2 Symmetric and orthogonal matrices IfT ... det , has eigenvalues and cos sin When we have antisymmetric matrices, we get into complex numbers. And those matrices have eigenvalues of size 1, possibly complex. There’s already few answers provided in this thread. A matrix A is diagonalisable if there is an invertible matrix Q such that QAQ 1 is diagonal. Since you want P and \(\displaystyle P^{-1}\) to be orthogonal, the columns must be "orthonormal". 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.

Then det(A−λI) is called the characteristic polynomial of A. Will just add a proof here for completeness on Quora. Note that A and QAQ 1 always have the same eigenvalues and the same characteristic polynomial. But the magnitude of the number is 1. Some Basic Matrix Theorems Richard E. Quandt Princeton University Definition 1. Homework Statement If a 3 x 3 matrix A is diagonalizable with eigenvalues -1, and +1, then it is an orthogonal matrix. Can't help it, even if the matrix is real. Orthogonal matrix polynomials on the real line First we need to introduce a matrix … Orthogonal Vectors and Subspaces; Projections onto Subspaces; Projection Matrices and Least Squares; Orthogonal Matrices and Gram-Schmidt; Properties of Determinants; Determinant Formulas and Cofactors; Cramer's Rule, Inverse Matrix and Volume; Eigenvalues and Eigenvectors; Diagonalization and Powers of A; Differential Equations and exp(At)

T8‚8 T TœTSince is square and , we have " X "œ ÐTT Ñœ ÐTTќРTÑÐ TќРTÑ Tœ„"Þdet det det det det , so det " X X # Theorem Suppose is orthogonal. And again, the eigenvectors are orthogonal. Let A be a squarematrix of ordern and let λ be a scalarquantity. Orthogonal Matrices#‚# Suppose is an orthogonal matrix.

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It is easy to see that <1, 1> and <1, -1> are orthogonal. Theorem 2 The matrix A is diagonalisable if and only if its minimal polynomial has no repeated roots. Taking eigenvectors as columns gives a matrix P such that \(\displaystyle P^-1AP\) is the diagonal matrix with the eigenvalues 1 and .6. Thanks for the A2A. Orthogonal matrix polynomials We are particularly interested in orthogonal matrix polynomials and we will restrict our attention to orthogonal matrix polynomials on the real line [9] and on the unit circle [5, 8, 10, 28, 35-1. And then finally is the family of orthogonal matrices.