Can You Solve for the Positive Definite Matrix with Eigenvalues 1 and 2?

In summary: The product of eigenvalues is equal to the determinant of a matrix, and the sum of eigenvalues is equal to the trace. and the determinant of a 2 × 2 matrix A − λI can be written as:λ^2 − (trace)λ + determinant = 0In summary, the matrix has eigenvalues of 1 and 2 and the determinant of the matrix is 0.
  • #1
johnny blaz
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1

Homework Statement


a positive definite matrix has eigenvalues λ=1 and λ=2. find the matrix

Homework Equations

The Attempt at a Solution


I've used a 2x2 matrix with entries a0,a1,a2,a3 as the unknown matrix but no use. (As little as i know a0 and a3 should be 1 and 2 respectively; corresponding to the eigenvalues) Any ideas on how to solve this?
 
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  • #2
johnny blaz said:

Homework Statement


a positive definite matrix has eigenvalues λ=1 and λ=2. find the matrix

Homework Equations

The Attempt at a Solution


I've used a 2x2 matrix with entries a0,a1,a2,a3 as the unknown matrix but no use. (As little as i know a0 and a3 should be 1 and 2 respectively; corresponding to the eigenvalues) Any ideas on how to solve this?
The fact, that the matrix is ##2\times 2## should be in the list of conditions I guess.
Start at the beginning: What does it mean for ##\begin{pmatrix}a && b \\ c && d \end{pmatrix}## to be positive definite?
(I find this notation easier in this case for I'll have to type less indices.)
Can you really assume already ##a=1## and ##d = 2##? What do you really know about ##λ_1 \cdot λ_2 = 2## and ##λ_1 + λ_2=3## with respect to the characteristic polynomial?
 
  • #3
fresh_42 said:
The fact, that the matrix is ##2\times 2## should be in the list of conditions I guess.
Start at the beginning: What does it mean for ##\begin{pmatrix}a && b \\ c && d \end{pmatrix}## to be positive definite?
(I find this notation easier in this case for I'll have to type less indices.)
Can you really assume already ##a=1## and ##d = 2##? What do you really know about ##λ_1 \cdot λ_2 = 2## and ##λ_1 + λ_2=3## with respect to the characteristic polynomial?
I've assumed it to be a 2x2 matrix as a positive 2x2 matrix would have only two eigenvalues. No other conditions are provided for this problem...
 
  • #4
johnny blaz said:
I've assumed it to be a 2x2 matrix as a positive 2x2 matrix would have only two eigenvalues. No other conditions are provided for this problem...
Yes, but you have not mentioned whether ##1## and ##2## are the only eigenvalues nor their multiplicity. Anyway, can you answer my questions?
 
  • #5
johnny blaz said:

Homework Statement


a positive definite matrix has eigenvalues λ=1 and λ=2. find the matrix

Homework Equations

The Attempt at a Solution


I've used a 2x2 matrix with entries a0,a1,a2,a3 as the unknown matrix but no use. (As little as i know a0 and a3 should be 1 and 2 respectively; corresponding to the eigenvalues) Any ideas on how to solve this?

The concepts of "definiteness" (positive, negative, semi-, etc) apply to symmetric matrices, so your matrix has three unknown parameters in it:
[tex] A = \pmatrix{a & b \\b & c} [/tex]
Its characteristic polynomial is a quadratic function of ##\lambda## having roots 1 and 2. What does that tell you about ##a,b,c##?

Why would you think that your ##a_0## and ##a_3## should be 1 and 2 (or 2 and 1)?
 
  • #6
Ray Vickson said:
The concepts of "definiteness" (positive, negative, semi-, etc) apply to symmetric matrices, ...
This is not necessary to be positive definite. The definition applies to all bilinear forms and every quadratic matrix defines one.
(I'm not quite sure whether I have made some mistake or not, but my solution isn't symmetric. The elements on the second diagonal may cancel out.)
 
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  • #7
fresh_42 said:
The fact, that the matrix is ##2\times 2## should be in the list of conditions I guess.
Start at the beginning: What does it mean for ##\begin{pmatrix}a && b \\ c && d \end{pmatrix}## to be positive definite?
(I find this notation easier in this case for I'll have to type less indices.)
Can you really assume already ##a=1## and ##d = 2##? What do you really know about ##λ_1 \cdot λ_2 = 2## and ##λ_1 + λ_2=3## with respect to the characteristic polynomial?
The product of eigenvalues is equal to the determinant of a matrix, and the sum of eigenvalues is equal to the trace. and the determinant of a 2 × 2 matrix A − λI can be written as:
λ^2 − (trace)λ + determinant = 0
 
  • #8
johnny blaz said:
The product of eigenvalues is equal to the determinant of a matrix, and the sum of eigenvalues is equal to the trace. and the determinant of a 2 × 2 matrix A − λI can be written as:
λ^2 − (trace)λ + determinant = 0
Correct. So the characteristic polynomial has to be ##char(t) = t^2-3t+2 = \det\begin{pmatrix}a-t && b \\ c && d-t\end{pmatrix}.##
(I've changed the variable to ##t## because you already used ##λ## for the eigenvalues, i.e. the solutions of ##char(t)=0.##)
Furthermore the definition of positive definiteness means, that ##(x,y)\begin{pmatrix}a && b \\ c && d\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} > 0## for all possible values of ##x,y## where not both are ##0.##
These conditions, if written out, can be used to calculate possible values of ##a,b,c,d##. And there is an obvious solution to it and another one.
 
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  • #9
fresh_42 said:
Correct. So the characteristic polynomial has to be ##char(t) = t^2-3t+2 = \det\begin{pmatrix}a-t && b \\ c && d-t\end{pmatrix}.##
(I've changed the variable to ##t## because you already used ##λ## for the eigenvalues, i.e. the solutions of ##char(t)=0.##)
Furthermore the definition of positive definiteness means, that ##(x,y)\begin{pmatrix}a && b \\ c && d\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} > 0## for all possible values of ##x,y## where not both are ##0.##
These conditions, if written out, can be used to calculate possible values of ##a,b,c,d##. And there is an obvious solution to it and another one.

Whether or not the 'definiteness' concept applies generally, or only to symmetric matrices, is a point of disagreement in the literature.

Anyway, one can reduce the general form for matrix ##B## to the equivalent symmetric form ##A = (1/2) B + (1/2) B^T##, get the general symmetric solution, then deal with the multitude of solutions to ##b_{12} + b_{21} = 2 a_{12}##. There are still infinitely many symmetric solutions, but all satisfying some specific relationships between ##a_{11} , a_{12} = a_{21}## and ##a_{22}##..
 
  • #10
Thank you all for helping me out :smile:
 

What is a positive definite matrix?

A positive definite matrix is a square matrix in which all of its eigenvalues are positive. This means that when the matrix is multiplied by any non-zero vector, the result will always be a positive number.

How can a positive definite matrix be identified?

A positive definite matrix can be identified through several methods, including checking its eigenvalues, performing a Cholesky decomposition, or using the Sylvester's criterion.

What are the applications of positive definite matrices?

Positive definite matrices have many applications in mathematics, physics, and engineering. They are commonly used in optimization problems, statistical analysis, and in solving differential equations.

Can a matrix be both positive definite and positive semi-definite?

No, a matrix cannot be both positive definite and positive semi-definite. A positive definite matrix has all positive eigenvalues, while a positive semi-definite matrix has all non-negative eigenvalues.

How are positive definite matrices used in machine learning?

In machine learning, positive definite matrices are used to define the covariance matrix in multivariate Gaussian distributions. They are also used in algorithms such as the Cholesky decomposition and the conjugate gradient method for solving optimization problems.

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