the Lie group that leaves this matrix invariant?

What is the group that leaves \begin{equation} Y = \bigoplus_{j=1}^{k} \alpha_j \mathbb{I}_{2n_j\times 2n_j} \end{equation} invariant under congruence ($Y = XYX^T$) where $\alpha_j \in \mathbb{R}$ ...
0
votes
1answer
32 views

Solve matrix equation $AXB+CX=D$

How to solve matrix equation $AXB+CX=D$ for $X$? If it is not solvable, are there any numerical methods to do it?
1
vote
2answers
24 views

find the smallest interval in which the eigen value of the matrix lie

$$ \begin{bmatrix} 3 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 3 \\ \end{bmatrix} $$ I was practicing questions on Matrices & Determinants ...
0
votes
2answers
46 views

if $rank{(A - \lambda I)^k} = rank{(B - \lambda I)^k}$ then $A$ is similar $B$

Let $A,B \in M_n(\mathbb{R}).$ Suppose for all $\lambda \in \sigma (A)$ and for all $k \geq 0,$ we have $\mathrm{rank}(A - \lambda I)^k = \mathrm{rank}(B - \lambda I)^k.$ Then why are $A$ and $B $ ...
0
votes
1answer
58 views

If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
0
votes
1answer
27 views

Eigenvalues of an 2x2 matrix [closed]

How do i calculate the EigenValues of an Hessian Matrix which is 2x2.? And what is EigenValues. It is used in imagesProc. when i have to find goodFeatures is found when both eigenvalues are high.. but ...
1
vote
1answer
36 views

Jordan form of different matrices

Suppose you have a 4x4 matrix with the characteristic polynomial equal to the minimal polynomial $m_F(x)=C_F(x)=(x-3)^2(x+2)^2$. Find the Jordan form. Is this the correct solution? $$ ...
1
vote
1answer
17 views

A question about non-linear least square method…

I am trying to fit a set of points into a sine function, using nonlinear least square method. The final step to obtain the derivative of its parameters is given by the equation (8) of: ...
0
votes
0answers
36 views

How to calculate a person's Latitudinal and Longitudinal location based off of Sun and time

INTRO I remembered hearing about it being possible to calculate a person's position or the position a picture was taken, based on time of day and the position of the Sun, position meaning latitudinal ...
2
votes
0answers
14 views

Cartan matrices: motivation and intuitive examples?

could anyone provide me with a sketch of the motivation that gave rise to Cartan matrices in abstract (homological) algebra, Lie algebrae and so on? Which was the trigger or the need for them? It ...
0
votes
0answers
17 views

Can matrix generated by ith power of adjacency matrix, have -ve value?

I read that - The uv-entry of the k-th power $$A^k$$ counts the number of walks of length k from the vertex u to the vertex v. I wanted to know if such a matrix can have negative values, and ...
-3
votes
0answers
31 views

$A=UMW^*$ and $B=VMO^*$.Is this true that $A$ and $B$ are unitarily equivalent? [closed]

Let $U,W,V,O$ are unitary matrises and $A=UMW^*$ and $B=VMO^*$.Is this true that $A$ and $B$ are unitarily equivalent?(all matrises n-by-n)
-1
votes
1answer
45 views

$A,B $ are both normal and $A$ and $B$ commute.Why $AB$ is normal? [closed]

Let $A,B \in {M_n}$ are both normal and $A$ and $B$ commute.Why $AB$ is normal?
0
votes
1answer
25 views

Complete misunderstanding of Lie groups and representations

Consider a particular representation of $\operatorname{SO}(2,\mathbb{R})$: \begin{equation} \begin{pmatrix} \operatorname{cos}(\theta) & \operatorname{sin}(\theta) \\ ...
0
votes
1answer
37 views

How can we find if a matrix is full column rank

If $A$ is an $n*k$ matrix with complicated form of elements. How can I show this matrix is full column rank? By complicated form I mean there is no known form for the elements of $A$.
1
vote
1answer
42 views

Find Jordan Form of αA (α is a scalar, A a matrix)

In my linear algebra course I have a problem which goes as follows: Suppose A is an nxn matrix over field (R) And J(A) is the jordan form of A. Given α belongs to field R, what is the jordan form of ...
0
votes
0answers
19 views

Signature Defect of a Matrix

Let $A,D$ be symmetric real matrices and let $B,C$ be real matrices such that $B$ and $C$ have the same number of rows, $A$ has the same number of columns as $B$, and $C$ has the same number of ...
-1
votes
1answer
24 views

Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)? .

Let $A \in {M_n}$.Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)?
1
vote
0answers
24 views

Formula for powering a matrix not working for all matrices

I'm currently learning about matrices and was asked to show that this formula works for powers of $M$. $$M^n = nM-(n-1)I$$ Where $M$ is the matrix (show below), $n$ is the exponent an $I$ is the ...
0
votes
2answers
21 views

proof a theorem in linear algebra

prove that if λ1 and λ2 are two distinct eigenvalues of a matrix A and λ1 , λ2 are corresponding eigenvectors, respectively, then α1 and α2 are linearly independent please help... thank you...
0
votes
1answer
40 views

If ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = \sum\limits_{i = 1}^n {{\sigma _i}^2} \Rightarrow$A is normal matrix

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$ and suppose ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = ...
1
vote
2answers
48 views

Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$. Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?
1
vote
2answers
89 views

How is the vector space of abstract “tuples” isomorphic to vector space of $n \times 1$ or $1 \times n $ matrices?

I read that the vector space of abstract "tuples" is isomorphic to vector space of $n \times 1$ or $1 \times n $ matrices. Where can I find a good explanation of this or can someone explain it here?
1
vote
2answers
60 views

If $\dim_F F[A] <n$, then $A$ is not cyclic. [closed]

Suppose that $A \in M_n(F)$ and the minimal polynomial of $A$ is irreducible. S0 $F[A]$ is a field extension of $F$. I have to show: 1) If $\dim_F F[A] <n$, then $A$ is not cyclic. 2) If $\dim_F ...
1
vote
1answer
20 views

finding matrix represention for linear transformation for field extension

need some clarification. given an extension field K over F with F-linear transformation, for $\alpha \in K$, $f_\alpha(k) = \alpha \cdot k$ i.e. multiplication on the left. I need to find the ...
1
vote
0answers
12 views

Similarity of orthogonal matrices

Prove that for any $M$ in $SO(3)$, there is a matrix $P$ in $SO(3)$ and a real $\alpha$ such that $$PMP^{-1} = \left[ \begin{matrix} \cos\alpha & \sin\alpha & 0 \\ ...
-1
votes
1answer
22 views

How to find the matrix of the transformation relative to the basis?

Let $T:P_2\to P_2$ be the linear operator defined by $$T(a+bx+cx^2)=(3a+2b+4c)+(2a+2c)x+(4a+2b+3c)x^2$$ Find the matrix of the transformation $T$ relative to the basis $B=\{1,x,x^2\}$.
-1
votes
1answer
41 views

Image of the product of a Matrix and its transpose

If $A$ is an $n\times m$ matrix, is it necessarily true that $\text{im}(A)=\text{im}(AA^T)$ where $A^T$ is the transpose of $A$.
0
votes
1answer
46 views

Eigen vectors for matrix with unknown constants?

I have the following matrix: $$\begin{bmatrix}\alpha&0&0\\\beta-\alpha&\beta&0\\1-\beta&1-\beta&1\end{bmatrix}$$ So far I have worked out the polynomial to be: ...
0
votes
3answers
25 views

How can I show that the dimension of span is . . .?

The Krylov subspace generated by $n$-by-$n$ matrix $A$ is defined by : $K_k(A,x)=span\{x,Ax,A^2x,...,A^{k-1}x\}$ How do I show its dimension is at most $k$? I only know that $dim(span (V))=rank ...
2
votes
1answer
19 views

Proving matrix exponent property [closed]

How can I prove the following equation. I have tried but i couldn't. $$\exp(A(t_2+t_1))=\exp(At_2)\cdot \exp(At_1)$$ $A$ is a matrix Will I use state-transition matrix or what ? Thank you...
3
votes
2answers
83 views

Show a matrix satisfying $A^2 − 8A + 15I = 0$ is diagonalisable.

A square matrix $A$ (of some size $n × n$) satisfies the condition $A^2 − 8A + 15I = 0$. Show that this matrix is similar to a diagonal matrix. I know that we must show that 5 and 3 are the ...
0
votes
1answer
43 views

How to see that $A = A^{-1}$ and $A^2 = A$ as quick and easy as possible without computer aid

I'm wondering which are the quickest/easiest methods to identify that the following relations hold for any given matrix: $A = A^{-1}$ and $A^2 = A$ On a computer it's easy and quick to identify if ...
2
votes
0answers
15 views

Constrained zero diagonal low rank approximation of a matrix with zero diagonal

Suppose that you have a $n\times n$ matrix $A$ that is symmetric and has zero diagonal, such as for example $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0 & 1\\ 2 & 1 & 0}, $$ and you want ...
-1
votes
0answers
24 views

In Markov chains, does $(I-N)^{-1}$ always exist? [duplicate]

Spins-off from these two questions. Under what conditions does $(I-N)^{-1}$ exist? If the entries of an invertible matrix N are between -1 and 1, is its operator norm less than 1? Apparently, in ...
4
votes
1answer
47 views

Question about eigenvalue of Hermitian matrix

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
0
votes
4answers
34 views

Solving for a vector $x$ given $Ax=b$

This is a dumb question I know. If I have matrix equation $Ax = b$ where $A$ is a square matrix and $x,b$ are vectors, and I know $A$ and $b$, I am solving for $x$. But multiplication is not ...
1
vote
0answers
45 views

Moore–Penrose Pseudoinverse of Squared Matrix Product

We want to show the following equation: \begin{align} \left(C'R'RC\right)^+=\frac{1}{T-1}\left(A^{-1}-\frac{A^{-1}\iota_N\iota_N'A^{-1}}{\iota_N'A^{-1}\iota_N}\right), \end{align} where ...
0
votes
4answers
35 views

Linear algebra, inverse of a matrix

Prove that if $A$ and $B$ are square matrices such that $AB = I$ then $B$ is invertible and $A$ is inverse of $B$. Basically can you help me prove the uniqueness of the inverse of matrix?
0
votes
1answer
27 views

Simple question in Rank

Let $A,U \in {M_n}$ and $U$ is unitary matrix.Is this true that $Rank(AU)=Rank(A)$?
1
vote
0answers
19 views

How to find value of an unknown in matrix to make system of linear equations consistent

I'm currently stuck on this question relating to finding the unknown in a matrix so that the system of linear equations is consistent. I need to solve for $\lambda$. My first instinct is to try and ...
0
votes
0answers
12 views

A Rayleigh quotient-related eigenvalue problem

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
0
votes
0answers
10 views

the SVD (singular value decomposition) of an augmented matrix

Suppose we have a $4\times 3$ dimensional matrix $A$. Denote the SVD of $A$ by $USV^T$, where $U\in R^{4\times 3}, S\in R^{3\times 3}, V\in R^{3\times 3}$. Then, we construct a new matrix $B=[A;0]\in ...
0
votes
0answers
7 views

Singular vectors of matrix products

Consider a matrix $M = U V^T$, where $U,V$ are the singular vectors. Is there a way to relate the matrices $U,V$ to the singular vectors of the product $Z = AMB^T$ ?