Eigen value ppt Microsoft power point

1. Name- Tanmay Bera
Roll No-17700123053
Reg No-231770110062(2023-24)
Sub-Mathematics(BS-M101)
Dept-CSE , SEM-1st
Calcutta Institute Of Technology

2. Discoveringthe Eigen
Values& EigenVectorsin
Math
In math,eigenvaluesareasetof scalarvaluesassociatedwithasetof linear
equations.Eigenvectorsarealsoknownascharacteristicroots.Theyarenon-zero
vectorsthatcanbe changedby theirscalarfactoraftertheapplicationoflinear
transformations.
by TanmayBera

3. Definitionof the Eigen Values
& Eigen Vectors
Eigenvaluesandeigenvectorsarescalarand vectorquantitiesassociatedwith
matricesusedforlineartransformations.
Herearesomedefinitionsofeigenvaluesandeigenvectors:
• Eigenvalues: A special set of scalar values associated with a set of linear
equations, usually in matrix equations.
• Eigenvectors: Also known as characteristic roots, these are non-zero vectors
that can only change by their scalar factor after the applicationof linear
transformations.
• Linear transformation: A transformation that rotates, stretches, or shears the
vectors it acts upon.
• Eigenvectors: Vectors that are only stretched, with no rotation or shear.
• Eigenvalues: The factor by which an eigenvector is stretched or squished.

4. Propertiesand Characteristicsof the
Eigen Values & Eigen Vectors
Eigen
Properties of Eigenvalues:
• Eigenvectors with Distinct Eigenvalues are Linearly Independent.
• Singular Matrices have Zero Eigenvalues.
• If A is a square matrix, then λ = 0 is not an eigenvalue of A.
• For a scalar multiple of a matrix: If A is a square matrix and λ is an
eigenvalue of A. Then, aλ is an eigenvalue of aA.

5. The characteristic equation of a square matrix A is given by det(A – λI) = 0. In this
equation, det denotes the determinant, λ is an eigenvalue of A, and I is the identity
matrix of the same size as A.
• The roots of the characteristic equation are called characteristic roots or
eigenvalues. The associated eigenvectors of A are nonzero solutions of the equation
(A – λI)x = 0.
• Every square matrix has its characteristic equation.
CharacteristicEquationOf A SquareMatrix:
Matrix:

6. Cayley Hamilton Theorem:
• The Cayley–Hamilton theorem states that every square matrix is a root of its own
characteristic polynomial. The Cayley-Hamilton theorem can be used to solve problems
involving the computation of eAt, where A is a constant n x n matrix.
• Here are some resourcesfor Cayley-Hamilton theorem presentations:
1. This presentation includes topics such as the Cayley-Hamilton theorem, eigenvalues,
eigenvectors,and eigenspace.
2. This presentation includes the Cayley–Hamilton theorem, which states that every square
matrix is a root of its own characteristic polynomial.
3. This presentation includes over 100 PowerPoint presentationson the Cayley-Hamilton
theorem.
4. Generalized cayley hamilton theorem
5. This presentation includes PowerPoint presentationson the generalized Cayley-Hamilton
theorem.
6. The Cayley-Hamilton theorem is only true for square matrices. The Cayley-converse
Hamilton’s theorem is false for nn matrices with n>1, and it fails for 2×2 matrices.

7. Properties of Eigen Vectors For Symmetric
and Skew-Symmetric Matrix
Herearesomepropertiesofeigenvectorsforsymmetricand
skew-symmetricmatrices:
• Symmetric matrices: The eigenvalues of a symmetric matrix
are real.
• Skew-symmetric matrices: The eigenvalues of a skew-
symmetric matrix are either zero or purely imaginary. The
eigenvectors corresponding to distinct eigenvalues are
orthogonal.
A skew-symmetricmatrixis a squarematrixwhosetransposeisequal to its
negative.For example,if A is a skew-symmetricmatrix,thenA^T = -A.

8. Propertyof Eigen Vectorfor OrthogonalMatrix
Hereare someproperties of eigenvectorsforanorthogonalmatrix:
• Magnitude:Theeigenvaluesofanorthogonalmatrixalwayshavea
magnitudeof 1.
• Orthonormal: The eigenvectors are also orthonormal. This means that if A is
an orthogonal matrix and λ is an eigenvalue of A with corresponding
eigenvector x, then |λ|= 1 and x is a unit vector.
• Orthogonal: The eigenvectors would also be orthogonal and real.
• Product: The product of the transpose of x and x is zero.
Herearesomeother properties ofeigenvectors:
• Real numbers: All the eigenvaluesare real numbers.
• Orthogonal to each other: All the eigenvectorsrelated to distinct eigenvalues are
orthogonal to each other.
• Zero eigenvalues: Singular matrices or zero matrices always have zero eigenvalues.

9. Examples of the Eigen Values & Eigen Vectors in Math
Math Problems

11. THANK YOU