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), ...
2
votes
2answers
126 views
simplify this expression
I want to know how to simplify the following expression by using the fact that $\sum_{i=0}^\infty \frac{X^i}{i!}=e^X$. The expression to be simplified is as follows:
$$\sum_{i=0}^{\infty} ...
3
votes
1answer
116 views
$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant
Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the ...
1
vote
3answers
129 views
Some properties of a $2\times 2$ matrix with repeated eigenvalues
I got a problem in my exam
Consider the matrix $ A =\left(
\begin{array}{cc}
a & b \\
c & ...
1
vote
1answer
138 views
What is this linear operator/matrix?
I have a linear operator with its matrix in certain coordinates to be
$$
\begin{pmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & \frac{1}{2} & 0 ...
21
votes
3answers
406 views
Multiplying by a $1\times 1$ matrix?
For matrix multiplication to work, you have to multiply an $m \times n$ matrix by an $n \times p$ matrix, so we have $$\bigg(m \times n\bigg)\bigg( n\times p \bigg).$$
But what about a $1 \times 1$ ...
1
vote
4answers
116 views
Determinant of a matrix $A$ is zero when its has a zero submatrix of dimentions $p \times q$ and …
Let $A$ be a $n \times n$ matrix and suppose $A$ has a zero submatrix of order $p \times q$ where $p + q \ge n+1$. Then $\det(A) = 0$.
I can see this happening when doing Laplace expansion. I can ...
0
votes
1answer
88 views
Counting number of linear transformations
Let $v_{1} = (1, 0)$, $v_{2} = (1, -1)$ and $v_{3} = (0, 1)$. How many linear transformations $T :\mathbb {R^2}\rightarrow \mathbb {R^2} $ are there such that $T(v_{1} ) = v_{2}$, $T(v_{2} ) = ...
1
vote
2answers
70 views
Checking diagonalizability of a given $2\times 2$ matrix
Let $A$ be the matrix $ A = \left(
\begin{array}{cc}
a & c\\
0 & a \\
...
0
votes
0answers
161 views
Normal matrices
Let $A = (a_{ij})$ be an $n \times n$ normal matrix. Let $B = (b_{ij})$ be an $m \times m$ normal matrix. Consider the $(n + m) \times (n + m)$ matrix $C = (c_{ij})$ with entries given by the matrix: ...
3
votes
1answer
950 views
Matrix for rotation around a vector
I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
2
votes
1answer
119 views
Linear Mapping/Matrices Proof
At first look a rather logical question which has till date stumped many of us attempting to solve it. Hmm, hope you guys could offer some brain power here :)
$A$ is a matrix from $\mathbb{R}^{2,2}$, ...
3
votes
2answers
120 views
Finding the dimension of a given vector space
What is the dimension of the space of all $n \times n$ matrices with real entries
which are such that the sum of the entries in the first row and the sum of
the diagonal entries are both zero?
...
2
votes
2answers
911 views
Finding the dimension of real symmetric matrices with trace zero
What is the dimension of the vector space of all symmetric matrices of order $n\times n$ $(n\geq 2)$ with real entries and trace equal to zero?
0
votes
1answer
4k views
orthogonal eigenvectors
I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see ...
3
votes
2answers
101 views
How to find out the dimension of a given vector space?
What will be the dimension of a vector space $ V =\{ a_{ij}\in \mathbb{C_{n\times n}} : a_{ij}=-a_{ji} \}$ over field $\mathbb{R}$ and over field $\mathbb{C}$?
0
votes
0answers
147 views
Householder Transformation
Let $\mathbf{a}\in\mathbb{R}^{n}$
be a non-zero vector. Develop a numerically stable procedure to compute a Householder transformation P such that $$P\mathbf{a}=\left(\begin{array}{c}
...
1
vote
1answer
62 views
Nonlinear system
We are given a non-linear system:
$4x_1 − x_2 + x_3 = x_1x_4,$
$−x_1 + 3x_2 − 2x_3 = x_2x_4$
$x_1 − 2x_2 + 3x_3 = x_3x_4$
$x_1^2 + x_2^2 + x_3^2 = 1$
And the question asks:
Show how to solve the ...
3
votes
1answer
147 views
Calculating the inertia of a real symmetric (or tridiagonal) matrix
I'm trying to find a quick method for evaluating the inertia of a real symmetric matrix, though I don't need to evaluate eigenvalues directly. The inertia of a matrix is a triple of the number of ...
2
votes
1answer
97 views
eigenvector computation
Given a full-rank matrix $X$, and assume that the eigen-decomposition of $X$ is known as $X=V \cdot D \cdot V^{-1}$, where $D$ is a diagonal matrix.
Now let $C$ be a full-rank diagonal matrix, now I ...
3
votes
1answer
109 views
If $null(A) \subset null(B)$ can we draw any conclusion about range spaces of A and B
A and B are given $n\times$ m matrices
If $null(A) \subset null(B)$ what conclusion can we draw about range Space of $A$ and $B$.
Can we conclude that range space of B is contained in a range space of ...
1
vote
1answer
227 views
difference between matrices
Is it possible to compute a distance between two matrices of different rank, and different dimension?
In particular I'm interested in the following case. Suppose $[K]_{ij}=\exp[-(x_i-x_j)^2]$, and ...
2
votes
1answer
241 views
Derivation of the derivative of a square matrix w.r.t. a vector
So I have gotten stumped on something that seems like it (should?) be easy. I am trying to find the following derivative shown below. I have scoured the wiki link on matrix derivatives, and I think my ...
1
vote
1answer
89 views
Matrix-valued ODE - nonsingularity of solution
I have a matrix-valued inhomogenous linear ODE
$X' = F(t)X + G(t)$, $X(0) = I_{n \times n}$,
$F(t),G(t) \in \mathbb{R}^{n \times n}$,
and the entries of $f$ and $g$ are continuous functions. What ...
1
vote
1answer
103 views
Gradient vector function using sum and scalar
Could someone take a look on my attempt to compute the gradient for:
$$f(x) = \lambda \sum_{x = 1}^n g(x_i)$$
Where $x \in \mathbb{R^d}$, $\lambda \in \mathbb{R}$ and
$$g(x_i) = \begin{cases}
x_i - ...
3
votes
3answers
168 views
Proving two results about the spectral radius
How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces?
Theorem 1.
Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon ...
3
votes
3answers
111 views
What is an example of a linear function that maps a matrix to a scalar? What makes it a 'function'?
I suppose this is part terminology question and part math, but I am trying to untangle what we mean when we say "The linear function $f$ maps $R_{m\times n}$ space to $R_{m}$ space", and "The linear ...
3
votes
1answer
379 views
What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?
What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?
The rotation matrix is a skew-symmetric matrix when $\theta$ is some multiple of $\frac{\pi}{2}$. But it cannot be ...
0
votes
3answers
88 views
Prove that a space is a subspace
I have 2 subspaces of $M_2({\bf R})$:
$U =\left\{ \pmatrix{a&b\cr c&d\cr} : c \ge 0 \right\}$
$V = \left\{ \pmatrix{a&b\cr c&d\cr} : c + 2d = 0, a + b - 2c = 0 \right\}$
I need to
...
3
votes
3answers
171 views
What does an inverse matrix abstracts?
I am trying to understand inverse matrixes more in depth.
I took the simplest example: 2 points in a 2d space and put it into a matrix.
5 7
-2 3
Calculating the ...
3
votes
3answers
107 views
Find the inverse of a $4\times4$ matrix
My matrix looks like this:
$$\left(\begin{array}{rrrr}
1& 1 & 1 & 1\\
1& -1 & 1 & 0\\
1& 1 & 0 & 0\\
1& 0 & 0 & 0
...
2
votes
0answers
125 views
Average transformation matrix?
I have several estimates of the transformation matrix between two planes and some values that give some indication of the error involved in the estimate.
How can I use this information to gain the ...
1
vote
1answer
312 views
Conditional Probability Matrices
If I have a conditional probability matrix for binary variables $A,B,C$ with entries of the form
$$
P(A | B \cap C) =
\left(
\begin{matrix}
P(a_1 | b_1 \cap c_1) & P(a_1| b_1 \cap c_2) & ...
4
votes
3answers
788 views
Can the product of two non-zero symmetric matrices be anti-symmetric?
I'm trying to find an example to show that the product of two non-zero symmetric matrices can be anti-symmetric.
I've proven that this is impossible for 2x2 matrices.
For 3x3 matrices, I've ...
1
vote
1answer
25 views
Determine matrix of a set in a certain base
I have a set
$S = \{ x^2 + 1, x + 1, 1 - x, x^3 \}$
in a polynomial vector space.
How do I write a vector matrix of $S$ in the base $B = \{ 1, x, x^2, x^3 \}$?
I attempted this using the formula: ...
1
vote
0answers
779 views
What is the algorithm for LU factorization in MATLAB?
What is the algorithm for LU factorization in MATLAB, i.e. [L,U] = lu(a)?
After searching for many examples and trying to compare the result with MATLAB,
they are ...
2
votes
1answer
98 views
Find two vectors that don't belong to a vector space
I have a vector space $L$, which is a subspace of $\mathbb{R}^4$, spanned by these vectors:
$$(4, 1, 1, 2), (2, 3, 1, 0), (-10, 35, 5, -20), (2, 13, 3, -4).$$
I need to find two vectors from ...
1
vote
1answer
95 views
Vector derivative with inner function
I want to compute the gradient for the following function:
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2 + \sum_{j = 1}^k l(\beta_j)$$
where $l(\beta_j) = \begin{cases}
\beta_j - ...
1
vote
0answers
83 views
Calculation of stopping condition for Conjugate Gradient
I am a person with programming background and need some math help.
I am looking at the source code for an implementation of the Conjugate Gradient iterative solver ...
1
vote
0answers
126 views
Minimizing L1 Regularization
I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
2
votes
4answers
3k views
Matrices: Anyone have a real-life problem that uses matrices / linear systems of equations?
Looking for something beyond a contrived textbook problem concerning jelly beans. Not just matrix manipulation for it's own sake.
I know matrix math is used in real life applications (finance, ...
1
vote
0answers
52 views
Gradient of a dot product with a matrix
I have a function:
$$ f(\vec x) = \frac 12 \vec x \cdot (A \vec x) - \vec x \cdot \vec b $$
Now I have to find the gradient of the function: $\vec \nabla f(\vec x)$.
Is there some easy way to do ...
4
votes
3answers
168 views
Rings with isomorphic proper subrings
Rings will be unital here but I don't require that subrings share the identity elements with superrings.
I just accidentally came up with an example of a ring $R$ with a proper subring $S$ such that ...
1
vote
1answer
39 views
Vector derivative with power of two in it
I want to compute the gradient of the following function with respect to $\beta$
$$L(\beta) = \sum_{i=1}^n (y_i - \phi(x_i)^T \cdot \beta)^2$$
Where $\beta$, $y_i$ and $x_i$ are vectors. The ...
1
vote
1answer
208 views
How do I prove that if $\det(A) < 0$, then $A\in\mathbb R^{2\times 2}$ is a diagonalizable matrix?
Suppose $A$ is a $2\times2$ matrix. How do I prove that, if $\det(A) < 0$, then $A$ is a diagonalizable matrix over $\mathbb{R}$?
1
vote
2answers
78 views
Determining the dimension and a basis for a vector space
I have the following problem:
Let $W$ be a vector space of all solutions to these homogenous equations:
$$\begin{matrix} x &+& 2y &+& 2z &-& s &+& 3t ...
8
votes
4answers
1k views
Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]
Possible Duplicate:
Units and Nilpotents
Hi all wondering if I could get a bit of help with this, given $A^{2012}=0$ prove $(A+I)$ is invertible and find an expression for $(A+I)^{-1}$ in ...
3
votes
1answer
555 views
Are complex determinants for matrices possible and if so, how can they be interpreted?
I've been asked to compute the determinant of a 3x3 matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My ...
0
votes
1answer
99 views
What Does It Say About the Eigenvectors of a Matrix if…
What does it say about the eigenvectors of a matrix $A$ if the row-reduced form of the characteristic polynomal in coefficient matrix form has a row of 0's?
I know that it indicates something about ...
1
vote
2answers
162 views
Linear Algebra - Finding Eigenvalues of a Matrix
$A=\begin{bmatrix}3 & -2 & 5\\ 1 & 0 & 7\\ 0 & 0 & 2\end{bmatrix}$, Find the eigenvalues of A.
I realized that if I swap columns I and II then I can make it an upper ...
1
vote
2answers
294 views
Determinant of a 3x3 matrix with 6 unknowns given the determinants of two 3x3 matrices with same unknowns?
Given:
$$
det(A) = 3 \\ det(B) = -4
$$
$$
A =
\begin{pmatrix}
a & b & c \\
1 & 1 & 1\\
d & e & f
\end{pmatrix} \\
B =
\begin{pmatrix}
a & b & c \\
1 & 2 & 3 \\
...
