Why is orthogonal basis useful? The special thing about an orthonormal basis is thatit makes those last two equalities hold. With an orthonormal basis, the coordinate representations have the same lengths as the original vectors, and make the same angles with each other. ...
What is the determinant of an orthogonal matrix? What makes a square matrix singular? What is the determinant of the transpose of a matrix? If A = \bigl(\begin{smallmatrix} 1 & 2\\ 2 & 1 \end{smallmatrix}\bigr) , find a matrix S such that S^{-1}AS is a diagonal matrix ...
Given an {eq}n\times n {/eq} matrix {eq}A {/eq} the image of the matrix is the image of the linear transformation {eq}T_A:\mathbb{R}^n\to... Learn more about this topic: Multiplicative Inverse of a Matrix | Overview & Examples ...
As astounding as it may still seem to many, Bell’s theorems do not prove nonlocality. Non separable multipartite objects exist classically, meaning w
This minor of a GUE matrix is basically again a GUE matrix, so the above theorem applies verbatim to the ; but it turns out to be necessary to control the joint distribution of the and , and also of the interlacing gaps between the and . For fixed energy, these gaps are in principle...
This combines well with the mathematical distinction between vector, scalar, and matrix quantities, which among other things prohibits one from adding together two such quantities if their vector or matrix type are different (e.g. one cannot add a scalar to a vector, or a vector to a matrix...
What makes the computation a little less straightforward is the fact the we are usually not used to view matrix-vector products as linear maps in but in . So let’s rewrite the thing: There are two particular notions which come in handy here: The Kronecker product of matrices and the ...
A Closer Look at PCA (Principal Component Analysis) Let’s examine the model to help us further describe PCA. It assumes that we have a high-dimensional representation of data that is, in fact, embedded in a low-dimensional space. We assume thatL≈XVwhereLis some low-rank matrix,Xis the...
Suppose that A is a matrix with real entries. Let v and w be right eigenvectors for different eigenvalues of A. Prove or disprove that v is orthogonal to w. Find the determinants of the matrix [A] = [4 7 8 -3]. Suppose that A, B, and C are 3 times 3 matrices with det(A)...
For the intended application to GUE hives, it is important to not just control gaps of the eigenvalues of the GUE matrix , but also the gaps of the eigenvalues of the top left minor of . This minor of a GUE matrix is basically again a GUE matrix, so the above theorem applies verbatim...