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### When does a real symmetric matrix have $LDL^{T}$ decomposition? And when is the $LDL^{T}$ decomposition unique?

Let us answer the questions one by one: Point1: The factorization $LDU$ relies on the factorization $LU$ and then we use $D$ to make $U$ as an upper triangular matrix with $1$'s in the main diagonal. ...
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### Decomposition for non symmetric matrix using left and right eigenvectors

For a square matrix $A$, $x$ is a right-eigenvector if $Ax = \lambda x$ for some scalar $\lambda$ and a left-eigenvector if $A^Tx = \lambda x$ for some scalar $\lambda$. $A^T$ denotes the transpose of ...
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The sticking point in my proof is the assumption that the 𝑅 matrices are invertible (or simply that they have nonzero diagonals). Notice for any choice of $k$ $0\lt \det\big(A^T A\big) = \det\big(... • 7,743 1 vote Accepted ### To compute Hermite Normal Form H and U To undestand how the process works, it is best illustrated with a non-squared matrix. As square matrices are easier to work with, it doesn't show the full extend of the process. Process: The intuitive ... • 116 1 vote ### Gram matrix of QR decomposition The projection$\hat x$of$x$onto the image of$A$is $$\hat x = A(A^T A)^{-1} A^T x. \tag{1}$$ so$y = x - 2 A(A^T A)^{-1} A^T x $. This is a well known formula in the topic "linear ... • 5,848 0 votes Accepted ### Simple eigendecomposition of matrix In general, you can't. We know the eigenvectors of$\bf P D P^{-1}$are the columns$\bf v_i$of$\bf P$,$i = 1, \ldots, n$. Let's call the corresponding eigenvalues$\lambda_i$. Now, observe that, ... • 330 0 votes ### Cholesky decomposition of the inverse of a matrix To add to previous answers, if we view$X$as covariance matrix of data, the relationship between two decompositions reduces to relationship between coefficients of "right-to-left" ... • 4,372 1 vote ### Show that no$n \times n$real matrix$A^2$can be a diagonal matrix with distinct negative entries Here is a simpler proof that requires only that$a_1$be different from the other$a_j$'s. Supposing that such a matrix$A$exists, let$p$be a polynomial such that$p(-a_1)=1$, and$p(-a_j)=0$, for ... • 17.6k 2 votes ### Show that no$n \times n$real matrix$A^2$can be a diagonal matrix with distinct negative entries$A^2=\begin{bmatrix} -a_1 \\ &\ddots \\ & & -a_n\end{bmatrix}$implies, working over$\mathbb C$, that each eigenvalue of$A$satisfies$\lambda_k^2 \lt 0$, i.e. each eigenvalue must be ... • 7,743 3 votes ### Show that no$n \times n$real matrix$A^2$can be a diagonal matrix with distinct negative entries Let's suppose that $$A^2=\begin{pmatrix} -a_1 \\ &\ddots \\ & & -a_n\end{pmatrix}$$ Then the characteristic polynomial of$A^2$is$P(X)= (-1)^n\prod_{k=1}^n (X+a_k)$, so by Cayley-... • 24.7k 3 votes Accepted ### Could the product of a skew-symmetric matrix and an invertible matrix be nilpotent? In fact, for any matrix$A$, skew-symmetric or otherwise, there is some invertible matrix$B$such that$AB$is nilpotent if and only if$A=0$or$A$is not invertible. First and foremost, it's an ... 1 vote Accepted ### Rotation matrix decomposition into mixed local/global Euler angles In any 3 angle rotation scheme, there are actually four reference frames at play. Lets define: Frame Description$G$Global frame before rotations$L_1$Local frame after 1st rotation$L_2$Local ... • 86 1 vote Accepted ### Cholesky inverse To solve$PDP^T = LXL^T$with matrices as specified, compute the diagonal matrix$R$such that$RR^T = D$by taking the square roots of the diagonal entries. then you have:$PRR^TP^T = LCC^TL^T\$ where ...
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