2
votes
2answers
38 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
1
vote
5answers
85 views

What does it mean for $AA^T$ to be symmetric?

What does it mean for $AA^T$ to be symmetric? A question in my book says to show that $AA^T$ is symmetric so I took a very simple matrix to try and understand this: $A=\begin{bmatrix} 2 \\ 8 \\ ...
5
votes
3answers
61 views

Matrix $A=B+C$ with $B$ symmetric and $C$ antisymmetric

I am stumped on a question and am looking for some guidance on how to get it done. The problem gives you: $x_1 = \begin{bmatrix}9&-4&-2 \\-9&6&-3 \\10&-3&9\end{bmatrix}$ ...
2
votes
3answers
121 views

$A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$

I'm having trouble with this question, I'd like someone to point me in the right direction. let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that ...
1
vote
3answers
133 views

short question regarding convention - symmetric matrices and transpose

I have a short question because wikipedia is extremly vague on this subject. Suppose I have the matrix $A=\begin{pmatrix} i & 1 \\ 1 & -i\end{pmatrix}$. Is it symmetric? I mean, in the ...
0
votes
1answer
122 views

Why inner product on R^n have uniform prototype with symmetric matrix A and positive eigenvalues?

Details of the problems are given below. Assume A is a n*n symmetric matrix. Show that any inner product on R^n has this formula for some symmetric matrix A with all positive eigenvalues. The formula ...
1
vote
0answers
42 views

Constraints on eigenvalues due to the form (symmetry) of Matrix

So I have to deal with a square matrix with complex elements $A$ given by - $$A_{ij}(k) = \underbrace{\frac{1}{2}(c_{ij}+c_ji)}_{\text{symmetric under interchange of $i$ and $j$}} \exp(Ik(i-j))$$ ...
0
votes
3answers
59 views

Showing symmetry involving a matrix and its transposed matrix

I'd appreciate if someone could find a better title for this question, for I'm short of ideas right now. Given a matrix $A \in R^{n,n}$, show that $$ \frac{1}{2}(A + A^t) $$ is symmetric. I see that ...
1
vote
0answers
46 views

Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
4
votes
2answers
282 views

What's the intuition of the transpose of a matrix?

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
0
votes
1answer
133 views

Diagonalizing a matrix with a symmetric matrix

Ok. The question was, find a real matrix $U$ with $U^{-1} = U^T$ Such that $A = UDU^T$ Where $D$ is diagonal matrix. and $$A=\begin{bmatrix}1/2 & -3/2 \\ -3/2 & 1/2\end{bmatrix}$$ I get how ...