You are watching: In each of problems 16 through 25, find all eigenvalues and eigenvectors of the given matrix.
The eigenvalue problem is to identify the solution to the equation Av = λv,where A is one n-by-n matrix, v isa shaft vector of size n, and λ isa scalar. The worths of λ that meet theequation are the eigenvalues. The equivalent values of v thatsatisfy the equation room the best eigenvectors. The left eigenvectors, w,satisfy the equation w’A = λw’.
e = eig(A,B) returnsa tower vector containing the generalised eigenvalues that square matrices A and B.
The generalized eigenvalue difficulty is to identify the solutionto the equation Av = λBv,where A and also B are n-by-n matrices, v isa obelisk vector of size n, and also λ isa scalar. The worths of λ that fulfill theequation are the generalized eigenvalues. The matching valuesof v room the generalized right eigenvectors. Theleft eigenvectors, w, meet the equation w’A = λw’B.
<___> = eig(A,balanceOption), where balanceOption is "nobalance", disables the preliminary balancing action in the algorithm. The default for balanceOption is "balance", which enables balancing. The eig duty can return any type of of the output arguments in ahead syntaxes.
<___> = eig(A,B,algorithm),where algorithm is "chol", usesthe Cholesky administer of B to compute thegeneralized eigenvalues. The default because that algorithm dependson the properties of A and also B,but is typically "qz", which offers the QZ algorithm.
If A is Hermitian and also B isHermitian hopeful definite, climate the default because that algorithm is "chol".
<___> = eig(___,outputForm) return the eigenvalues in the form specified by outputForm using any kind of of the input or output debates in previous syntaxes. Clues outputForm together "vector" to return the eigenvalues in a column vector or together "matrix" come return the eigenvalues in a diagonal line matrix.
A = 4×4 1.0000 0.5000 0.3333 0.2500 0.5000 1.0000 0.6667 0.5000 0.3333 0.6667 1.0000 0.7500 0.2500 0.5000 0.7500 1.0000
V = 3×3 complex -0.5774 + 0.0000i 0.5774 + 0.0000i 0.5774 + 0.0000i -0.5774 + 0.0000i -0.2887 - 0.5000i -0.2887 + 0.5000i -0.5774 + 0.0000i -0.2887 + 0.5000i -0.2887 - 0.5000i
D = 3×3 complex 6.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -1.5000 + 0.8660i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -1.5000 - 0.8660i
ans = 3×3 complex10-14 × -0.2665 + 0.0000i -0.0888 - 0.0111i -0.0888 + 0.0111i 0.0888 + 0.0000i 0.0000 + 0.0833i 0.0000 - 0.0833i -0.0444 + 0.0000i -0.1157 + 0.0666i -0.1157 - 0.0666i
Ideally, the eigenvalue decomposition satisfies the relationship. Since eig performs the decomposition utilizing floating-point computations, climate A*V can, in ~ best, strategy V*D. In various other words, A*V - V*D is near to, but not exactly, 0.
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By default eig does not always return the eigenvalues and eigenvectors in sorted order. Use the sort role to placed the eigenvalues in ascending order and also reorder the corresponding eigenvectors.