Question

The sum of the squares of the eigen values of \(\left[\begin{array}{lll}3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5\end{array}\right]\) is ........

Short answer

The sum is 38.

Step-by-step solution

Identify Eigenvalues

For an upper triangular matrix, the eigenvalues are the elements on the main diagonal. Therefore, the eigenvalues for the matrix \[\begin{bmatrix}3 & 1 & 4 \0 & 2 & 6 \0 & 0 & 5\end{bmatrix}\]are 3, 2, and 5.

Calculate Squares of the Eigenvalues

Square each eigenvalue individually: \(3^2 = 9\), \(2^2 = 4\), \(5^2 = 25\).

Sum the Squares of the Eigenvalues

Add the squared eigenvalues: \(9 + 4 + 25 = 38\).

Key concepts

Upper Triangular Matrices

Understanding upper triangular matrices can significantly simplify solving problems involving eigenvalues. Upper triangular matrices are special because all elements below the main diagonal are zero.
This creates a clear structure where the entries on its diagonal have a distinct role.

Here's what defines them:

• The diagonals contain elements, while all elements below them are zero.
• Non-zero elements can appear only on or above the main diagonal.
• For example, in the matrix \[\begin{bmatrix}3 & 1 & 4 \0 & 2 & 6 \0 & 0 & 5\end{bmatrix}\]the elements 3, 2, and 5 are on the main diagonal and the rest below are zero.

This simplicity leads us to an important characteristic: the eigenvalues of an upper triangular matrix are its diagonal elements themselves. Therefore, recognizing this matrix form can save time as there's no need to perform complex calculations to find eigenvalues.

Matrix Diagonalization

Matrix diagonalization is an advanced but critical process in linear algebra, transforming a matrix into a diagonal form, making computations more straightforward.
It involves expressing a matrix in terms of its eigenvalues and eigenvectors.

Essential components of matrix diagonalization include:

• Eigenvalues: As previously mentioned, for matrices like the upper triangular one, these are the diagonal elements, such as in our example 3, 2, and 5.
• Eigenvectors: Vectors associated with each eigenvalue that satisfy the equation \(A\mathbf{v} = \lambda\mathbf{v}\), where \(A\) is the matrix, \(\lambda\) is the eigenvalue, and \(\mathbf{v}\) is the eigenvector.

When a matrix is diagonalizable, it can be expressed as \(PDP^{-1}\), where \(P\) contains the eigenvectors, \(D\) is a diagonal matrix of eigenvalues, and \(P^{-1}\) is the inverse of \(P\).
This representation simplifies many operations, such as matrix powers, making them easier to calculate.

Linear Algebra

Linear algebra is an essential area of mathematics with vast applications in various fields, from computer science to engineering.
It focuses on concepts like vectors, matrices, and linear transformations.

Key elements of linear algebra include:

• Vectors and Matrices: Vectors are elements of vector spaces, while matrices are rectangular arrays of numbers that represent linear transformations.
• Eigenvalues and Eigenvectors: These are foundational to understanding matrix behaviors, helping solve systems of linear equations efficiently.
• Determinants, Rank, and Systems of Equations: These concepts help in understanding the properties of matrices, systems' solvability, and transformations' impacts.

In our example, linear algebra principles allowed for identifying eigenvalues from an upper triangular matrix easily, simply by observing its diagonal.
Recognizing such patterns can greatly reduce computational complexity.
This blend of theory and practice demonstrates linear algebra's power in unraveling complex mathematical problems.