Complex Eigenvalues Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Numbers When solving for the roots of a quadratic equation, real solutions can not be found when the discriminant is negative. In these cases we use complex numbers to write down the solutions. For example, if we try to solve x2-2x+2=0 we run into trouble: With the definition i2=-1, we can write the solutions as COMPLEX numbers, with a REAL part and an IMAGINARY part. In general, complex numbers look like a+bi, where a and b are real. We call a the “real part” and b the “imaginary part” of the complex number. Every complex number has a “conjugate” a-bi. Multiplying conjugates yields a real number: (a+bi)(a-bi)=a2+b2. This is a very useful trick! Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: We will find the eigenvalues for this matrix. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: We will find the eigenvalues for this matrix. Next we find eigenvectors for the eigenvalues. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: At this point we can write down the equation and find the vector. We could do row reduction but it’s not really necessary (multiplying one row by the conjugate of its complex entry should make it clear that the rows are multiples of each other. Let’s use the second row: Here is an eigenvector for λ=1+4i Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: We could repeat the same work to find the other eigenvector, but we don’t need to. Since the e-values are conjugates, so are their eigenvectors. Here are the eigenvalues with their eigenvectors. At this point we could use our diagonalization technique to obtain a diagonal matrix with the (complex) eigenvalues on the diagonal, just like we did when the eigenvalues were real. Instead, we will get a matrix C that shows the rotation and stretch that we mentioned previously. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: We form matrix P whose columns are the real and imaginary parts of the eigenvector. The book uses the e-value with the negative imaginary part, so we will do the same. The process will also work if you choose the other e-value. When we use this matrix to do a similarity transformation on the original matrix A (just like we did for diagonalization) we get a new matrix C that will have the stretch and rotation properties we are looking for. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Use the shortcut for 2x2 inverse matrices Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Use the shortcut for 2x2 inverse matrices For eigenvalue a-bi, the C matrix will always take the form Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Matrix C is similar to Matrix A. It has the same eigenvalues. Matrix C splits into 2 parts. One is just a stretch (multiplication by the length of the eigenvalue). The other part is a rotation. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Matrix C is similar to Matrix A. It has the same eigenvalues. Matrix C splits into 2 parts. One is just a stretch (multiplication by the length of the eigenvalue). The other part is a rotation. In our example, the eigenvalue is 4+1i, so r=√17 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Matrix C is similar to Matrix A. It has the same eigenvalues. Matrix C splits into 2 parts. One is just a stretch (multiplication by the length of the eigenvalue). The other part is a rotation. In our example, the eigenvalue is 4+1i, so r=√17 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues When a matrix has complex eigenvalues they will always come in conjugate pairs. The effect of multiplying by the matrix will be a stretch and a rotation. The example below should help clarify the situation: Matrix C is similar to Matrix A. It has the same eigenvalues. Matrix C splits into 2 parts. One is just a stretch (multiplication by the length of the eigenvalue). The other part is a rotation. In our example, the eigenvalue is 4+1i, so r=√17 So matrix C will stretch a vector by √17, and rotate it counter-clockwise by 76°. This will correspond to an outward spiral when we look at the effect of matrix A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB