127130 use the following discussion: The loudness L(x), measured in decibels (dB), of a sound of intensity x, measured in watts per square meter, is defined as L(x) = 10 log x I0 , where I0 = 10-12 watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds

4.5 Complex Eigenvalues/Eigenvectors Recall: • Complex eigenvalues always have complex eigenvectors. • Complex eigenvalues occur in conjugate pairs. • Definition: Au= u λhere A is the matrix, u is the nonzero eigenvector associated with the eigenvalue, λ . There are always infinitely many eigenvectors for each eigenvalue. These eigenvectors are multiples of each other. One problem that arises when dealing with complex vectors is that it is difficult to recognize multiples. We have to perform complex arithmetic to see if vectors are multiples. ⎡ 1 ⎤ ⎡ 2 ⎤ ⎡ 5 ⎤ We can easily tell that ⎢ 2 ⎥ , ⎢ 4 ⎥ and ⎢ 10 ⎥ are multiples. ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ i +1 ⎤ ⎡ −1+i ⎤ ⎡ 1+ 3i ⎤ It is difficult, however to recognize that ⎢ ⎥, ⎢ ⎥ and ⎢ ⎥ ⎣ 2i − 3 ⎦ ⎣ −2 − 3i ⎦ ⎣ −8 +i ⎦ ⎡ i +1 ⎤ ⎡ −1+i ⎤ are multiples. But if we check i ⎢ ⎥ = ⎢ ⎥ ⎣ 2i − 3 ⎦ ⎣ −2 − 3i ⎦ and if we check (2+i) ⎡ i +1 ⎤ = ⎡ 1+ 3i ⎤. ⎣ 2i − 3 ⎦ ⎣ −8 +i ⎦ To find eigenvalues, solve det(A- I) λ 0. To find corresponding eigenvectors, reduce [A- I | 0]λ What if we’re given an eigenvector and a matrix and asked for the eigenvalue The above isn’t our only option.