# Eigenvalues and Eigenvectors of Matrices

• ### Mathematical Formulation

Consider a transformation matrix ${A}$, such that it transforms a vector ${X}$ into ${Y}$. Thus, we can write,

${Y = AX}$

Recall that 2 vectors directed along same direction are simply scalar multiples of each other. e.g. ${\vec P = 3 \hat i + 4 \hat j = (3,4)}$ and ${\vec Q = 6 \hat i + 8 \hat j}$ (Check their direction). We can write ${\vec Q = 2 \vec P}$.

Now, suppose there exists a scalar ${\lambda}$, such that

${Y = \lambda X}$

Now this is interesting, because ${Y}$ and ${X}$ are now related via. a matrix ${A}$ and a scalar (a number) ${\lambda}$. i.e. ${Y = AX}$ as well as ${Y = \lambda X}$.

Thus,

${Y = AX = \lambda X = \lambda IX}$

So,

${AX - \lambda IX = 0 \ or \ \ (A - \lambda I)X =0}$

Since RHS is a null vector, this forms a homogeneous system, which will have non-trivial solutions, when ${|A - \lambda I| =0}$. On expanding the determinant, we will get ${n}$ values of ${\lambda}$, if ${A}$ is of the order ${n \times n}$. These values are known as the eigenvalues. In German, eigen means special.

I) The determinant ${|A- \lambda I|}$ is known as the characteristic determinant and the polynomial obtained on expanding the determinant is known as the characteristic polynomial. If matrix ${A}$ is of the order ${n}$, the degree of polynomial is ${n}$ and hence, the matrix has ${n}$ eigenvalues, which may be distinct or identical.

The set of eigenvalues is known as the spectrum.

II) ${\sum \limits_{i=1}^n \lambda_i = \sum \limits_{i=1}^n a_{ii} =}$ Trace

III) ${\prod \limits_{i=1}^n \lambda_i = |A|}$

This implies, if any one of the eigenvalues is ${0}$, then ${|A|}$ is ${0}$.

IV) Eigenvalues of ${A^{n}}$ are ${\lambda^n}$, where ${n}$ is a non-negative integer.

V) Eigenvalues of ${A}$ and ${A^T}$ are same.

VI) Eigenvalues of ${A- KI}$ are ${\lambda_i - K}$, where ${K}$ is any number.

VII) If ${A}$ is symmetric, then its eigenvalues are real.

• ### Eigenvectors

Corresponding to each of the eigenvalues ${\lambda}$, there will be a vector, known as eigenvector. This is obtained by solving the system ${(A - \lambda I) X =0}$. As stated earlier, ${AX = \lambda X}$.

This forms a system of homogeneous equations, and to get ${X}$, we solve the system.

• ### Properties of Eigenvectors

These are the vectors, whose direction does not change under the transformation ${AX}$.

I) For an eigenvalue ${\lambda}$, if ${X}$ is an eigenvector, then ${KX, K \ne 0}$ is also an eigenvector.

II) If the eigenvalues are distinct, the eigenvectors are linearly independent.

III) If ${A}$ is symmetric, then the eigenvectors corresponding to 2 distinct eigenvalues are orthogonal.

• ### Use of Eigenvalues and Eigenvectors

There are many applications of eigenvalues and eigenvectors. Finding natural frequencies of a system with multiple degrees of freedom is an example which is a typical mechanical engineering application. The equations of motion form a system of the kind ${AX = B = \lambda X}$. The eigenvalues of ${A}$ tell us its natural frequencies. The corresponding eigenvectors indicate the mode shapes.

• ### Cayley Hamilton Theorem

It states that every matrix satisfies its characteristic equation, i.e. if ${|A - \lambda I|=0}$, or ${f(\lambda) = 0}$ , then

${f(A)=0}$

This theorem can be used to find higher powers of a matrix as well as its inverse.