Tagged Questions
0
votes
1answer
28 views
Intrinsic Definition of Orthogonal/Unitary Transformations
Can someone point me to a good explanation of the intrinsic definition of an orthogonal/unitary transformation? By this I mean one which does not make reference to matrices or matrix operations like ...
1
vote
1answer
40 views
Recovering a set of vectors from their Gram matrix
Suppose $\{ v_1, \ldots, v_k \}$ is a set of vectors in $\mathbb{R}^n$. The associated $k\times k$ Gram matrix is given by
$$
G = [v_i \cdot v_j]_{i,j}
$$
It's (apparently) well known that the Gram ...
2
votes
0answers
75 views
Good introductory book for matrix calculus
Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
0
votes
0answers
19 views
examples of or references for the computation of norm of circulant matrices
I search examples of circulant matrices such that the entries are roots of unity and such that the spectral norm is known.
Recall that the spectral norm of $A$ is the square root of the largest ...
0
votes
1answer
22 views
Eigenvalues for Hessian evaluated at nondegenerate critical points
Let $f \colon M \to \mathbb{R}$ be a smooth function on a manifold $M$. If $f$ is Morse then all the critical points of $f$ are non-degenerate; that is, if $p$ is a critical point of $f$, then $\det ...
1
vote
0answers
17 views
Bibliographic References on the number of invariant subspaces of a linear transformation
I need bibliographic references on the number of invariant subspaces of a linear transformation. Is there any application for counting the invariant subspaces?
2
votes
1answer
48 views
Direct decomposition of vector space in image of map plus kernel of adjoint
Let $A:V\to W$ be a linear map with $V,W$ finite dimensional Hilbert spaces. Is it always true that
$$ \dim(\mathrm{Im}(A)) + \dim(\ker(A^*)) = \dim(W),$$
i.e. (since $\mathrm{Im}(A) \cap \ker(A^*) ...
4
votes
4answers
177 views
Book on matrix computation
I'm taking a machine learning course and it involves a lot of matrix computation like compute the derivatives of a matrix with respect to a vector term. In my linear algebra course these material is ...
3
votes
2answers
68 views
Textbook determinant convention
My text book is called "Linear Algebra and its applications" by David C. Lay.
I am just wondering why the textbook uses the absolute value symbol when it wants us to compute determinants. For ...
2
votes
2answers
90 views
How to prove that $\langle v, v \rangle$ is the only scalar invariant for a vector under the isometry?
Consider an inner product space $V$ and $v \in V$.
It seems that the only scalar invariants a vector can have under the isometry come from $\langle v, v \rangle$. For example $\langle v, v \rangle$, ...
2
votes
1answer
24 views
Skew-triangular (?) matrices and their properties
I'm asking this just out of curiosity because a brief googling failed to give me the answer.
By skew-triangular matrices I mean matrices with this
$$
\begin{bmatrix}
\times & \times & \times ...
3
votes
0answers
99 views
Difference between Gilbert Strang's “Introduction to Linear Algebra” and his “Linear Algebra and Its Applications”?
Could someone please explain the difference between Gilbert Strang's "Introduction to Linear Algebra" and his "Linear Algebra and Its Applications"? Thank you.
1
vote
1answer
82 views
Trace and identity are the only linear matrix invariants?
This question is obviously related to that recent question of mine, but I feel it’s sufficiently different to be posted as a separate question. Let $V$ be a finite-dimensional space. Let ${\cal L}(V)$ ...
3
votes
1answer
49 views
Describe the invariant bilinear maps on the linear group
Apologies if this is a stupid question ; it is at least a natural question. Let $V$ be a finite dimensional space over $\mathbb R$ or $\mathbb C$. Denote by ${\mathcal L}(V)$ the vector space of all ...
1
vote
2answers
119 views
Linear Algebra presentation
Ok, so let me give you the background story.
In my country, unlike the US, there are different high schools for different levels of 'intelligence'. I believe there are 5 or 6 types, the toughest ...
