I know this is one of the most repetitive question but bear with me please. I am trying to gain an intuitive understanding of eigenvectors. I had this example in my mind where there is a matrix A, the columns correspond to features of house such as floors, rooms etc and rows correspond to houses. Now if I compute the eigenvectors of this matrix what information will those eigenvectors give me ?
1306
In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman?
18758
Question:     Let $A$ be a $3 \times 3$ Matrix such that $[-3,4,1]$   is the eigenvector corresponding to eigenvalue $3$, and $[6,-3,2]$ is an eigenvector corresponding to the eigenvalue $2$. If $v$ = $[0,5,4]$, compute $A^{2}v$.      Hint: $[6,-3,2]+2[-3,4,1]$ = $v$   Where I'm stuck:  I know that we can  solve A with the equation:   A = $P^{-1}DP$, where $P$ is the matrix consisting of the eigenvectors and $D$ is the matrix consisting of the eigenvalues.  However, since $A$ is a $3$ x $3$, we should have a $3 \times 3$ matrix $D$ and $ 3 \times 3 $ matrix $P$ as well, consisting of the eigenvalues and eigenvectors respectively - and in this case, we only have two eigenvalues and two eigenvectors. Where can I obtain the other one of each?  Any help will be very appreciated.  Thank you.
18435
Given a column vector $v$ in $\mathbb{R}^n$, what are the eigenvalues of matrix $vv^T$ and associated eigenvectors?  PS: not homework even though it may look like so.
6492
I have been working on this problem for a couple hours and am completely stuck. Any help would be greatly appreciated.  Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenvectors of $A$.
30476
What do you think is the most efficient algorithm for checking whether a graph represented by an adjacency matrix is connected? In my case I'm also given the weights of each edge.   There is another question very similar to mine:   That answer seems to be good, except I don't really understand it. How does repeatedly squaring the matrix give information about its connectivity? There is another an answer that claims eigenvectors also give information about the connectivity of the graph, could anyone explain that as well?   I'm asking this because I don't have the background to understand the answers given, I'm just solving a problem that has to do with these topics. Searching around on google didn't give me an answer either, so hopefully someone can clear it up.
53254
Let $A$ be a real symmetric matrix. Assume $A^{2}v=\lambda Av$ , can we somehow deduce $v$ is an eigenvector of $A$ with $\lambda$ as its corresponding eigenvalue? i.e $Av=\lambda v$. Thanks Edit: Also assume $\lambda \ne 0$ and we do not know if it is singular or nonsingular.
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Is Jordan basis unique? I have a 4x4 matrix, find eigenvectors and one generalized eigenvector, also trying different linearly independent eigenvectors but the matrix P so that PJP^-1 =A only works for certain vector. \begin{pmatrix}-\frac{1}{2}&amp;0&amp;-\frac{1}{2}&amp;0\\ \frac{3}{2}&amp;0&amp;\frac{1}{2}&amp;0\\ -\frac{3}{2}&amp;0&amp;-\frac{3}{2}&amp;-1\\ 1&amp;1&amp;1&amp;1\end{pmatrix}\begin{pmatrix}1&amp;-1&amp;0&amp;-1\\ 0&amp;2&amp;0&amp;1\\ -2&amp;1&amp;-1&amp;1\\ 2&amp;-1&amp;2&amp;0\end{pmatrix}\begin{pmatrix}1&amp;1&amp;0&amp;0\\ -1&amp;0&amp;1&amp;1\\ -3&amp;-1&amp;0&amp;0\\ 3&amp;0&amp;-1&amp;0\end{pmatrix}=\begin{pmatrix}-1&amp;0&amp;0&amp;0\\ 0&amp;1&amp;0&amp;-1\\ 0&amp;0&amp;1&amp;1\\ 0&amp;0&amp;0&amp;1\end{pmatrix}
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I have been studying linear algebra for a while now, and I still can't understand the basic concept of linear transformation and the easy ''translation'' of them the matrices.  I understand that every proof we make can be done with both concepts, and that they are connected pretty close, and there is a basis which ''moves'' us from one concept to another.  If anyone can write a brief explanation, or some reading material, It would be a big step for me for further understanding of the subject.  Thanks,  Alan
57446
For a given data matrix $A$ (with variables in columns and data points in rows), it seems like $A^TA$ plays an important role in statistics. For example, it is an important part of the analytical solution of ordinary least squares. Or, for PCA, its eigenvectors are the principal components of the data.  I understand how to calculate $A^TA$, but I was wondering if there's an intuitive interpretation of what this matrix represents, which leads to its important role?
9485
Define $T : M_{n×n}(\mathbb{R}) → M_{n×n}(\mathbb{R})$ by $T(A) := A^t$.   I know that the corresponding eigenvalues are $+1$ and $-1$, but I'm not sure how to find the eigenvectors of this transformation, in the case of a $2\times 2$ matrix it's simple, but not in the general case.
78274
The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding to this singular value?   Usually I found the columns of U with this relation, so let's say that we refer to $\vec{u_2}$ as the U column vector corresponding to singular value equal to 0:  $\vec{u_2} = (1/s_2)Av_2$  Now the as we can see the 0 is in an illegal place, since you can't divide by zero. But if you move it to the left side, I still can't quite see how to find the column of U corresponding to it. Any suggestions?  Here is an example of a matrix that provided this specific challenge  $ A = \left(\begin{array}{rrr} -3 &amp; 1 \\ 6 &amp; -2 \\ 6 &amp; -2 \end{array}\right). $  $ \lambda{_1} = 90$,  $\lambda{_2} = 0$,  $s_1 = 3 \sqrt{10}$,  $s_2 = 0$,  How can one find $\vec{v_2}$ when $\lambda{_2} = 0$? In this case, you know that the eigenvectors of $A^{t} A$ are orthogonal, solve for $\lambda{_2}$ and find the eigenvector if it's orthogonal to the eigenvector corresponding to the first eigenvalue you have your solution.  So we end up with the following normalized column vectors for V, which we can see are orthogonal:  $ \vec{v_1} = \left(\begin{array}{rrr} -3/\sqrt{10} \\ 1/\sqrt{10} \\ \end{array}\right). $  $ \vec{v_2} = \left(\begin{array}{rrr} 1/\sqrt{10} \\ 3/\sqrt{10} \\ \end{array}\right). $
95701
I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman.  Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to understand.I found that there are many application of eigenvalue and eigenvector in multivariate analysis. For example     In principal components, proportion of total population variance due   to kth principal component equal    $$\frac{\lambda_k}{\lambda_1+\lambda_2+...\lambda_k}$$   I think eigenvalue product corresponding eigenvector has same effect as the matrix product eigenvector geometrically.   I think my former understanding may be too naive so that I cannot find the link between eigenvalue and its application in principal components and others.   I know how to induce almost every step form the assumption to the result mathematically. I'd like to know how to intuitively or geometrically understand eigenvalue and eigenvector in the context of multivariate analysis(in linear algebra is also good).  Thank you!
54959
so I understand row space of a matrix X to mean the subspace of $R^n$ spanned by the row vectors of A  Here is the text that is in question:    Can the rows of B be written as a linear combination of A just because they are row-reduced from A?  Say we have matrix:  $$\begin{bmatrix} 1 &amp; 2 \\ 3 &amp; 8 \end{bmatrix}$$ $$ -&gt; \begin{bmatrix} 1 &amp; 2 \\ 0 &amp; 2 \end{bmatrix}$$  How is the row-reduced matrix a linear combination of A?
43154
Consider a set $X=(X_1; \dots; X_n)$ of $n$ data points such that $X_i \in \mathbb{R}^d$ is a column vector. Let $Y = \text{pca_proj}(X)$ denote the projection of points in $X$ according to the PCA components i.e.  $$Y_i \in \mathbb{R}^d\\ Y_i = W X_i$$  where $W \in \mathbb{R}^{d \times d}$ is the projection matrix obtained from PCA, and each row of $W$ is an eigen vector of $X^T X$.   Now let's define $Z = \text{pca_proj}(Y)$. My questions is: does $Y = Z$? If not, what happens if we keep applying PCA recursively on a matrix? Does it converge to a specific matrix?
70255
I'm a student in an elementary linear algebra course. Without bashing on my professor, I must say that s/he is very poor at answering questions, often not addressing the question itself. Throughout the course, there have been multiple questions that have gone unanswered, but I must have the answer to this one question.  "Why are some vectors written as row vectors and others as column vectors?"  I understand that if I transpose one, it becomes the other. However, I'd like to understand the purpose behind writing a vector in a certain format.  Taking examples from my lectures, I see that when I'm trying to prove linear independence of a group of vectors, the vectors are written as column vectors in a matrix, and the row reduced form is found.  Other times, like trying to find the cross product or just solving a matrix, I see the vectors written as row vectors.  My professor is very vague on the notations and explanations, and it bugs me as a person who needs to know the reason behind every small thing, why this variation occurs in the format. Any input is greatly appreciated.
58133
Firstly, I know this question might seem slightly vague, but I can't find any examples or specific questions to specify it.   My question is in regards to finding the Jordan Canonical Form of a matrix, more specifically when you have two Jordan chain of length greater than $1$ for a particular eigenvalue. After you've found the eigenvectors, are these eigenvectors always linearly independent? Also, if they are, is it possible to find eigenvectors which form Jordan chains which are 'wrong' because they aren't linear independent?
60439
I try to understand and visualize myself covariance matrix. Supposing I have a matrix A = [ 2 3 4; 5 5 6 ], how do I calculate its covariance matrix, and what is its practical meaning? (All I was able to understand by now is that on the diagonal of the covariance matrix, variances for particular variables are placed, and on the upper and lower fields correlations between those variables.)