(2) Ais orthogonally diagonalizable: A= PDPT where P … A is a symmetric 2 × 2 matrix. The rst step of the proof is to show that all the roots of the characteristic polynomial of A(i.e. A is a general 2 × 2 matrix. The matrix U is called an orthogonal matrix if UTU= I. A proof of the spectral theorem for symmetric matrices (Optional) Math 419 In class we have covered - and by now seen some applications of - the following result Theorem 1 (The spectral theorem { for symmetric matrices).

It is clear that the characteristic polynomial is an nth degree polynomial in λ and det(A−λI) = 0 will have n (not necessarily distinct) solutions for λ. Let λi 6=λj. Let Abe a real, symmetric matrix of size d dand let Idenote the d didentity matrix. A symmetric matrix is a square matrix that satisfies A^(T)=A, (1) where A^(T) denotes the transpose, so a_(ij)=a_(ji). The matrix product uTAv is a real number (a 1×1 matrix). I To show these two properties, we need to consider complex matrices of type A 2Cn n, where C is the set of the eigenvalues of A) are real numbers. It is a beautiful story which carries the beautiful name the spectral theorem: Theorem 1 (The spectral theorem). Real symmetric matrices 1 Eigenvalues and eigenvectors We use the convention that vectors are row vectors and matrices act on the right. a. It is a beautiful story which carries the beautiful name the spectral theorem: Theorem 1 (The spectral theorem).

The proof of this is a bit tricky.

A proof of the spectral theorem for symmetric matrices (Optional) Math 419 In class we have covered - and by now seen some applications of - the following result Theorem 1 (The spectral theorem { for symmetric matrices). For column vectors v;wthe inner product is de ned in terms of transpose and matrix multiplication: hv;wi= vTw.

this theorem is saying that eigenvectors of a real symmetric matrix that correspond to different eigenvalues are orthogonal to each other under the usual scalar product. Theorem: Any symmetric matrix 1) has only real eigenvalues; 2) is always diagonalizable; 3) has orthogonal eigenvectors. Some Basic Matrix Theorems Richard E. Quandt Princeton University Definition 1. Corollary: If matrix A then there exists Q TQ = I such that A = Q ΛQ. Theorem: Any symmetric matrix 1) has only real eigenvalues; 2) is always diagonalizable; 3) has orthogonal eigenvectors. So this proof shows that the eigenvalues has to be REAL numbers in order to satisfy the comparison. Properties of real symmetric matrices I Recall that a matrix A 2Rn n is symmetric if AT = A. I For real symmetric matrices we have the following two crucial properties: I All eigenvalues of a real symmetric matrix are real. Where possible, determine the unknown matrix that solves the following matrix equations. I Eigenvectors corresponding to distinct eigenvalues are orthogonal.

This implies that UUT = I, by uniqueness of inverses. Linear Algebra 101 — Part 7: Eigendecomposition when symmetric.

symmetric matrix A, meaning A= AT. Perhaps the most important and useful property of symmetric matrices is that their eigenvalues behave very nicely. Then Av = … Question 10.3. A skew-symmetric matrix is determined by (−) scalars (the number of entries above the main diagonal); a symmetric matrix is determined by (+) scalars (the number of entries on or above the main diagonal). To avoid being too abstract we will rely on coordinates for the following two de ni-tions. Theorem 2. De nition 1 Let U be a d dmatrix. Proposition 4 If Q is a real symmetric matrix, its eigenvectors correspond-ing to different eigenvalues are orthogonal.

If Ais an n nsym-metric matrix then (1)All eigenvalues of Aare real.


Then det(A−λI) is called the characteristic polynomial of A. In order to be symmetric then A = A T then K = A A and since by definition we have that K = A n is symmetric …