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# HP 48g Series Calculators - Verifying Eigenvalues and Eigenvectors

## Description

Eigenvalues are important in many branches of the physical sciences; they make it possible to find coordinate systems in which the transformation in question takes on a more simple form. A number LAMBDA is called an eigenvalue of a linear transformation of A if there exists a vector x not equal to 0 such that A(x)= (LAMBDA)x. The vector x is then called an eigenvector of the transformation A belonging to LAMBDA.
The following instructions will verify an eigenvalue LAMBDA and a corresponding eigenvector x if Ax - (LAMBDA)x = 0.

## Calculator symbol key

The procedures in this document use the following text to represent symbol keys:
 Key Description Text Representation Right shift key RS Left-shift key LS Move cursor left cursor-left Move cursor right cursor-right Move cursor up cursor-up Move cursor down cursor-down
note:
The menu keys are the top row of white keys, with the corresponding function displayed in the blue boxes at the bottom of the display.

## Instructions

1. Clear the display and go into the Matrix Writer application to enter the square matrix n x n.
2. Enter in the first row of the matrix. Then, press cursor-down to end the first row and enter the rest of the matrix.
3. Press ENTER two times to place the matrix on the stack and make a copy of the original matrix.
4. Compute the eigenvalues and eigenvectors for the square matrix on level one.
5. Separate the matrix into its components onto the stack and drop the number of components off level one
6. Store the eigenvalues from the stack into variables called LAMBDAn through LAMBDA1, (n is the number of columns).
7. Transform the matrix into a series of column vectors, and drop the component count from level one.
8. Store the eigenvectors from the stack into variables called Xn through Xn, (n is the number of columns). Then store the original matrix in variable A.
9. From the stack, perform the math to prove whether the eigenvalues and eigenvector supports the statement Ax - (LAMBDA)x = 0. To aid in recalling the variables back to the stack, press the variable key [ - ]. The variable should appear in the bottom part of the screen. Press the corresponding white menu key, (the top row), to recall the appropriate variable onto the stack. If the result returns a vector with zeros [0 0 0], or very close to zero then the eigenvalues and eigenvector are correct.

## Example of verifying eigenvalues and eigenvectors

1. Clear the display and go into the Matrix Writer.
2. Enter the matrix: 2 4 6 3 6 9 5 4 3.
3. Place the matrix on the stack and make a copy.
4. Compute the eigenvalues and eigenvectors: Press MATR, then EGV or E, G, then V.
5. Separate the eigenvalues: Press TYPE, then OBJ.
6. Store the eigenvalues: 3, 2, 1 (Press L to type in the character).
7. Transform the matrix into a series of column vectors: Press MATR, COL, then COL.
8. Store the eigenvectors from the stack into variables called Xn through X1, (n is the number of columns). Then store the original matrix in variable A. X3, X2, X1, then A'.
9. From the stack, press the following keystrokes.
• A, X1, 1, X1.
• If the result returns a vector with zeros, [0 0 0], or very close to zero, then the eigenvalues and eigenvector are correct.
• Repeat the keystrokes for eigenvector 2 and eigenvector 3: Press A, X2, 2, X2 and A, X3, 3, then X3.

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