I am currently in a Modern Systems Controls Class, a Vibration Analysis class, a

Question

I am currently in a Modern Systems Controls Class, a Vibration Analysis class, and other linear algebra heavy classes (despite lin alg not being a prerequisite...)

The math is pretty similar and the concepts are almost the same

(n=1 is the smallest to incrementing to the biggest)

(and for clarification, i is row and j is column of matrix P, so I place the eigenvectors vertically in matrix P)

My problem is more fundamental...
I have seen these two equations pop up all the time, dealing with eigenvector transformation matrices...

Here's more knowledge that I know from my experience in all my class, and I appreciate it if you could fill in the blanks, correct it, and expand.

P is orthogonal (and I know what orthogonally is)
P is square

Also in linear algebra they teach you that, you guaranteed that there is a unique solution to x

Explanation / Answer

1) When you talk of P-1AP, you are talking in general about the diagonalisation of Matrix A.

Take the matrix A, Find its eigenvalues and then its eigenvectors. The eigenvectors can be used to construct the Matrix P by using each eigenvectors of A as the column of the Matrix P.

After we find P, we compute its inverse by usual use of minors and determinants. After that if we compute P-1AP, we end up with a matrix, which is in a diagonal form. This works in general for any invertible matrix.

Now consider an orthogonal matrix. For example, consider a rotation in two dimensions.

The matrix is given by:

If we take such a matrix P and take its transpose, we will find PTP=1. So its this case the inverse is like the Transpose of the original matrix. So like the previous case, now PTAP will now carry the operation of Diagonalising A.

2) The second question is nothing but using a matrix and operating it with a vector state x.

P-1 will come in general, but for orthogonal case, PT will come instead of P-1.

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