Question

If \(A=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 0 & 3 & 5 \\ 0 & 0 & -2\end{array}\right]\), then the eigen values of \(A^{2}\) are

Short answer

The eigenvalues of \( A^2 \) are 1, 9, and 4.

Step-by-step solution

Identify the matrix structure

The matrix \( A \) is an upper triangular matrix. An upper triangular matrix is a matrix where all the elements below the main diagonal are zero.

Determine Eigenvalues of A

For an upper triangular matrix, the eigenvalues are the elements on the main diagonal. Therefore, the eigenvalues of \( A \) are \( -1, 3, -2 \).

Relate Eigenvalues of A to A^2

If \( \lambda \) is an eigenvalue of \( A \), then \( \lambda^2 \) is an eigenvalue of \( A^2 \). This means we need to square each eigenvalue of \( A \).

Calculate Eigenvalues of A^2

Square each of the eigenvalues of \( A \):1. \( (-1)^2 = 1 \)2. \( 3^2 = 9 \)3. \( (-2)^2 = 4 \)Thus, the eigenvalues of \( A^2 \) are 1, 9, and 4.

Key concepts

Upper Triangular Matrix

An upper triangular matrix is a special kind of square matrix. It is neatly organized, with all the non-zero elements positioned above the main diagonal. This means every element below the diagonal is zero. This special structure simplifies many matrix-related operations.

In a mathematical sense, if you imagine a matrix as a grid of numbers, the main diagonal runs from the top left corner to the bottom right. An upper triangular matrix looks somewhat like a triangle, with its non-zero values perched above this diagonal line. This simplicity is not just for aesthetics; it provides helpful shortcuts when performing calculations.

• Quick Eigenvalues: The eigenvalues of an upper triangular matrix are simply the entries on the main diagonal. This makes determining eigenvalues much more straightforward compared to a non-triangular matrix.
• Easy Multiplications: Multiplying upper triangular matrices is generally easier, as it preserves the triangular form and helps maintain simpler calculations.

Recognizing and working with upper triangular matrices is an essential skill. It helps in simplifying calculations and understanding deeper properties of matrices.

Squaring Eigenvalues

Eigenvalues are inherently tied to the concept of matrix transformation. When we square a matrix, we are essentially applying the transformation it represents twice.

For an upper triangular matrix like the one given in the exercise, understanding how its eigenvalues relate to the squared matrix is straightforward. If a matrix has an eigenvalue denoted by \(\lambda\), then when you square the matrix, each eigenvalue of this new matrix becomes \(\lambda^2\).

• Simplified Process: Squaring the eigenvalues simplifies the calculation significantly. You just square each original eigenvalue, rather than recalculating from scratch.
• Remember the Sign: Squaring changes positive to positive and negative to positive, which can be critical in identifying matrix behavior changes.

For the specific exercise, this principle allows us to rapidly conclude that if the eigenvalues of \( A \) are \(-1, 3,\) and \(-2\), then the eigenvalues of \( A^2 \) become \(1, 9,\) and \(4\), transitioning directly from the raw values of \(-1, 3,\) and \(-2\). This paints an entirely different picture for the higher-order system modeled by the matrix.

Matrix Diagonalization

Matrix diagonalization involves representing a matrix in its simplest form, a diagonal matrix, which is much easier to work with. A diagonal matrix is another type of matrix, where all entries are zero except for those on the main diagonal.

Diagonalization has many benefits. It simplifies powers and functions of matrices, making computations more manageable. Not every matrix is diagonalizable, but when they are, it offers a powerful way to understand and manipulate them more effectively.

• Connection to Eigenvalues: A matrix can often be diagonalized if it has distinct eigenvalues. The diagonal entries of the diagonal matrix correspond directly to these eigenvalues.
• Transformation Clarity: Diagonal matrices retain the core transformation properties of the original matrix while making calculations such as raising the matrix to a power more direct.

In practical terms, diagonalization is a pivotal concept in simplifying linear transformations and systems. Knowing when and how to apply it can lead to significant simplifications in solving complex matrix problems.