Linear independence of eigen vectors- Let λ1, …, λk be k-distinct eigen-values of square matrox of order n, then corresponding eigen vectors x1, …, xk from a linearly independence set.
Basic of eigen vectors- If n-order square matrix has n-distinct eigen values, then A has a basis of eigen vectors for ℂn (or ℝn). Where ℂ (and ℝ) are set of complex (and real) numbers.
Diagonalization of matrix- If n-order square matrix A has a basis of eigen vector, then D= X-1AX is diagonal, with the eigen values of A as the entries on the main diagonal. Here X is the matrix with these eigen vectors as column vectors.
Inverse of Matrix:
If A is a square matrix, then inverse of A, A-1 exists if AA-1 = A-1A = I.
A-1 exist ⇔ det A ≠ 0
⇔ A is non-singular matrix.
⇔ Columns (rows) of A are linearly independent.
Elementary Canonical Forms:
Eigen values and eigen vectors- if V is a vector space over the field F and T is a linear operator on V. An eigen value of T is a scalar c in F such that there is a non-zero vector
α ∈ V with Tα = cα.
If c is an eigen value of T, then
(a) Any α such that Tα = cα is called eigen vector of T associated with the eigen value c.
(b) The collection of all c such that Tα = cα is called the eigen space associated with c.
Eigen value of matrix A over F- If A is an n × n matrix over the field F, an eigen value of A over F is a scalar c in F such that the matrix (A - cI) is singular (not invertible).
Eigen polynomial- f(c) = |A - cI|
Diagonalizable- If T is a linear operator on the finite dimensional space V. The T is diagonalizable if there is a basis for V each vector of which is an eigen vector of T.
Basic of eigen vectors- If n-order square matrix has n-distinct eigen values, then A has a basis of eigen vectors for ℂn (or ℝn). Where ℂ (and ℝ) are set of complex (and real) numbers.
Diagonalization of matrix- If n-order square matrix A has a basis of eigen vector, then D= X-1AX is diagonal, with the eigen values of A as the entries on the main diagonal. Here X is the matrix with these eigen vectors as column vectors.
Inverse of Matrix:
If A is a square matrix, then inverse of A, A-1 exists if AA-1 = A-1A = I.
A-1 exist ⇔ det A ≠ 0
⇔ A is non-singular matrix.
⇔ Columns (rows) of A are linearly independent.
Elementary Canonical Forms:
Eigen values and eigen vectors- if V is a vector space over the field F and T is a linear operator on V. An eigen value of T is a scalar c in F such that there is a non-zero vector
α ∈ V with Tα = cα.
If c is an eigen value of T, then
(a) Any α such that Tα = cα is called eigen vector of T associated with the eigen value c.
(b) The collection of all c such that Tα = cα is called the eigen space associated with c.
Eigen value of matrix A over F- If A is an n × n matrix over the field F, an eigen value of A over F is a scalar c in F such that the matrix (A - cI) is singular (not invertible).
Eigen polynomial- f(c) = |A - cI|
Diagonalizable- If T is a linear operator on the finite dimensional space V. The T is diagonalizable if there is a basis for V each vector of which is an eigen vector of T.
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